Replaced group_ and ring_simps by algebra_simps;
removed compare_rls - use algebra_simps now
--- a/src/HOL/Algebra/abstract/Ring2.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Algebra/abstract/Ring2.thy Wed Jan 28 16:29:16 2009 +0100
@@ -224,10 +224,6 @@
{* Method.no_args (Method.SIMPLE_METHOD' (full_simp_tac ring_ss)) *}
{* computes distributive normal form in rings *}
-lemmas ring_simps =
- l_zero r_zero l_neg r_neg minus_minus minus0
- l_one r_one l_null r_null l_minus r_minus
-
subsection {* Rings and the summation operator *}
--- a/src/HOL/Algebra/poly/UnivPoly2.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Algebra/poly/UnivPoly2.thy Wed Jan 28 16:29:16 2009 +0100
@@ -189,7 +189,7 @@
lemma coeff_smult [simp]: "coeff (a *s p) n = (a::'a::ring) * coeff p n"
proof -
have "!!f. f : UP ==> (%n. a * f n) : UP"
- by (unfold UP_def) (force simp add: ring_simps)
+ by (unfold UP_def) (force simp add: algebra_simps)
*) (* this force step is slow *)
(* then show ?thesis
apply (simp add: coeff_def smult_def Abs_UP_inverse Rep_UP)
@@ -198,7 +198,7 @@
lemma coeff_smult [simp]: "coeff (a *s p) n = (a::'a::ring) * coeff p n"
proof -
have "Rep_UP p : UP ==> (%n. a * Rep_UP p n) : UP"
- by (unfold UP_def) (force simp add: ring_simps)
+ by (unfold UP_def) (force simp add: algebra_simps)
(* this force step is slow *)
then show ?thesis
by (simp add: coeff_def smult_def Abs_UP_inverse Rep_UP)
@@ -220,7 +220,7 @@
fix i
assume "max n m < i"
with boundn and boundm show "f i + g i = 0"
- by (fastsimp simp add: ring_simps)
+ by (fastsimp simp add: algebra_simps)
qed
then show "(%i. (f i + g i)) : UP"
by (unfold UP_def) fast
@@ -251,15 +251,15 @@
have "f i * g (k-i) = 0"
proof cases
assume "n < i"
- with `bound n f` show ?thesis by (auto simp add: ring_simps)
+ with `bound n f` show ?thesis by (auto simp add: algebra_simps)
next
assume "~ (n < i)"
with bound have "m < k-i" by arith
- with `bound m g` show ?thesis by (auto simp add: ring_simps)
+ with `bound m g` show ?thesis by (auto simp add: algebra_simps)
qed
}
then show "setsum (%i. f i * g (k-i)) {..k} = 0"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
qed
then show "(%n. setsum (%i. f i * g (n-i)) {..n}) : UP"
by (unfold UP_def) fast
@@ -270,7 +270,7 @@
qed
lemma coeff_uminus [simp]: "coeff (-p) n = (-coeff p n::'a::ring)"
-by (unfold up_uminus_def) (simp add: ring_simps)
+by (unfold up_uminus_def) (simp add: algebra_simps)
(* Other lemmas *)
@@ -388,7 +388,7 @@
proof (cases k)
case 0 then show ?thesis by simp ring
next
- case Suc then show ?thesis by (simp add: ring_simps) ring
+ case Suc then show ?thesis by (simp add: algebra_simps) ring
qed
then show "coeff (monom a 0 * p) k = coeff (a *s p) k" by ring
qed
--- a/src/HOL/Complex.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Complex.thy Wed Jan 28 16:29:16 2009 +0100
@@ -237,9 +237,9 @@
show "scaleR 1 x = x"
by (simp add: expand_complex_eq)
show "scaleR a x * y = scaleR a (x * y)"
- by (simp add: expand_complex_eq ring_simps)
+ by (simp add: expand_complex_eq algebra_simps)
show "x * scaleR a y = scaleR a (x * y)"
- by (simp add: expand_complex_eq ring_simps)
+ by (simp add: expand_complex_eq algebra_simps)
qed
end
@@ -312,7 +312,7 @@
(simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
show "norm (x * y) = norm x * norm y"
by (induct x, induct y)
- (simp add: real_sqrt_mult [symmetric] power2_eq_square ring_simps)
+ (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def)
qed
--- a/src/HOL/Dense_Linear_Order.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Dense_Linear_Order.thy Wed Jan 28 16:29:16 2009 +0100
@@ -552,7 +552,7 @@
lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
proof-
assume H: "c < 0"
- have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
+ have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
also have "\<dots> = (0 < x)" by simp
finally show "(c*x < 0) == (x > 0)" by simp
qed
@@ -560,7 +560,7 @@
lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
proof-
assume H: "c > 0"
- hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
+ hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
also have "\<dots> = (0 > x)" by simp
finally show "(c*x < 0) == (x < 0)" by simp
qed
@@ -569,7 +569,7 @@
proof-
assume H: "c < 0"
have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_simps)
+ also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t < x)" by simp
finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
qed
@@ -578,7 +578,7 @@
proof-
assume H: "c > 0"
have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_simps)
+ also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t > x)" by simp
finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
qed
@@ -589,7 +589,7 @@
lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
proof-
assume H: "c < 0"
- have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
+ have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
also have "\<dots> = (0 <= x)" by simp
finally show "(c*x <= 0) == (x >= 0)" by simp
qed
@@ -597,7 +597,7 @@
lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
proof-
assume H: "c > 0"
- hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
+ hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
also have "\<dots> = (0 >= x)" by simp
finally show "(c*x <= 0) == (x <= 0)" by simp
qed
@@ -606,7 +606,7 @@
proof-
assume H: "c < 0"
have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_simps)
+ also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t <= x)" by simp
finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
qed
@@ -615,7 +615,7 @@
proof-
assume H: "c > 0"
have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_simps)
+ also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t >= x)" by simp
finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
qed
@@ -628,7 +628,7 @@
proof-
assume H: "c \<noteq> 0"
have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
- also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_simps)
+ also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)
finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
qed
lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
--- a/src/HOL/Deriv.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Deriv.thy Wed Jan 28 16:29:16 2009 +0100
@@ -141,7 +141,7 @@
lemma inverse_diff_inverse:
"\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
\<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
-by (simp add: ring_simps)
+by (simp add: algebra_simps)
lemma DERIV_inverse_lemma:
"\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_field)\<rbrakk>
@@ -206,7 +206,7 @@
case (Suc k)
from DERIV_mult' [OF f Suc] show ?case
apply (simp only: of_nat_Suc ring_distribs mult_1_left)
- apply (simp only: power_Suc right_distrib mult_ac add_ac)
+ apply (simp only: power_Suc algebra_simps)
done
qed
@@ -724,7 +724,7 @@
from isCont_bounded [OF le this]
obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
- by (simp add: M3 compare_rls)
+ by (simp add: M3 algebra_simps)
have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
by (auto intro: order_le_less_trans [of _ k])
with Minv
@@ -1398,7 +1398,7 @@
have "?h b - ?h a =
((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
- by (simp add: mult_ac add_ac right_diff_distrib)
+ by (simp add: algebra_simps)
hence "?h b - ?h a = 0" by auto
}
ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
@@ -1427,7 +1427,7 @@
lemma coeff_pderiv: "coeff (pderiv p) n = of_nat (Suc n) * coeff p (Suc n)"
apply (induct p arbitrary: n, simp)
- apply (simp add: pderiv_pCons coeff_pCons ring_simps split: nat.split)
+ apply (simp add: pderiv_pCons coeff_pCons algebra_simps split: nat.split)
done
lemma pderiv_eq_0_iff: "pderiv p = 0 \<longleftrightarrow> degree p = 0"
@@ -1451,21 +1451,21 @@
by (simp add: pderiv_pCons)
lemma pderiv_add: "pderiv (p + q) = pderiv p + pderiv q"
-by (rule poly_ext, simp add: coeff_pderiv ring_simps)
+by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
lemma pderiv_minus: "pderiv (- p) = - pderiv p"
by (rule poly_ext, simp add: coeff_pderiv)
lemma pderiv_diff: "pderiv (p - q) = pderiv p - pderiv q"
-by (rule poly_ext, simp add: coeff_pderiv ring_simps)
+by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
lemma pderiv_smult: "pderiv (smult a p) = smult a (pderiv p)"
-by (rule poly_ext, simp add: coeff_pderiv ring_simps)
+by (rule poly_ext, simp add: coeff_pderiv algebra_simps)
lemma pderiv_mult: "pderiv (p * q) = p * pderiv q + q * pderiv p"
apply (induct p)
apply simp
-apply (simp add: pderiv_add pderiv_smult pderiv_pCons ring_simps)
+apply (simp add: pderiv_add pderiv_smult pderiv_pCons algebra_simps)
done
lemma pderiv_power_Suc:
@@ -1475,7 +1475,7 @@
apply (subst power_Suc)
apply (subst pderiv_mult)
apply (erule ssubst)
-apply (simp add: smult_add_left ring_simps)
+apply (simp add: smult_add_left algebra_simps)
done
lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
--- a/src/HOL/Divides.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Divides.thy Wed Jan 28 16:29:16 2009 +0100
@@ -38,10 +38,10 @@
by (simp only: add_ac)
lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
- by (simp add: mod_div_equality)
+by (simp add: mod_div_equality)
lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
- by (simp add: mod_div_equality2)
+by (simp add: mod_div_equality2)
lemma mod_by_0 [simp]: "a mod 0 = a"
using mod_div_equality [of a zero] by simp
@@ -63,7 +63,7 @@
by (simp add: mod_div_equality)
also from False div_mult_self1 [of b a c] have
"\<dots> = (c + a div b) * b + (a + c * b) mod b"
- by (simp add: left_distrib add_ac)
+ by (simp add: algebra_simps)
finally have "a = a div b * b + (a + c * b) mod b"
by (simp add: add_commute [of a] add_assoc left_distrib)
then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
@@ -72,7 +72,7 @@
qed
lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
- by (simp add: mult_commute [of b])
+by (simp add: mult_commute [of b])
lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
using div_mult_self2 [of b 0 a] by simp
@@ -217,7 +217,7 @@
have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
by (simp only: mod_div_equality)
also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
- by (simp only: left_distrib right_distrib add_ac mult_ac)
+ by (simp only: algebra_simps)
also have "\<dots> = (a mod c * b) mod c"
by (rule mod_mult_self1)
finally show ?thesis .
@@ -228,7 +228,7 @@
have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
by (simp only: mod_div_equality)
also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
- by (simp only: left_distrib right_distrib add_ac mult_ac)
+ by (simp only: algebra_simps)
also have "\<dots> = (a * (b mod c)) mod c"
by (rule mod_mult_self1)
finally show ?thesis .
@@ -552,7 +552,7 @@
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
let (q, r) = divmod (m - n) n in (Suc q, r))"
- by (simp add: divmod_zero divmod_base divmod_step)
+by (simp add: divmod_zero divmod_base divmod_step)
(simp add: divmod_div_mod)
code_modulename SML
@@ -568,16 +568,16 @@
subsubsection {* Quotient *}
lemma div_geq: "0 < n \<Longrightarrow> \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
- by (simp add: le_div_geq linorder_not_less)
+by (simp add: le_div_geq linorder_not_less)
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
- by (simp add: div_geq)
+by (simp add: div_geq)
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
- by simp
+by simp
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
- by simp
+by simp
subsubsection {* Remainder *}
@@ -597,13 +597,13 @@
qed
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
- by (simp add: le_mod_geq linorder_not_less)
+by (simp add: le_mod_geq linorder_not_less)
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
- by (simp add: le_mod_geq)
+by (simp add: le_mod_geq)
lemma mod_1 [simp]: "m mod Suc 0 = 0"
- by (induct m) (simp_all add: mod_geq)
+by (induct m) (simp_all add: mod_geq)
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
apply (cases "n = 0", simp)
@@ -614,11 +614,11 @@
done
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
- by (simp add: mult_commute [of k] mod_mult_distrib)
+by (simp add: mult_commute [of k] mod_mult_distrib)
(* a simple rearrangement of mod_div_equality: *)
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
- by (cut_tac a = m and b = n in mod_div_equality2, arith)
+by (cut_tac a = m and b = n in mod_div_equality2, arith)
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
apply (drule mod_less_divisor [where m = m])
@@ -630,7 +630,7 @@
lemma divmod_rel_mult1_eq:
"[| divmod_rel b c q r; c > 0 |]
==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
-by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
+by (auto simp add: split_ifs divmod_rel_def algebra_simps)
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
apply (cases "c = 0", simp)
@@ -638,19 +638,19 @@
done
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
- by (rule mod_mult_right_eq)
+by (rule mod_mult_right_eq)
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
- by (rule mod_mult_left_eq)
+by (rule mod_mult_left_eq)
lemma mod_mult_distrib_mod:
"(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
- by (rule mod_mult_eq)
+by (rule mod_mult_eq)
lemma divmod_rel_add1_eq:
"[| divmod_rel a c aq ar; divmod_rel b c bq br; c > 0 |]
==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
-by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
+by (auto simp add: split_ifs divmod_rel_def algebra_simps)
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
lemma div_add1_eq:
@@ -660,7 +660,7 @@
done
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
- by (rule mod_add_eq)
+by (rule mod_add_eq)
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
apply (cut_tac m = q and n = c in mod_less_divisor)
@@ -671,7 +671,7 @@
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r; 0 < b; 0 < c |]
==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
- by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
+by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
apply (cases "b = 0", simp)
@@ -690,7 +690,7 @@
lemma div_mult_mult_lemma:
"[| (0::nat) < b; 0 < c |] ==> (c*a) div (c*b) = a div b"
- by (auto simp add: div_mult2_eq)
+by (auto simp add: div_mult2_eq)
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
apply (cases "b = 0")
@@ -706,7 +706,7 @@
subsubsection{*Further Facts about Quotient and Remainder*}
lemma div_1 [simp]: "m div Suc 0 = m"
- by (induct m) (simp_all add: div_geq)
+by (induct m) (simp_all add: div_geq)
(* Monotonicity of div in first argument *)
@@ -780,10 +780,10 @@
done
lemma nat_mod_div_trivial: "m mod n div n = (0 :: nat)"
- by simp
+by simp
lemma nat_mod_mod_trivial: "m mod n mod n = (m mod n :: nat)"
- by simp
+by simp
subsubsection {* The Divides Relation *}
@@ -792,7 +792,7 @@
unfolding dvd_def by simp
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
- by (simp add: dvd_def)
+by (simp add: dvd_def)
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
unfolding dvd_def
@@ -813,7 +813,7 @@
done
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
- by (drule_tac m = m in dvd_diff, auto)
+by (drule_tac m = m in dvd_diff, auto)
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
apply (rule iffI)
@@ -839,7 +839,7 @@
done
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
- by (blast intro: dvd_mod_imp_dvd dvd_mod)
+by (blast intro: dvd_mod_imp_dvd dvd_mod)
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
unfolding dvd_def
@@ -894,7 +894,7 @@
done
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
- by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
+by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
--- a/src/HOL/Finite_Set.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Finite_Set.thy Wed Jan 28 16:29:16 2009 +0100
@@ -1033,12 +1033,12 @@
lemma setsum_Un_nat: "finite A ==> finite B ==>
(setsum f (A Un B) :: nat) = setsum f A + setsum f B - setsum f (A Int B)"
-- {* For the natural numbers, we have subtraction. *}
-by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)
+by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
lemma setsum_Un: "finite A ==> finite B ==>
(setsum f (A Un B) :: 'a :: ab_group_add) =
setsum f A + setsum f B - setsum f (A Int B)"
-by (subst setsum_Un_Int [symmetric], auto simp add: ring_simps)
+by (subst setsum_Un_Int [symmetric], auto simp add: algebra_simps)
lemma setsum_diff1_nat: "(setsum f (A - {a}) :: nat) =
(if a:A then setsum f A - f a else setsum f A)"
@@ -1711,7 +1711,7 @@
lemma setsum_constant [simp]: "(\<Sum>x \<in> A. y) = of_nat(card A) * y"
apply (cases "finite A")
apply (erule finite_induct)
-apply (auto simp add: ring_simps)
+apply (auto simp add: algebra_simps)
done
lemma setprod_constant: "finite A ==> (\<Prod>x\<in> A. (y::'a::{recpower, comm_monoid_mult})) = y^(card A)"
@@ -2220,16 +2220,16 @@
thus ?case
proof
assume "a = x" thus ?thesis using insert
- by (simp add: mult_ac_idem)
+ by (simp add: mult_ac)
next
assume "a \<in> F"
hence bel: "fold1 inf F \<le> a" by (rule insert)
have "inf (fold1 inf (insert x F)) a = inf x (inf (fold1 inf F) a)"
- using insert by (simp add: mult_ac_idem)
+ using insert by (simp add: mult_ac)
also have "inf (fold1 inf F) a = fold1 inf F"
using bel by (auto intro: antisym)
also have "inf x \<dots> = fold1 inf (insert x F)"
- using insert by (simp add: mult_ac_idem)
+ using insert by (simp add: mult_ac)
finally have aux: "inf (fold1 inf (insert x F)) a = fold1 inf (insert x F)" .
moreover have "inf (fold1 inf (insert x F)) a \<le> a" by simp
ultimately show ?thesis by simp
--- a/src/HOL/Fundamental_Theorem_Algebra.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Fundamental_Theorem_Algebra.thy Wed Jan 28 16:29:16 2009 +0100
@@ -45,7 +45,7 @@
have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
unfolding power2_eq_square
- by (simp add: ring_simps real_sqrt_divide sqrt4)
+ by (simp add: algebra_simps real_sqrt_divide sqrt4)
from y0 xy have ?thesis apply (simp add: csqrt_def power2_eq_square)
apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric])
using th1 th2 ..}
@@ -109,7 +109,7 @@
lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
apply (induct p)
apply (simp add: offset_poly_0)
-apply (simp add: offset_poly_pCons ring_simps)
+apply (simp add: offset_poly_pCons algebra_simps)
done
lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
@@ -350,7 +350,7 @@
from md z have xy: "x^2 + y^2 = 1" by (simp add: cmod_def)
{assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
- by (simp_all add: cmod_def power2_eq_square ring_simps)
+ by (simp_all add: cmod_def power2_eq_square algebra_simps)
hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"
by - (rule power_mono, simp, simp)+
@@ -391,9 +391,9 @@
1"
apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric])
using right_inverse[OF b']
- by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] ring_simps)
+ by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] algebra_simps)
have th0: "cmod (complex_of_real (cmod b) / b) = 1"
- apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse ring_simps )
+ apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse algebra_simps )
by (simp add: real_sqrt_mult[symmetric] th0)
from o have "\<exists>m. n = Suc (2*m)" by presburger+
then obtain m where m: "n = Suc (2*m)" by blast
@@ -667,10 +667,10 @@
from h have z1: "cmod z \<ge> 1" by arith
from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
- unfolding norm_mult by (simp add: ring_simps)
+ unfolding norm_mult by (simp add: algebra_simps)
from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)"
- by (simp add: diff_le_eq ring_simps)
+ by (simp add: diff_le_eq algebra_simps)
from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)" by arith}
hence ?case by blast}
moreover
@@ -685,7 +685,7 @@
have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
from complex_mod_triangle_sub[of "z*c" a ]
have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"
using cs0' by simp}
then have ?case by blast}
@@ -850,7 +850,7 @@
with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=0" by auto
{fix w
have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
- using kas(4)[rule_format, of w] s0 r01 by (simp add: ring_simps)}
+ using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
note hth = this [symmetric]
from reduce_poly_simple[OF kas(1,2)]
have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
@@ -866,7 +866,7 @@
from H[rule_format, OF k1n th01 th02]
obtain w where w: "1 + w^k * a = 0"
unfolding poly_pCons poly_monom
- using kas(2) by (cases k, auto simp add: ring_simps)
+ using kas(2) by (cases k, auto simp add: algebra_simps)
from poly_bound_exists[of "cmod w" s] obtain m where
m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
@@ -879,7 +879,7 @@
t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
let ?ct = "complex_of_real t"
let ?w = "?ct * w"
- have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: ring_simps power_mult_distrib)
+ have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib)
also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
unfolding wm1 by (simp)
finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
@@ -898,7 +898,7 @@
have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
apply - apply (rule mult_strict_left_mono) by simp_all
have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))" using w0 t(1)
- by (simp add: ring_simps power_mult_distrib norm_of_real norm_power norm_mult)
+ by (simp add: algebra_simps power_mult_distrib norm_of_real norm_power norm_mult)
then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
using t(1,2) m(2)[rule_format, OF tw] w0
apply (simp only: )
@@ -1308,14 +1308,14 @@
{assume l: ?lhs
then obtain u where u: "q = p * u" ..
have "r = p * (smult a u - t)"
- using u qrp' [symmetric] t by (simp add: ring_simps mult_smult_right)
+ using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right)
then have ?rhs ..}
moreover
{assume r: ?rhs
then obtain u where u: "r = p * u" ..
from u [symmetric] t qrp' [symmetric] a0
have "q = p * smult (1/a) (u + t)"
- by (simp add: ring_simps mult_smult_right smult_smult)
+ by (simp add: algebra_simps mult_smult_right smult_smult)
hence ?lhs ..}
ultimately have "?lhs = ?rhs" by blast }
thus "?lhs \<equiv> ?rhs" by - (atomize(full), blast)
--- a/src/HOL/Groebner_Basis.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Groebner_Basis.thy Wed Jan 28 16:29:16 2009 +0100
@@ -165,7 +165,7 @@
interpretation class_semiring!: gb_semiring
"op +" "op *" "op ^" "0::'a::{comm_semiring_1, recpower}" "1"
- proof qed (auto simp add: ring_simps power_Suc)
+ proof qed (auto simp add: algebra_simps power_Suc)
lemmas nat_arith =
add_nat_number_of
@@ -345,13 +345,13 @@
interpretation class_ringb!: ringb
"op +" "op *" "op ^" "0::'a::{idom,recpower,number_ring}" "1" "op -" "uminus"
-proof(unfold_locales, simp add: ring_simps power_Suc, auto)
+proof(unfold_locales, simp add: algebra_simps power_Suc, auto)
fix w x y z ::"'a::{idom,recpower,number_ring}"
assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
hence ynz': "y - z \<noteq> 0" by simp
from p have "w * y + x* z - w*z - x*y = 0" by simp
- hence "w* (y - z) - x * (y - z) = 0" by (simp add: ring_simps)
- hence "(y - z) * (w - x) = 0" by (simp add: ring_simps)
+ hence "w* (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
+ hence "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
with no_zero_divirors_neq0 [OF ynz']
have "w - x = 0" by blast
thus "w = x" by simp
@@ -361,20 +361,20 @@
interpretation natgb!: semiringb
"op +" "op *" "op ^" "0::nat" "1"
-proof (unfold_locales, simp add: ring_simps power_Suc)
+proof (unfold_locales, simp add: algebra_simps power_Suc)
fix w x y z ::"nat"
{ assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
hence "y < z \<or> y > z" by arith
moreover {
assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
then obtain k where kp: "k>0" and yz:"z = y + k" by blast
- from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz ring_simps)
+ from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
hence "x*k = w*k" by simp
hence "w = x" using kp by (simp add: mult_cancel2) }
moreover {
assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
then obtain k where kp: "k>0" and yz:"y = z + k" by blast
- from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz ring_simps)
+ from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
hence "w*k = x*k" by simp
hence "w = x" using kp by (simp add: mult_cancel2)}
ultimately have "w=x" by blast }
--- a/src/HOL/Int.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Int.thy Wed Jan 28 16:29:16 2009 +0100
@@ -157,13 +157,13 @@
show "i - j = i + - j"
by (simp add: diff_int_def)
show "(i * j) * k = i * (j * k)"
- by (cases i, cases j, cases k) (simp add: mult ring_simps)
+ by (cases i, cases j, cases k) (simp add: mult algebra_simps)
show "i * j = j * i"
- by (cases i, cases j) (simp add: mult ring_simps)
+ by (cases i, cases j) (simp add: mult algebra_simps)
show "1 * i = i"
by (cases i) (simp add: One_int_def mult)
show "(i + j) * k = i * k + j * k"
- by (cases i, cases j, cases k) (simp add: add mult ring_simps)
+ by (cases i, cases j, cases k) (simp add: add mult algebra_simps)
show "0 \<noteq> (1::int)"
by (simp add: Zero_int_def One_int_def)
qed
@@ -301,36 +301,35 @@
lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
proof -
have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
- by (simp add: congruent_def compare_rls of_nat_add [symmetric]
+ by (simp add: congruent_def algebra_simps of_nat_add [symmetric]
del: of_nat_add)
thus ?thesis
by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
qed
lemma of_int_0 [simp]: "of_int 0 = 0"
- by (simp add: of_int Zero_int_def)
+by (simp add: of_int Zero_int_def)
lemma of_int_1 [simp]: "of_int 1 = 1"
- by (simp add: of_int One_int_def)
+by (simp add: of_int One_int_def)
lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
- by (cases w, cases z, simp add: compare_rls of_int OrderedGroup.compare_rls add)
+by (cases w, cases z, simp add: algebra_simps of_int add)
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
- by (cases z, simp add: compare_rls of_int minus)
+by (cases z, simp add: algebra_simps of_int minus)
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
- by (simp add: OrderedGroup.diff_minus diff_minus)
+by (simp add: OrderedGroup.diff_minus diff_minus)
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
apply (cases w, cases z)
-apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
- mult add_ac of_nat_mult)
+apply (simp add: algebra_simps of_int mult of_nat_mult)
done
text{*Collapse nested embeddings*}
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
- by (induct n) auto
+by (induct n) auto
end
@@ -339,7 +338,7 @@
lemma of_int_le_iff [simp]:
"of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
- by (cases w, cases z, simp add: of_int le minus compare_rls of_nat_add [symmetric] del: of_nat_add)
+by (cases w, cases z, simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add)
text{*Special cases where either operand is zero*}
lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
@@ -511,7 +510,7 @@
also have "\<dots> = (\<exists>n. z - w = of_nat n)"
by (auto elim: zero_le_imp_eq_int)
also have "\<dots> = (\<exists>n. z = w + of_nat n)"
- by (simp only: group_simps)
+ by (simp only: algebra_simps)
finally show ?thesis .
qed
@@ -829,7 +828,7 @@
next
case (neg n)
thus ?thesis by (simp del: of_nat_Suc of_nat_add
- add: compare_rls of_nat_1 [symmetric] of_nat_add [symmetric])
+ add: algebra_simps of_nat_1 [symmetric] of_nat_add [symmetric])
qed
lemma bin_less_0_simps:
@@ -1067,16 +1066,16 @@
lemma eq_number_of_eq:
"((number_of x::'a::number_ring) = number_of y) =
iszero (number_of (x + uminus y) :: 'a)"
- unfolding iszero_def number_of_add number_of_minus
- by (simp add: compare_rls)
+unfolding iszero_def number_of_add number_of_minus
+by (simp add: algebra_simps)
lemma iszero_number_of_Pls:
"iszero ((number_of Pls)::'a::number_ring)"
- unfolding iszero_def numeral_0_eq_0 ..
+unfolding iszero_def numeral_0_eq_0 ..
lemma nonzero_number_of_Min:
"~ iszero ((number_of Min)::'a::number_ring)"
- unfolding iszero_def numeral_m1_eq_minus_1 by simp
+unfolding iszero_def numeral_m1_eq_minus_1 by simp
subsubsection {* Comparisons, for Ordered Rings *}
@@ -1162,7 +1161,7 @@
next
case (neg n)
thus ?thesis by (simp del: of_nat_Suc of_nat_add
- add: compare_rls of_nat_1 [symmetric] of_nat_add [symmetric])
+ add: algebra_simps of_nat_1 [symmetric] of_nat_add [symmetric])
qed
text {* Less-Than or Equals *}
@@ -1249,9 +1248,7 @@
lemma add_number_of_diff2 [simp]:
"number_of v + (c - number_of w) =
number_of (v + uminus w) + (c::'a::number_ring)"
-apply (subst diff_number_of_eq [symmetric])
-apply (simp only: compare_rls)
-done
+by (simp add: algebra_simps diff_number_of_eq [symmetric])
subsection {* The Set of Integers *}
--- a/src/HOL/IntDiv.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/IntDiv.thy Wed Jan 28 16:29:16 2009 +0100
@@ -831,12 +831,13 @@
lemma zmult2_lemma_aux1: "[| (0::int) < c; b < r; r \<le> 0 |] ==> b*c < b*(q mod c) + r"
apply (subgoal_tac "b * (c - q mod c) < r * 1")
-apply (simp add: right_diff_distrib)
+ apply (simp add: algebra_simps)
apply (rule order_le_less_trans)
-apply (erule_tac [2] mult_strict_right_mono)
-apply (rule mult_left_mono_neg)
-apply (auto simp add: compare_rls add_commute [of 1]
- add1_zle_eq pos_mod_bound)
+ apply (erule_tac [2] mult_strict_right_mono)
+ apply (rule mult_left_mono_neg)
+ using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
+ apply (simp)
+apply (simp)
done
lemma zmult2_lemma_aux2:
@@ -854,12 +855,13 @@
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
-apply (simp add: right_diff_distrib)
+ apply (simp add: right_diff_distrib)
apply (rule order_less_le_trans)
-apply (erule mult_strict_right_mono)
-apply (rule_tac [2] mult_left_mono)
-apply (auto simp add: compare_rls add_commute [of 1]
- add1_zle_eq pos_mod_bound)
+ apply (erule mult_strict_right_mono)
+ apply (rule_tac [2] mult_left_mono)
+ apply simp
+ using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
+apply simp
done
lemma zmult2_lemma: "[| quorem ((a,b), (q,r)); b \<noteq> 0; 0 < c |]
@@ -1251,7 +1253,7 @@
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
using zmod_zdiv_equality[where a="m" and b="n"]
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
apply (subgoal_tac "m mod n = 0")
--- a/src/HOL/Library/Abstract_Rat.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/Abstract_Rat.thy Wed Jan 28 16:29:16 2009 +0100
@@ -240,7 +240,7 @@
by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
then have "of_int x / of_int d = ?t / of_int d"
using cong[OF refl[of ?f] eq] by simp
- then show ?thesis by (simp add: add_divide_distrib ring_simps prems)
+ then show ?thesis by (simp add: add_divide_distrib algebra_simps prems)
qed
lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
--- a/src/HOL/Library/BigO.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/BigO.thy Wed Jan 28 16:29:16 2009 +0100
@@ -349,7 +349,7 @@
apply (drule set_plus_imp_minus)
apply (rule set_minus_imp_plus)
apply (drule bigo_mult3 [where g = g and j = g])
- apply (auto simp add: ring_simps)
+ apply (auto simp add: algebra_simps)
done
lemma bigo_mult5: "ALL x. f x ~= 0 ==>
@@ -799,14 +799,14 @@
apply simp
apply (subst abs_of_nonneg)
apply (drule_tac x = x in spec) back
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (subst diff_minus)+
apply (rule add_right_mono)
apply (erule spec)
apply (rule order_trans)
prefer 2
apply (rule abs_ge_zero)
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
done
lemma bigo_lesso3: "f =o g +o O(h) ==>
@@ -823,7 +823,7 @@
apply simp
apply (subst abs_of_nonneg)
apply (drule_tac x = x in spec) back
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (subst diff_minus)+
apply (rule add_left_mono)
apply (rule le_imp_neg_le)
@@ -831,7 +831,7 @@
apply (rule order_trans)
prefer 2
apply (rule abs_ge_zero)
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
done
lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
@@ -844,7 +844,7 @@
apply assumption
apply (erule bigo_lesseq2) back
apply (rule allI)
- apply (auto simp add: func_plus fun_diff_def compare_rls
+ apply (auto simp add: func_plus fun_diff_def algebra_simps
split: split_max abs_split)
done
@@ -856,7 +856,7 @@
apply (rule allI)
apply (drule_tac x = x in spec)
apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0")
- apply (clarsimp simp add: compare_rls add_ac)
+ apply (clarsimp simp add: algebra_simps)
apply (rule abs_of_nonneg)
apply (rule le_maxI2)
done
--- a/src/HOL/Library/Commutative_Ring.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/Commutative_Ring.thy Wed Jan 28 16:29:16 2009 +0100
@@ -173,11 +173,11 @@
text {* mkPinj preserve semantics *}
lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
- by (induct B) (auto simp add: mkPinj_def ring_simps)
+ by (induct B) (auto simp add: mkPinj_def algebra_simps)
text {* mkPX preserves semantics *}
lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
- by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add ring_simps)
+ by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add algebra_simps)
text {* Correctness theorems for the implemented operations *}
@@ -192,13 +192,13 @@
show ?case
proof (rule linorder_cases)
assume "x < y"
- with 6 show ?case by (simp add: mkPinj_ci ring_simps)
+ with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
next
assume "x = y"
with 6 show ?case by (simp add: mkPinj_ci)
next
assume "x > y"
- with 6 show ?case by (simp add: mkPinj_ci ring_simps)
+ with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
qed
next
case (7 x P Q y R)
@@ -206,7 +206,7 @@
moreover
{ assume "x = 0" with 7 have ?case by simp }
moreover
- { assume "x = 1" with 7 have ?case by (simp add: ring_simps) }
+ { assume "x = 1" with 7 have ?case by (simp add: algebra_simps) }
moreover
{ assume "x > 1" from 7 have ?case by (cases x) simp_all }
ultimately show ?case by blast
@@ -225,20 +225,20 @@
show ?case
proof (rule linorder_cases)
assume a: "x < y" hence "EX d. d + x = y" by arith
- with 9 a show ?case by (auto simp add: mkPX_ci power_add ring_simps)
+ with 9 a show ?case by (auto simp add: mkPX_ci power_add algebra_simps)
next
assume a: "y < x" hence "EX d. d + y = x" by arith
- with 9 a show ?case by (auto simp add: power_add mkPX_ci ring_simps)
+ with 9 a show ?case by (auto simp add: power_add mkPX_ci algebra_simps)
next
assume "x = y"
- with 9 show ?case by (simp add: mkPX_ci ring_simps)
+ with 9 show ?case by (simp add: mkPX_ci algebra_simps)
qed
-qed (auto simp add: ring_simps)
+qed (auto simp add: algebra_simps)
text {* Multiplication *}
lemma mul_ci: "Ipol l (P \<otimes> Q) = Ipol l P * Ipol l Q"
by (induct P Q arbitrary: l rule: mul.induct)
- (simp_all add: mkPX_ci mkPinj_ci ring_simps add_ci power_add)
+ (simp_all add: mkPX_ci mkPinj_ci algebra_simps add_ci power_add)
text {* Substraction *}
lemma sub_ci: "Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q"
@@ -247,7 +247,7 @@
text {* Square *}
lemma sqr_ci: "Ipol ls (sqr P) = Ipol ls P * Ipol ls P"
by (induct P arbitrary: ls)
- (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci ring_simps power_add)
+ (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci algebra_simps power_add)
text {* Power *}
lemma even_pow:"even n \<Longrightarrow> pow n P = pow (n div 2) (sqr P)"
--- a/src/HOL/Library/Float.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/Float.thy Wed Jan 28 16:29:16 2009 +0100
@@ -38,7 +38,7 @@
show ?thesis
proof (induct a)
case (1 n)
- from pos show ?case by (simp add: ring_simps)
+ from pos show ?case by (simp add: algebra_simps)
next
case (2 n)
show ?case
@@ -59,7 +59,7 @@
show ?case by simp
next
case (Suc m)
- show ?case by (auto simp add: ring_simps pow2_add1 prems)
+ show ?case by (auto simp add: algebra_simps pow2_add1 prems)
qed
next
case (2 n)
@@ -72,7 +72,7 @@
apply (subst pow2_neg[of "-1"])
apply (simp)
apply (insert pow2_add1[of "-a"])
- apply (simp add: ring_simps)
+ apply (simp add: algebra_simps)
apply (subst pow2_neg[of "-a"])
apply (simp)
done
@@ -83,7 +83,7 @@
apply (auto)
apply (subst pow2_neg[of "a + (-2 - int m)"])
apply (subst pow2_neg[of "-2 - int m"])
- apply (auto simp add: ring_simps)
+ apply (auto simp add: algebra_simps)
apply (subst a)
apply (subst b)
apply (simp only: pow2_add1)
@@ -91,13 +91,13 @@
apply (subst pow2_neg[of "int m + 1"])
apply auto
apply (insert prems)
- apply (auto simp add: ring_simps)
+ apply (auto simp add: algebra_simps)
done
qed
qed
lemma "float (a, e) + float (b, e) = float (a + b, e)"
-by (simp add: float_def ring_simps)
+by (simp add: float_def algebra_simps)
definition
int_of_real :: "real \<Rightarrow> int" where
@@ -254,7 +254,7 @@
lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)"
apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])
- apply (simp_all add: pow2_def even_def real_is_int_def ring_simps)
+ apply (simp_all add: pow2_def even_def real_is_int_def algebra_simps)
apply (auto)
proof -
fix q::int
@@ -319,7 +319,7 @@
"float (a1, e1) + float (a2, e2) =
(if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
else float (a1*2^(nat (e1-e2))+a2, e2))"
- apply (simp add: float_def ring_simps)
+ apply (simp add: float_def algebra_simps)
apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
done
--- a/src/HOL/Library/Nat_Infinity.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/Nat_Infinity.thy Wed Jan 28 16:29:16 2009 +0100
@@ -223,10 +223,10 @@
end
lemma mult_iSuc: "iSuc m * n = n + m * n"
- unfolding iSuc_plus_1 by (simp add: ring_simps)
+ unfolding iSuc_plus_1 by (simp add: algebra_simps)
lemma mult_iSuc_right: "m * iSuc n = m + m * n"
- unfolding iSuc_plus_1 by (simp add: ring_simps)
+ unfolding iSuc_plus_1 by (simp add: algebra_simps)
lemma of_nat_eq_Fin: "of_nat n = Fin n"
apply (induct n)
--- a/src/HOL/Library/Pocklington.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/Pocklington.thy Wed Jan 28 16:29:16 2009 +0100
@@ -254,7 +254,7 @@
apply (simp add: nat_mod)
apply (rule_tac x="q" in exI)
apply (rule_tac x="q + q" in exI)
- by (auto simp: ring_simps)
+ by (auto simp: algebra_simps)
lemma cong_to_1: "[a = 1] (mod n) \<longleftrightarrow> a = 0 \<and> n = 1 \<or> (\<exists>m. a = 1 + m * n)"
proof-
@@ -780,7 +780,7 @@
of "(n - 1) div m * m"]
have yn: "coprime ?y n" by (simp add: coprime_commute)
have "?y mod n = (a^m)^((n - 1) div m) mod n"
- by (simp add: ring_simps power_mult)
+ by (simp add: algebra_simps power_mult)
also have "\<dots> = (a^m mod n)^((n - 1) div m) mod n"
using power_mod[of "a^m" n "(n - 1) div m"] by simp
also have "\<dots> = 1" using m(3)[unfolded modeq_def onen] onen
@@ -1239,7 +1239,7 @@
from bqn psp qrn
have bqn: "a ^ (n - 1) mod n = 1"
and psp: "list_all (\<lambda>p. prime p \<and> coprime (a^(r *(q div p)) mod n - 1) n) ps" unfolding arnb[symmetric] power_mod
- by (simp_all add: power_mult[symmetric] ring_simps)
+ by (simp_all add: power_mult[symmetric] algebra_simps)
from n have n0: "n > 0" by arith
from mod_div_equality[of "a^(n - 1)" n]
mod_less_divisor[OF n0, of "a^(n - 1)"]
@@ -1248,7 +1248,7 @@
apply -
apply (rule exI[where x="0"])
apply (rule exI[where x="a^(n - 1) div n"])
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
{fix p assume p: "prime p" "p dvd q"
from psp psq have pfpsq: "primefact ps q"
by (auto simp add: primefact_variant list_all_iff)
--- a/src/HOL/Library/Primes.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/Primes.thy Wed Jan 28 16:29:16 2009 +0100
@@ -204,7 +204,7 @@
from z have z': "?g > 0" by simp
from bezout_gcd_strong[OF az, of b]
obtain x y where xy: "a*x = b*y + ?g" by blast
- from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: ring_simps)
+ from xy a b have "?g * a'*x = ?g * (b'*y + 1)" by (simp add: algebra_simps)
hence "?g * (a'*x) = ?g * (b'*y + 1)" by (simp add: mult_assoc)
hence "a'*x = (b'*y + 1)"
by (simp only: nat_mult_eq_cancel1[OF z'])
--- a/src/HOL/Library/SetsAndFunctions.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/SetsAndFunctions.thy Wed Jan 28 16:29:16 2009 +0100
@@ -101,8 +101,8 @@
instance "fun" :: (type,comm_ring_1)comm_ring_1
apply default
- apply (auto simp add: func_plus func_times fun_Compl_def fun_diff_def ext
- func_one func_zero ring_simps)
+ apply (auto simp add: func_plus func_times fun_Compl_def fun_diff_def
+ func_one func_zero algebra_simps)
apply (drule fun_cong)
apply simp
done
--- a/src/HOL/Library/Univ_Poly.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Library/Univ_Poly.thy Wed Jan 28 16:29:16 2009 +0100
@@ -217,7 +217,7 @@
from Cons.hyps[rule_format, of x]
obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
- using qr by(cases q, simp_all add: ring_simps diff_def[symmetric]
+ using qr by(cases q, simp_all add: algebra_simps diff_def[symmetric]
minus_mult_left[symmetric] right_minus)
hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
thus ?case by blast
@@ -303,7 +303,7 @@
from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
by blast
from y have "y = a \<or> poly q y = 0"
- by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: ring_simps)
+ by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
with i[rule_format, of y] y(1) y(2) have False apply auto
apply (erule_tac x="m" in allE)
apply auto
@@ -414,8 +414,8 @@
lemma (in idom_char_0) poly_entire:
"poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
-using poly_entire_lemma2[of p q]
-by auto (rule ext, simp add: poly_mult)+
+using poly_entire_lemma2[of p q]
+by (auto simp add: expand_fun_eq poly_mult)
lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
by (simp add: poly_entire)
@@ -424,7 +424,7 @@
by (auto intro!: ext)
lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
-by (auto simp add: ring_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric])
+by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric])
lemma (in comm_ring_1) poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult right_distrib minus_mult_left[symmetric] minus_mult_right[symmetric])
@@ -550,7 +550,7 @@
"[| p divides q; p divides (q +++ r) |] ==> p divides r"
apply (simp add: divides_def, auto)
apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
-apply (auto simp add: poly_add fun_eq poly_mult poly_minus right_diff_distrib compare_rls add_ac)
+apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
done
lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
@@ -892,15 +892,15 @@
proof
assume eq: ?lhs
hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
- by (simp only: poly_minus poly_add ring_simps) simp
- hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by - (rule ext, simp)
+ by (simp only: poly_minus poly_add algebra_simps) simp
+ hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add:expand_fun_eq)
hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
- unfolding poly_zero by (simp add: poly_minus_def ring_simps minus_mult_left[symmetric])
+ unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
unfolding poly_zero[symmetric] by simp
- thus ?rhs apply (simp add: poly_minus poly_add ring_simps) apply (rule ext, simp) done
+ thus ?rhs by (simp add: poly_minus poly_add algebra_simps expand_fun_eq)
next
- assume ?rhs then show ?lhs by - (rule ext,simp)
+ assume ?rhs then show ?lhs by(simp add:expand_fun_eq)
qed
lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
@@ -1008,7 +1008,7 @@
from p have ap: "poly ([a,1] *** p) \<noteq> poly []"
using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
- by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult mult_ac add_ac right_distrib)
+ by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
from ap have ap': "(poly ([a,1] *** p) = poly []) = False" by blast
have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
--- a/src/HOL/Lim.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Lim.thy Wed Jan 28 16:29:16 2009 +0100
@@ -52,7 +52,7 @@
apply (drule_tac r="r" in LIM_D, safe)
apply (rule_tac x="s" in exI, safe)
apply (drule_tac x="x + k" in spec)
-apply (simp add: compare_rls)
+apply (simp add: algebra_simps)
done
lemma LIM_offset_zero: "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
--- a/src/HOL/Ln.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Ln.thy Wed Jan 28 16:29:16 2009 +0100
@@ -187,7 +187,7 @@
proof -
assume a: "0 <= (x::real)" and b: "x < 1"
have "(1 - x) * (1 + x + x^2) = (1 - x^3)"
- by (simp add: ring_simps power2_eq_square power3_eq_cube)
+ by (simp add: algebra_simps power2_eq_square power3_eq_cube)
also have "... <= 1"
by (auto simp add: a)
finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
@@ -325,10 +325,10 @@
done
also have "... <= 2 * x^2"
apply (subgoal_tac "- (-x) - 2 * (-x)^2 <= ln (1 - (-x))")
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (rule ln_one_minus_pos_lower_bound)
apply (insert prems, auto)
- done
+ done
finally show ?thesis .
qed
@@ -375,7 +375,7 @@
apply simp
apply (subst ln_div [THEN sym])
apply arith
- apply (auto simp add: ring_simps add_frac_eq frac_eq_eq
+ apply (auto simp add: algebra_simps add_frac_eq frac_eq_eq
add_divide_distrib power2_eq_square)
apply (rule mult_pos_pos, assumption)+
apply assumption
@@ -394,7 +394,7 @@
apply auto
done
have "x * ln y - x * ln x = x * (ln y - ln x)"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
also have "... = x * ln(y / x)"
apply (subst ln_div)
apply (rule b, rule a, rule refl)
--- a/src/HOL/Matrix/LP.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Matrix/LP.thy Wed Jan 28 16:29:16 2009 +0100
@@ -20,7 +20,7 @@
proof -
from prems have 1: "y * b <= y * b'" by (simp add: mult_left_mono)
from prems have 2: "y * (A * x) <= y * b" by (simp add: mult_left_mono)
- have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: ring_simps)
+ have 3: "y * (A * x) = c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x" by (simp add: algebra_simps)
from 1 2 3 have 4: "c * x + (y * (A - A') + (y * A' - c') + (c'-c)) * x <= y * b'" by simp
have 5: "c * x <= y * b' + abs((y * (A - A') + (y * A' - c') + (c'-c)) * x)"
by (simp only: 4 estimate_by_abs)
@@ -32,7 +32,7 @@
by (simp add: abs_triangle_ineq mult_right_mono)
have 9: "(abs (y * (A-A')) + abs (y*A'-c') + abs(c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x"
by (simp add: abs_le_mult mult_right_mono)
- have 10: "c'-c = -(c-c')" by (simp add: ring_simps)
+ have 10: "c'-c = -(c-c')" by (simp add: algebra_simps)
have 11: "abs (c'-c) = abs (c-c')"
by (subst 10, subst abs_minus_cancel, simp)
have 12: "(abs y * abs (A-A') + abs (y*A'-c') + abs (c'-c)) * abs x <= (abs y * abs (A-A') + abs (y*A'-c') + \<delta>c) * abs x"
@@ -85,7 +85,7 @@
apply simp
done
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
by (simp_all add: prems mult_mono)
moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
@@ -134,10 +134,10 @@
(is "_ <= _ + ?C")
proof -
from prems have "y * (A * x) <= y * b" by (simp add: mult_left_mono)
- moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: ring_simps)
+ moreover have "y * (A * x) = c * x + (y * A - c) * x" by (simp add: algebra_simps)
ultimately have "c * x + (y * A - c) * x <= y * b" by simp
then have "c * x <= y * b - (y * A - c) * x" by (simp add: le_diff_eq)
- then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: ring_simps)
+ then have cx: "c * x <= y * b + (c - y * A) * x" by (simp add: algebra_simps)
have s2: "c - y * A <= c2 - y * A1"
by (simp add: diff_def prems add_mono mult_left_mono)
have s1: "c1 - y * A2 <= c - y * A"
--- a/src/HOL/Matrix/Matrix.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Matrix/Matrix.thy Wed Jan 28 16:29:16 2009 +0100
@@ -1573,17 +1573,17 @@
show "A * B * C = A * (B * C)"
apply (simp add: times_matrix_def)
apply (rule mult_matrix_assoc)
- apply (simp_all add: associative_def ring_simps)
+ apply (simp_all add: associative_def algebra_simps)
done
show "(A + B) * C = A * C + B * C"
apply (simp add: times_matrix_def plus_matrix_def)
apply (rule l_distributive_matrix[simplified l_distributive_def, THEN spec, THEN spec, THEN spec])
- apply (simp_all add: associative_def commutative_def ring_simps)
+ apply (simp_all add: associative_def commutative_def algebra_simps)
done
show "A * (B + C) = A * B + A * C"
apply (simp add: times_matrix_def plus_matrix_def)
apply (rule r_distributive_matrix[simplified r_distributive_def, THEN spec, THEN spec, THEN spec])
- apply (simp_all add: associative_def commutative_def ring_simps)
+ apply (simp_all add: associative_def commutative_def algebra_simps)
done
qed
@@ -1793,7 +1793,7 @@
by (simp add: scalar_mult_def)
lemma scalar_mult_add: "scalar_mult y (a+b) = (scalar_mult y a) + (scalar_mult y b)"
-by (simp add: scalar_mult_def apply_matrix_add ring_simps)
+by (simp add: scalar_mult_def apply_matrix_add algebra_simps)
lemma Rep_scalar_mult[simp]: "Rep_matrix (scalar_mult y a) j i = y * (Rep_matrix a j i)"
by (simp add: scalar_mult_def)
--- a/src/HOL/Matrix/SparseMatrix.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Matrix/SparseMatrix.thy Wed Jan 28 16:29:16 2009 +0100
@@ -274,7 +274,7 @@
apply (rule conjI)
apply (intro strip)
apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp add: ring_simps sparse_row_matrix_cons)
+ apply (simp add: algebra_simps sparse_row_matrix_cons)
apply (simplesubst Rep_matrix_zero_imp_mult_zero)
apply (simp)
apply (intro strip)
@@ -296,7 +296,7 @@
apply (intro strip | rule conjI)+
apply (frule_tac as=arr in sorted_spvec_cons1)
- apply (simp add: ring_simps)
+ apply (simp add: algebra_simps)
apply (subst Rep_matrix_zero_imp_mult_zero)
apply (simp)
apply (rule disjI2)
@@ -310,7 +310,7 @@
apply (simp_all)
apply (frule_tac as=arr in sorted_spvec_cons1)
apply (frule_tac as=brr in sorted_spvec_cons1)
- apply (simp add: sparse_row_matrix_cons ring_simps sparse_row_vector_addmult_spvec)
+ apply (simp add: sparse_row_matrix_cons algebra_simps sparse_row_vector_addmult_spvec)
apply (rule_tac B1 = "sparse_row_matrix brr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
apply (auto)
apply (rule sorted_sparse_row_matrix_zero)
@@ -360,7 +360,7 @@
lemma sparse_row_mult_spmat[rule_format]:
"sorted_spmat A \<longrightarrow> sorted_spvec B \<longrightarrow> sparse_row_matrix (mult_spmat A B) = (sparse_row_matrix A) * (sparse_row_matrix B)"
apply (induct A)
- apply (auto simp add: sparse_row_matrix_cons sparse_row_mult_spvec_spmat ring_simps move_matrix_mult)
+ apply (auto simp add: sparse_row_matrix_cons sparse_row_mult_spvec_spmat algebra_simps move_matrix_mult)
done
lemma sorted_spvec_mult_spmat[rule_format]:
--- a/src/HOL/Matrix/cplex/Cplex.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Matrix/cplex/Cplex.thy Wed Jan 28 16:29:16 2009 +0100
@@ -34,8 +34,8 @@
add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1), mult_spmat (nprt_spmat s1) (nprt_spmat r1)))))))"
apply (simp add: Let_def)
apply (insert assms)
- apply (simp add: sparse_row_matrix_op_simps ring_simps)
- apply (rule mult_le_dual_prts[where A=A, simplified Let_def ring_simps])
+ apply (simp add: sparse_row_matrix_op_simps algebra_simps)
+ apply (rule mult_le_dual_prts[where A=A, simplified Let_def algebra_simps])
apply (auto)
done
--- a/src/HOL/MetisExamples/BigO.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/MetisExamples/BigO.thy Wed Jan 28 16:29:16 2009 +0100
@@ -486,7 +486,7 @@
have 2: "\<And>X3. (0\<Colon>'b) + X3 = X3"
by (metis diff_eq_eq right_minus_eq)
have 3: "\<not> (0\<Colon>'b) \<le> f x - lb x"
- by (metis 1 compare_rls(1))
+ by (metis 1 diff_minus)
have 4: "\<not> (0\<Colon>'b) + lb x \<le> f x"
by (metis 3 le_diff_eq)
show "False"
@@ -1197,7 +1197,7 @@
apply (erule thin_rl)
(*sledgehammer*);
apply (case_tac "0 <= f x - k x")
- apply (simp del: compare_rls diff_minus);
+ apply (simp)
apply (subst abs_of_nonneg)
apply (drule_tac x = x in spec) back
ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso3_simpler"*}
@@ -1216,7 +1216,7 @@
apply assumption
apply (erule bigo_lesseq2) back
apply (rule allI)
- apply (auto simp add: func_plus fun_diff_def compare_rls
+ apply (auto simp add: func_plus fun_diff_def algebra_simps
split: split_max abs_split)
done
--- a/src/HOL/Nat.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Nat.thy Wed Jan 28 16:29:16 2009 +0100
@@ -1254,7 +1254,7 @@
begin
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
- by (simp add: compare_rls of_nat_add [symmetric])
+by (simp add: algebra_simps of_nat_add [symmetric])
end
--- a/src/HOL/NumberTheory/IntPrimes.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/NumberTheory/IntPrimes.thy Wed Jan 28 16:29:16 2009 +0100
@@ -146,7 +146,7 @@
lemma zcong_trans:
"[a = b] (mod m) ==> [b = c] (mod m) ==> [a = c] (mod m)"
unfolding zcong_def
- apply (auto elim!: dvdE simp add: ring_simps)
+ apply (auto elim!: dvdE simp add: algebra_simps)
unfolding left_distrib [symmetric]
apply (rule dvd_mult dvd_refl)+
done
@@ -255,7 +255,7 @@
lemma zcong_iff_lin: "([a = b] (mod m)) = (\<exists>k. b = a + m * k)"
unfolding zcong_def
- apply (auto elim!: dvdE simp add: ring_simps)
+ apply (auto elim!: dvdE simp add: algebra_simps)
apply (rule_tac x = "-k" in exI) apply simp
done
--- a/src/HOL/OrderedGroup.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/OrderedGroup.thy Wed Jan 28 16:29:16 2009 +0100
@@ -22,17 +22,34 @@
\end{itemize}
*}
+ML{*
+structure AlgebraSimps =
+ NamedThmsFun(val name = "algebra_simps"
+ val description = "algebra simplification rules");
+*}
+
+setup AlgebraSimps.setup
+
+text{* The rewrites accumulated in @{text algebra_simps} deal with the
+classical algebraic structures of groups, rings and family. They simplify
+terms by multiplying everything out (in case of a ring) and bringing sums and
+products into a canonical form (by ordered rewriting). As a result it decides
+group and ring equalities but also helps with inequalities.
+
+Of course it also works for fields, but it knows nothing about multiplicative
+inverses or division. This is catered for by @{text field_simps}. *}
+
subsection {* Semigroups and Monoids *}
class semigroup_add = plus +
- assumes add_assoc: "(a + b) + c = a + (b + c)"
+ assumes add_assoc[algebra_simps]: "(a + b) + c = a + (b + c)"
class ab_semigroup_add = semigroup_add +
- assumes add_commute: "a + b = b + a"
+ assumes add_commute[algebra_simps]: "a + b = b + a"
begin
-lemma add_left_commute: "a + (b + c) = b + (a + c)"
- by (rule mk_left_commute [of "plus", OF add_assoc add_commute])
+lemma add_left_commute[algebra_simps]: "a + (b + c) = b + (a + c)"
+by (rule mk_left_commute [of "plus", OF add_assoc add_commute])
theorems add_ac = add_assoc add_commute add_left_commute
@@ -41,14 +58,14 @@
theorems add_ac = add_assoc add_commute add_left_commute
class semigroup_mult = times +
- assumes mult_assoc: "(a * b) * c = a * (b * c)"
+ assumes mult_assoc[algebra_simps]: "(a * b) * c = a * (b * c)"
class ab_semigroup_mult = semigroup_mult +
- assumes mult_commute: "a * b = b * a"
+ assumes mult_commute[algebra_simps]: "a * b = b * a"
begin
-lemma mult_left_commute: "a * (b * c) = b * (a * c)"
- by (rule mk_left_commute [of "times", OF mult_assoc mult_commute])
+lemma mult_left_commute[algebra_simps]: "a * (b * c) = b * (a * c)"
+by (rule mk_left_commute [of "times", OF mult_assoc mult_commute])
theorems mult_ac = mult_assoc mult_commute mult_left_commute
@@ -57,24 +74,20 @@
theorems mult_ac = mult_assoc mult_commute mult_left_commute
class ab_semigroup_idem_mult = ab_semigroup_mult +
- assumes mult_idem: "x * x = x"
+ assumes mult_idem[simp]: "x * x = x"
begin
-lemma mult_left_idem: "x * (x * y) = x * y"
+lemma mult_left_idem[simp]: "x * (x * y) = x * y"
unfolding mult_assoc [symmetric, of x] mult_idem ..
-lemmas mult_ac_idem = mult_ac mult_idem mult_left_idem
-
end
-lemmas mult_ac_idem = mult_ac mult_idem mult_left_idem
-
class monoid_add = zero + semigroup_add +
assumes add_0_left [simp]: "0 + a = a"
and add_0_right [simp]: "a + 0 = a"
lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
- by (rule eq_commute)
+by (rule eq_commute)
class comm_monoid_add = zero + ab_semigroup_add +
assumes add_0: "0 + a = a"
@@ -90,7 +103,7 @@
assumes mult_1_right [simp]: "a * 1 = a"
lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
- by (rule eq_commute)
+by (rule eq_commute)
class comm_monoid_mult = one + ab_semigroup_mult +
assumes mult_1: "1 * a = a"
@@ -108,11 +121,11 @@
lemma add_left_cancel [simp]:
"a + b = a + c \<longleftrightarrow> b = c"
- by (blast dest: add_left_imp_eq)
+by (blast dest: add_left_imp_eq)
lemma add_right_cancel [simp]:
"b + a = c + a \<longleftrightarrow> b = c"
- by (blast dest: add_right_imp_eq)
+by (blast dest: add_right_imp_eq)
end
@@ -142,7 +155,7 @@
begin
lemma minus_add_cancel: "- a + (a + b) = b"
- by (simp add: add_assoc[symmetric])
+by (simp add: add_assoc[symmetric])
lemma minus_zero [simp]: "- 0 = 0"
proof -
@@ -176,8 +189,7 @@
qed
lemma equals_zero_I:
- assumes "a + b = 0"
- shows "- a = b"
+ assumes "a + b = 0" shows "- a = b"
proof -
have "- a = - a + (a + b)" using assms by simp
also have "\<dots> = b" by (simp add: add_assoc[symmetric])
@@ -185,23 +197,22 @@
qed
lemma diff_self [simp]: "a - a = 0"
- by (simp add: diff_minus)
+by (simp add: diff_minus)
lemma diff_0 [simp]: "0 - a = - a"
- by (simp add: diff_minus)
+by (simp add: diff_minus)
lemma diff_0_right [simp]: "a - 0 = a"
- by (simp add: diff_minus)
+by (simp add: diff_minus)
lemma diff_minus_eq_add [simp]: "a - - b = a + b"
- by (simp add: diff_minus)
+by (simp add: diff_minus)
lemma neg_equal_iff_equal [simp]:
"- a = - b \<longleftrightarrow> a = b"
proof
assume "- a = - b"
- hence "- (- a) = - (- b)"
- by simp
+ hence "- (- a) = - (- b)" by simp
thus "a = b" by simp
next
assume "a = b"
@@ -210,11 +221,11 @@
lemma neg_equal_0_iff_equal [simp]:
"- a = 0 \<longleftrightarrow> a = 0"
- by (subst neg_equal_iff_equal [symmetric], simp)
+by (subst neg_equal_iff_equal [symmetric], simp)
lemma neg_0_equal_iff_equal [simp]:
"0 = - a \<longleftrightarrow> 0 = a"
- by (subst neg_equal_iff_equal [symmetric], simp)
+by (subst neg_equal_iff_equal [symmetric], simp)
text{*The next two equations can make the simplifier loop!*}
@@ -233,10 +244,12 @@
qed
lemma diff_add_cancel: "a - b + b = a"
- by (simp add: diff_minus add_assoc)
+by (simp add: diff_minus add_assoc)
lemma add_diff_cancel: "a + b - b = a"
- by (simp add: diff_minus add_assoc)
+by (simp add: diff_minus add_assoc)
+
+declare diff_minus[symmetric, algebra_simps]
end
@@ -257,43 +270,38 @@
then show "b = c" by simp
qed
-lemma uminus_add_conv_diff:
+lemma uminus_add_conv_diff[algebra_simps]:
"- a + b = b - a"
- by (simp add:diff_minus add_commute)
+by (simp add:diff_minus add_commute)
lemma minus_add_distrib [simp]:
"- (a + b) = - a + - b"
- by (rule equals_zero_I) (simp add: add_ac)
+by (rule equals_zero_I) (simp add: add_ac)
lemma minus_diff_eq [simp]:
"- (a - b) = b - a"
- by (simp add: diff_minus add_commute)
-
-lemma add_diff_eq: "a + (b - c) = (a + b) - c"
- by (simp add: diff_minus add_ac)
+by (simp add: diff_minus add_commute)
-lemma diff_add_eq: "(a - b) + c = (a + c) - b"
- by (simp add: diff_minus add_ac)
+lemma add_diff_eq[algebra_simps]: "a + (b - c) = (a + b) - c"
+by (simp add: diff_minus add_ac)
-lemma diff_eq_eq: "a - b = c \<longleftrightarrow> a = c + b"
- by (auto simp add: diff_minus add_assoc)
+lemma diff_add_eq[algebra_simps]: "(a - b) + c = (a + c) - b"
+by (simp add: diff_minus add_ac)
-lemma eq_diff_eq: "a = c - b \<longleftrightarrow> a + b = c"
- by (auto simp add: diff_minus add_assoc)
+lemma diff_eq_eq[algebra_simps]: "a - b = c \<longleftrightarrow> a = c + b"
+by (auto simp add: diff_minus add_assoc)
-lemma diff_diff_eq: "(a - b) - c = a - (b + c)"
- by (simp add: diff_minus add_ac)
+lemma eq_diff_eq[algebra_simps]: "a = c - b \<longleftrightarrow> a + b = c"
+by (auto simp add: diff_minus add_assoc)
-lemma diff_diff_eq2: "a - (b - c) = (a + c) - b"
- by (simp add: diff_minus add_ac)
+lemma diff_diff_eq[algebra_simps]: "(a - b) - c = a - (b + c)"
+by (simp add: diff_minus add_ac)
-lemmas compare_rls =
- diff_minus [symmetric]
- add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
- diff_eq_eq eq_diff_eq
+lemma diff_diff_eq2[algebra_simps]: "a - (b - c) = (a + c) - b"
+by (simp add: diff_minus add_ac)
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
- by (simp add: compare_rls)
+by (simp add: algebra_simps)
end
@@ -305,7 +313,7 @@
lemma add_right_mono:
"a \<le> b \<Longrightarrow> a + c \<le> b + c"
- by (simp add: add_commute [of _ c] add_left_mono)
+by (simp add: add_commute [of _ c] add_left_mono)
text {* non-strict, in both arguments *}
lemma add_mono:
@@ -322,11 +330,11 @@
lemma add_strict_left_mono:
"a < b \<Longrightarrow> c + a < c + b"
- by (auto simp add: less_le add_left_mono)
+by (auto simp add: less_le add_left_mono)
lemma add_strict_right_mono:
"a < b \<Longrightarrow> a + c < b + c"
- by (simp add: add_commute [of _ c] add_strict_left_mono)
+by (simp add: add_commute [of _ c] add_strict_left_mono)
text{*Strict monotonicity in both arguments*}
lemma add_strict_mono:
@@ -355,8 +363,7 @@
begin
lemma add_less_imp_less_left:
- assumes less: "c + a < c + b"
- shows "a < b"
+ assumes less: "c + a < c + b" shows "a < b"
proof -
from less have le: "c + a <= c + b" by (simp add: order_le_less)
have "a <= b"
@@ -381,23 +388,23 @@
lemma add_less_cancel_left [simp]:
"c + a < c + b \<longleftrightarrow> a < b"
- by (blast intro: add_less_imp_less_left add_strict_left_mono)
+by (blast intro: add_less_imp_less_left add_strict_left_mono)
lemma add_less_cancel_right [simp]:
"a + c < b + c \<longleftrightarrow> a < b"
- by (blast intro: add_less_imp_less_right add_strict_right_mono)
+by (blast intro: add_less_imp_less_right add_strict_right_mono)
lemma add_le_cancel_left [simp]:
"c + a \<le> c + b \<longleftrightarrow> a \<le> b"
- by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono)
+by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono)
lemma add_le_cancel_right [simp]:
"a + c \<le> b + c \<longleftrightarrow> a \<le> b"
- by (simp add: add_commute [of a c] add_commute [of b c])
+by (simp add: add_commute [of a c] add_commute [of b c])
lemma add_le_imp_le_right:
"a + c \<le> b + c \<Longrightarrow> a \<le> b"
- by simp
+by simp
lemma max_add_distrib_left:
"max x y + z = max (x + z) (y + z)"
@@ -416,8 +423,7 @@
begin
lemma add_pos_nonneg:
- assumes "0 < a" and "0 \<le> b"
- shows "0 < a + b"
+ assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
proof -
have "0 + 0 < a + b"
using assms by (rule add_less_le_mono)
@@ -425,13 +431,11 @@
qed
lemma add_pos_pos:
- assumes "0 < a" and "0 < b"
- shows "0 < a + b"
- by (rule add_pos_nonneg) (insert assms, auto)
+ assumes "0 < a" and "0 < b" shows "0 < a + b"
+by (rule add_pos_nonneg) (insert assms, auto)
lemma add_nonneg_pos:
- assumes "0 \<le> a" and "0 < b"
- shows "0 < a + b"
+ assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
proof -
have "0 + 0 < a + b"
using assms by (rule add_le_less_mono)
@@ -439,8 +443,7 @@
qed
lemma add_nonneg_nonneg:
- assumes "0 \<le> a" and "0 \<le> b"
- shows "0 \<le> a + b"
+ assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
proof -
have "0 + 0 \<le> a + b"
using assms by (rule add_mono)
@@ -448,8 +451,7 @@
qed
lemma add_neg_nonpos:
- assumes "a < 0" and "b \<le> 0"
- shows "a + b < 0"
+ assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
proof -
have "a + b < 0 + 0"
using assms by (rule add_less_le_mono)
@@ -457,13 +459,11 @@
qed
lemma add_neg_neg:
- assumes "a < 0" and "b < 0"
- shows "a + b < 0"
- by (rule add_neg_nonpos) (insert assms, auto)
+ assumes "a < 0" and "b < 0" shows "a + b < 0"
+by (rule add_neg_nonpos) (insert assms, auto)
lemma add_nonpos_neg:
- assumes "a \<le> 0" and "b < 0"
- shows "a + b < 0"
+ assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
proof -
have "a + b < 0 + 0"
using assms by (rule add_le_less_mono)
@@ -471,8 +471,7 @@
qed
lemma add_nonpos_nonpos:
- assumes "a \<le> 0" and "b \<le> 0"
- shows "a + b \<le> 0"
+ assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
proof -
have "a + b \<le> 0 + 0"
using assms by (rule add_mono)
@@ -500,31 +499,25 @@
lemma max_diff_distrib_left:
shows "max x y - z = max (x - z) (y - z)"
- by (simp add: diff_minus, rule max_add_distrib_left)
+by (simp add: diff_minus, rule max_add_distrib_left)
lemma min_diff_distrib_left:
shows "min x y - z = min (x - z) (y - z)"
- by (simp add: diff_minus, rule min_add_distrib_left)
+by (simp add: diff_minus, rule min_add_distrib_left)
lemma le_imp_neg_le:
- assumes "a \<le> b"
- shows "-b \<le> -a"
+ assumes "a \<le> b" shows "-b \<le> -a"
proof -
- have "-a+a \<le> -a+b"
- using `a \<le> b` by (rule add_left_mono)
- hence "0 \<le> -a+b"
- by simp
- hence "0 + (-b) \<le> (-a + b) + (-b)"
- by (rule add_right_mono)
- thus ?thesis
- by (simp add: add_assoc)
+ have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono)
+ hence "0 \<le> -a+b" by simp
+ hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono)
+ thus ?thesis by (simp add: add_assoc)
qed
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
proof
assume "- b \<le> - a"
- hence "- (- a) \<le> - (- b)"
- by (rule le_imp_neg_le)
+ hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
thus "a\<le>b" by simp
next
assume "a\<le>b"
@@ -532,19 +525,19 @@
qed
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
- by (subst neg_le_iff_le [symmetric], simp)
+by (subst neg_le_iff_le [symmetric], simp)
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
- by (subst neg_le_iff_le [symmetric], simp)
+by (subst neg_le_iff_le [symmetric], simp)
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
- by (force simp add: less_le)
+by (force simp add: less_le)
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
- by (subst neg_less_iff_less [symmetric], simp)
+by (subst neg_less_iff_less [symmetric], simp)
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
- by (subst neg_less_iff_less [symmetric], simp)
+by (subst neg_less_iff_less [symmetric], simp)
text{*The next several equations can make the simplifier loop!*}
@@ -573,7 +566,7 @@
qed
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
- by (auto simp add: le_less minus_less_iff)
+by (auto simp add: le_less minus_less_iff)
lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0"
proof -
@@ -583,56 +576,34 @@
finally show ?thesis .
qed
-lemma diff_less_eq: "a - b < c \<longleftrightarrow> a < c + b"
+lemma diff_less_eq[algebra_simps]: "a - b < c \<longleftrightarrow> a < c + b"
apply (subst less_iff_diff_less_0 [of a])
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
apply (simp add: diff_minus add_ac)
done
-lemma less_diff_eq: "a < c - b \<longleftrightarrow> a + b < c"
+lemma less_diff_eq[algebra_simps]: "a < c - b \<longleftrightarrow> a + b < c"
apply (subst less_iff_diff_less_0 [of "plus a b"])
apply (subst less_iff_diff_less_0 [of a])
apply (simp add: diff_minus add_ac)
done
-lemma diff_le_eq: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
- by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
-
-lemma le_diff_eq: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
- by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
+lemma diff_le_eq[algebra_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
+by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
-lemmas compare_rls =
- diff_minus [symmetric]
- add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
- diff_less_eq less_diff_eq diff_le_eq le_diff_eq
- diff_eq_eq eq_diff_eq
-
-text{*This list of rewrites simplifies (in)equalities by bringing subtractions
- to the top and then moving negative terms to the other side.
- Use with @{text add_ac}*}
-lemmas (in -) compare_rls =
- diff_minus [symmetric]
- add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
- diff_less_eq less_diff_eq diff_le_eq le_diff_eq
- diff_eq_eq eq_diff_eq
+lemma le_diff_eq[algebra_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
+by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
- by (simp add: compare_rls)
+by (simp add: algebra_simps)
-lemmas group_simps =
- add_ac
- add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
- diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
- diff_less_eq less_diff_eq diff_le_eq le_diff_eq
+text{*Legacy - use @{text algebra_simps} *}
+lemmas group_simps = algebra_simps
end
-lemmas group_simps =
- mult_ac
- add_ac
- add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
- diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
- diff_less_eq less_diff_eq diff_le_eq le_diff_eq
+text{*Legacy - use @{text algebra_simps} *}
+lemmas group_simps = algebra_simps
class ordered_ab_semigroup_add =
linorder + pordered_ab_semigroup_add
@@ -766,8 +737,7 @@
unfolding neg_le_0_iff_le by simp
lemma abs_of_nonneg [simp]:
- assumes nonneg: "0 \<le> a"
- shows "\<bar>a\<bar> = a"
+ assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
proof (rule antisym)
from nonneg le_imp_neg_le have "- a \<le> 0" by simp
from this nonneg have "- a \<le> a" by (rule order_trans)
@@ -775,8 +745,8 @@
qed (rule abs_ge_self)
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
- by (rule antisym)
- (auto intro!: abs_ge_self abs_leI order_trans [of "uminus (abs a)" zero "abs a"])
+by (rule antisym)
+ (auto intro!: abs_ge_self abs_leI order_trans [of "uminus (abs a)" zero "abs a"])
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
proof -
@@ -792,7 +762,7 @@
qed
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
- by simp
+by simp
lemma abs_0_eq [simp, noatp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
proof -
@@ -811,7 +781,7 @@
qed
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
- by (simp add: less_le)
+by (simp add: less_le)
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
proof -
@@ -834,11 +804,10 @@
qed
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
- by (rule abs_of_nonneg, rule less_imp_le)
+by (rule abs_of_nonneg, rule less_imp_le)
lemma abs_of_nonpos [simp]:
- assumes "a \<le> 0"
- shows "\<bar>a\<bar> = - a"
+ assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
proof -
let ?b = "- a"
have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
@@ -849,24 +818,24 @@
qed
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
- by (rule abs_of_nonpos, rule less_imp_le)
+by (rule abs_of_nonpos, rule less_imp_le)
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
- by (insert abs_ge_self, blast intro: order_trans)
+by (insert abs_ge_self, blast intro: order_trans)
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
- by (insert abs_le_D1 [of "uminus a"], simp)
+by (insert abs_le_D1 [of "uminus a"], simp)
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
- by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
+by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
- apply (simp add: compare_rls)
- apply (subgoal_tac "abs a = abs (plus (minus a b) b)")
+ apply (simp add: algebra_simps)
+ apply (subgoal_tac "abs a = abs (plus b (minus a b))")
apply (erule ssubst)
apply (rule abs_triangle_ineq)
- apply (rule arg_cong) back
- apply (simp add: compare_rls)
+ apply (rule arg_cong[of _ _ abs])
+ apply (simp add: algebra_simps)
done
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
@@ -879,12 +848,9 @@
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
proof -
- have "abs(a - b) = abs(a + - b)"
- by (subst diff_minus, rule refl)
- also have "... <= abs a + abs (- b)"
- by (rule abs_triangle_ineq)
- finally show ?thesis
- by simp
+ have "abs(a - b) = abs(a + - b)" by (subst diff_minus, rule refl)
+ also have "... <= abs a + abs (- b)" by (rule abs_triangle_ineq)
+ finally show ?thesis by simp
qed
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
@@ -999,23 +965,19 @@
qed
lemma neg_inf_eq_sup: "- inf a b = sup (-a) (-b)"
- by (simp add: inf_eq_neg_sup)
+by (simp add: inf_eq_neg_sup)
lemma neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
- by (simp add: sup_eq_neg_inf)
+by (simp add: sup_eq_neg_inf)
lemma add_eq_inf_sup: "a + b = sup a b + inf a b"
proof -
have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
hence "0 = (-a + sup a b) + (inf a b + (-b))"
- apply (simp add: add_sup_distrib_left add_inf_distrib_right)
- by (simp add: diff_minus add_commute)
- thus ?thesis
- apply (simp add: compare_rls)
- apply (subst add_left_cancel [symmetric, of "plus a b" "plus (sup a b) (inf a b)" "uminus a"])
- apply (simp only: add_assoc, simp add: add_assoc[symmetric])
- done
+ by (simp add: add_sup_distrib_left add_inf_distrib_right)
+ (simp add: algebra_simps)
+ thus ?thesis by (simp add: algebra_simps)
qed
subsection {* Positive Part, Negative Part, Absolute Value *}
@@ -1044,13 +1006,13 @@
qed
lemma prts: "a = pprt a + nprt a"
- by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
+by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
lemma zero_le_pprt[simp]: "0 \<le> pprt a"
- by (simp add: pprt_def)
+by (simp add: pprt_def)
lemma nprt_le_zero[simp]: "nprt a \<le> 0"
- by (simp add: nprt_def)
+by (simp add: nprt_def)
lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0" (is "?l = ?r")
proof -
@@ -1071,16 +1033,16 @@
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
lemma pprt_eq_id [simp, noatp]: "0 \<le> x \<Longrightarrow> pprt x = x"
- by (simp add: pprt_def le_iff_sup sup_ACI)
+by (simp add: pprt_def le_iff_sup sup_ACI)
lemma nprt_eq_id [simp, noatp]: "x \<le> 0 \<Longrightarrow> nprt x = x"
- by (simp add: nprt_def le_iff_inf inf_ACI)
+by (simp add: nprt_def le_iff_inf inf_ACI)
lemma pprt_eq_0 [simp, noatp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"
- by (simp add: pprt_def le_iff_sup sup_ACI)
+by (simp add: pprt_def le_iff_sup sup_ACI)
lemma nprt_eq_0 [simp, noatp]: "0 \<le> x \<Longrightarrow> nprt x = 0"
- by (simp add: nprt_def le_iff_inf inf_ACI)
+by (simp add: nprt_def le_iff_inf inf_ACI)
lemma sup_0_imp_0: "sup a (- a) = 0 \<Longrightarrow> a = 0"
proof -
@@ -1105,10 +1067,10 @@
done
lemma inf_0_eq_0 [simp, noatp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"
- by (rule, erule inf_0_imp_0) simp
+by (rule, erule inf_0_imp_0) simp
lemma sup_0_eq_0 [simp, noatp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"
- by (rule, erule sup_0_imp_0) simp
+by (rule, erule sup_0_imp_0) simp
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
"0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
@@ -1190,22 +1152,22 @@
qed
lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"
- by (simp add: le_iff_inf nprt_def inf_commute)
+by (simp add: le_iff_inf nprt_def inf_commute)
lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"
- by (simp add: le_iff_sup pprt_def sup_commute)
+by (simp add: le_iff_sup pprt_def sup_commute)
lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"
- by (simp add: le_iff_sup pprt_def sup_commute)
+by (simp add: le_iff_sup pprt_def sup_commute)
lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"
- by (simp add: le_iff_inf nprt_def inf_commute)
+by (simp add: le_iff_inf nprt_def inf_commute)
lemma pprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"
- by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
+by (simp add: le_iff_sup pprt_def sup_ACI sup_assoc [symmetric, of a])
lemma nprt_mono [simp, noatp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"
- by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
+by (simp add: le_iff_inf nprt_def inf_ACI inf_assoc [symmetric, of a])
end
@@ -1246,12 +1208,10 @@
show "0 \<le> \<bar>a\<bar>" by simp
next
fix a
- show "a \<le> \<bar>a\<bar>"
- by (auto simp add: abs_lattice)
+ show "a \<le> \<bar>a\<bar>" by (auto simp add: abs_lattice)
next
fix a
- show "\<bar>-a\<bar> = \<bar>a\<bar>"
- by (simp add: abs_lattice sup_commute)
+ show "\<bar>-a\<bar> = \<bar>a\<bar>" by (simp add: abs_lattice sup_commute)
next
fix a b
show "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" by (erule abs_leI)
@@ -1266,8 +1226,7 @@
have c:"?n <= sup ?m ?n" by (simp)
from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
have e:"-a-b = -(a+b)" by (simp add: diff_minus)
- from a d e have "abs(a+b) <= sup ?m ?n"
- by (drule_tac abs_leI, auto)
+ from a d e have "abs(a+b) <= sup ?m ?n" by (drule_tac abs_leI, auto)
with g[symmetric] show ?thesis by simp
qed
qed auto
@@ -1287,7 +1246,7 @@
lemma abs_if_lattice:
fixes a :: "'a\<Colon>{lordered_ab_group_add_abs, linorder}"
shows "\<bar>a\<bar> = (if a < 0 then - a else a)"
- by auto
+by auto
text {* Needed for abelian cancellation simprocs: *}
@@ -1333,7 +1292,7 @@
"a + b <= (c::'a::lordered_ab_group_add_abs) \<Longrightarrow> a <= c + abs b"
proof -
assume "a+b <= c"
- hence 2: "a <= c+(-b)" by (simp add: group_simps)
+ hence 2: "a <= c+(-b)" by (simp add: algebra_simps)
have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
show ?thesis by (rule le_add_right_mono[OF 2 3])
qed
--- a/src/HOL/PReal.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/PReal.thy Wed Jan 28 16:29:16 2009 +0100
@@ -718,7 +718,7 @@
case (Suc k)
from this obtain b where "b \<in> A" "b + of_nat k * u \<in> A" ..
hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems)
- thus ?case by (force simp add: left_distrib add_ac prems)
+ thus ?case by (force simp add: algebra_simps prems)
qed
next
case (neg n)
@@ -815,7 +815,7 @@
proof -
have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
also with ypos have "... = (r/y) * (y + ?d)"
- by (simp only: right_distrib divide_inverse mult_ac, simp)
+ by (simp only: algebra_simps divide_inverse, simp)
also have "... = r*x" using ypos
by (simp add: times_divide_eq_left)
finally show "r + ?d < r*x" .
--- a/src/HOL/Polynomial.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Polynomial.thy Wed Jan 28 16:29:16 2009 +0100
@@ -442,11 +442,11 @@
lemma smult_add_right:
"smult a (p + q) = smult a p + smult a q"
- by (rule poly_ext, simp add: ring_simps)
+ by (rule poly_ext, simp add: algebra_simps)
lemma smult_add_left:
"smult (a + b) p = smult a p + smult b p"
- by (rule poly_ext, simp add: ring_simps)
+ by (rule poly_ext, simp add: algebra_simps)
lemma smult_minus_right [simp]:
"smult (a::'a::comm_ring) (- p) = - smult a p"
@@ -458,11 +458,11 @@
lemma smult_diff_right:
"smult (a::'a::comm_ring) (p - q) = smult a p - smult a q"
- by (rule poly_ext, simp add: ring_simps)
+ by (rule poly_ext, simp add: algebra_simps)
lemma smult_diff_left:
"smult (a - b::'a::comm_ring) p = smult a p - smult b p"
- by (rule poly_ext, simp add: ring_simps)
+ by (rule poly_ext, simp add: algebra_simps)
lemmas smult_distribs =
smult_add_left smult_add_right
@@ -517,7 +517,7 @@
lemma mult_pCons_right [simp]:
"p * pCons a q = smult a p + pCons 0 (p * q)"
- by (induct p, simp add: mult_poly_0_left, simp add: ring_simps)
+ by (induct p, simp add: mult_poly_0_left, simp add: algebra_simps)
lemmas mult_poly_0 = mult_poly_0_left mult_poly_0_right
@@ -531,7 +531,7 @@
fixes p q r :: "'a poly"
shows "(p + q) * r = p * r + q * r"
by (induct r, simp add: mult_poly_0,
- simp add: smult_distribs group_simps)
+ simp add: smult_distribs algebra_simps)
instance proof
fix p q r :: "'a poly"
@@ -758,7 +758,7 @@
from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
unfolding pdivmod_rel_def by simp_all
from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
by (auto intro: degree_diff_less)
@@ -894,7 +894,7 @@
lemma poly_add [simp]: "poly (p + q) x = poly p x + poly q x"
apply (induct p arbitrary: q, simp)
- apply (case_tac q, simp, simp add: ring_simps)
+ apply (case_tac q, simp, simp add: algebra_simps)
done
lemma poly_minus [simp]:
@@ -911,10 +911,10 @@
by (cases "finite A", induct set: finite, simp_all)
lemma poly_smult [simp]: "poly (smult a p) x = a * poly p x"
- by (induct p, simp, simp add: ring_simps)
+ by (induct p, simp, simp add: algebra_simps)
lemma poly_mult [simp]: "poly (p * q) x = poly p x * poly q x"
- by (induct p, simp_all, simp add: ring_simps)
+ by (induct p, simp_all, simp add: algebra_simps)
lemma poly_power [simp]:
fixes p :: "'a::{comm_semiring_1,recpower} poly"
@@ -983,7 +983,7 @@
fixes c :: "'a::comm_ring_1"
shows "[:-c, 1:] * synthetic_div p c + [:poly p c:] = p"
using synthetic_div_correct [of p c]
- by (simp add: group_simps)
+ by (simp add: algebra_simps)
lemma poly_eq_0_iff_dvd:
fixes c :: "'a::idom"
--- a/src/HOL/Presburger.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Presburger.thy Wed Jan 28 16:29:16 2009 +0100
@@ -59,7 +59,7 @@
"(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (d dvd x + t) = (d dvd (x - k*D) + t)"
"(d::'a::{comm_ring,Ring_and_Field.dvd}) dvd D \<Longrightarrow> \<forall>x k. (\<not>d dvd x + t) = (\<not>d dvd (x - k*D) + t)"
"\<forall>x k. F = F"
-apply (auto elim!: dvdE simp add: ring_simps)
+apply (auto elim!: dvdE simp add: algebra_simps)
unfolding mult_assoc [symmetric] left_distrib [symmetric] left_diff_distrib [symmetric]
unfolding dvd_def mult_commute [of d]
by auto
@@ -101,7 +101,7 @@
{fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x > t" and ng: "\<not> (x - D) > t"
hence "x -t \<le> D" and "1 \<le> x - t" by simp+
hence "\<exists>j \<in> {1 .. D}. x - t = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: ring_simps)
+ hence "\<exists>j \<in> {1 .. D}. x = t + j" by (simp add: algebra_simps)
with nob tB have "False" by simp}
thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x > t) \<longrightarrow> (x - D > t)" by blast
next
@@ -109,7 +109,7 @@
{fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j" and g: "x \<ge> t" and ng: "\<not> (x - D) \<ge> t"
hence "x - (t - 1) \<le> D" and "1 \<le> x - (t - 1)" by simp+
hence "\<exists>j \<in> {1 .. D}. x - (t - 1) = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: ring_simps)
+ hence "\<exists>j \<in> {1 .. D}. x = (t - 1) + j" by (simp add: algebra_simps)
with nob tB have "False" by simp}
thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (x \<ge> t) \<longrightarrow> (x - D \<ge> t)" by blast
next
@@ -119,7 +119,7 @@
next
assume d: "d dvd D"
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not> d dvd (x - D) + t"
- by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: ring_simps)}
+ by (clarsimp simp add: dvd_def,erule_tac x= "ka + k" in allE,simp add: algebra_simps)}
thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>B. x \<noteq> b + j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x - D) + t)" by auto
qed blast
@@ -158,26 +158,26 @@
{fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x < t" and ng: "\<not> (x + D) < t"
hence "t - x \<le> D" and "1 \<le> t - x" by simp+
hence "\<exists>j \<in> {1 .. D}. t - x = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: ring_simps)
+ hence "\<exists>j \<in> {1 .. D}. x = t - j" by (auto simp add: algebra_simps)
with nob tA have "False" by simp}
thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x < t) \<longrightarrow> (x + D < t)" by blast
next
assume dp: "D > 0" and tA:"t + 1\<in> A"
{fix x assume nob: "\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j" and g: "x \<le> t" and ng: "\<not> (x + D) \<le> t"
- hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: ring_simps)
+ hence "(t + 1) - x \<le> D" and "1 \<le> (t + 1) - x" by (simp_all add: algebra_simps)
hence "\<exists>j \<in> {1 .. D}. (t + 1) - x = j" by auto
- hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: ring_simps)
+ hence "\<exists>j \<in> {1 .. D}. x = (t + 1) - j" by (auto simp add: algebra_simps)
with nob tA have "False" by simp}
thus "\<forall>x.(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (x \<le> t) \<longrightarrow> (x + D \<le> t)" by blast
next
assume d: "d dvd D"
{fix x assume H: "d dvd x + t" with d have "d dvd (x + D) + t"
- by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: ring_simps)}
+ by (clarsimp simp add: dvd_def,rule_tac x= "ka + k" in exI,simp add: algebra_simps)}
thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (d dvd x+t) \<longrightarrow> (d dvd (x + D) + t)" by simp
next
assume d: "d dvd D"
{fix x assume H: "\<not>(d dvd x + t)" with d have "\<not>d dvd (x + D) + t"
- by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: ring_simps)}
+ by (clarsimp simp add: dvd_def,erule_tac x= "ka - k" in allE,simp add: algebra_simps)}
thus "\<forall>(x::int).(\<forall>j\<in>{1 .. D}. \<forall>b\<in>A. x \<noteq> b - j)\<longrightarrow> (\<not>d dvd x+t) \<longrightarrow> (\<not>d dvd (x + D) + t)" by auto
qed blast
@@ -304,7 +304,7 @@
from ePeqP1 obtain z where P1eqP: "\<forall>x>z. P x = P' x" ..
let ?w' = "x + (abs(x-z)+1) * d"
let ?w = "x - (-(abs(x-z) + 1))*d"
- have ww'[simp]: "?w = ?w'" by (simp add: ring_simps)
+ have ww'[simp]: "?w = ?w'" by (simp add: algebra_simps)
from dpos have w: "?w > z" by(simp only: ww' incr_lemma)
hence "P' x = P' ?w" using P1eqP1 by blast
also have "\<dots> = P(?w)" using w P1eqP by blast
--- a/src/HOL/RComplete.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/RComplete.thy Wed Jan 28 16:29:16 2009 +0100
@@ -376,7 +376,7 @@
hence "real (Suc n) * (inverse (real (Suc n)) * x) < real (Suc n) * 1"
by (rule mult_strict_left_mono) simp
hence "x < real (Suc n)"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
thus "\<exists>(n::nat). x < real n" ..
qed
@@ -391,9 +391,9 @@
hence "y * inverse x * x < real n * x"
using x_greater_zero by (simp add: mult_strict_right_mono)
hence "x * inverse x * y < x * real n"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
hence "y < real (n::nat) * x"
- using x_not_zero by (simp add: ring_simps)
+ using x_not_zero by (simp add: algebra_simps)
thus "\<exists>(n::nat). y < real n * x" ..
qed
@@ -1084,9 +1084,7 @@
done
lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"
- apply (simp add: compare_rls)
- apply (rule real_natfloor_add_one_gt)
-done
+using real_natfloor_add_one_gt by (simp add: algebra_simps)
lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"
apply (subgoal_tac "z < real(natfloor z) + 1")
@@ -1279,7 +1277,7 @@
by simp
also have "... = real((natfloor x) div y) + real((natfloor x) mod y) /
real y + (x - real(natfloor x)) / real y"
- by (auto simp add: ring_simps add_divide_distrib
+ by (auto simp add: algebra_simps add_divide_distrib
diff_divide_distrib prems)
finally have "natfloor (x / real y) = natfloor(...)" by simp
also have "... = natfloor(real((natfloor x) mod y) /
@@ -1293,7 +1291,7 @@
apply simp
apply (simp add: prems)
apply (rule divide_nonneg_pos)
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (rule real_natfloor_le)
apply (insert prems, auto)
done
@@ -1306,7 +1304,7 @@
apply force
apply (force simp add: prems)
apply (rule divide_nonneg_pos)
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (rule real_natfloor_le)
apply (auto simp add: prems)
apply (insert prems, arith)
@@ -1314,8 +1312,8 @@
apply (subgoal_tac "real y = real y - 1 + 1")
apply (erule ssubst)
apply (rule add_le_less_mono)
- apply (simp add: compare_rls)
- apply (subgoal_tac "real(natfloor x mod y) + 1 =
+ apply (simp add: algebra_simps)
+ apply (subgoal_tac "1 + real(natfloor x mod y) =
real(natfloor x mod y + 1)")
apply (erule ssubst)
apply (subst real_of_nat_le_iff)
@@ -1323,9 +1321,8 @@
apply arith
apply (rule mod_less_divisor)
apply auto
- apply (simp add: compare_rls)
- apply (subst add_commute)
- apply (rule real_natfloor_add_one_gt)
+ using real_natfloor_add_one_gt
+ apply (simp add: algebra_simps)
done
finally show ?thesis by simp
qed
--- a/src/HOL/Rational.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Rational.thy Wed Jan 28 16:29:16 2009 +0100
@@ -172,7 +172,7 @@
by (cases q) (simp add: One_rat_def eq_rat)
next
fix q r s :: rat show "(q + r) + s = q + (r + s)"
- by (cases q, cases r, cases s) (simp add: eq_rat ring_simps)
+ by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
next
fix q r :: rat show "q + r = r + q"
by (cases q, cases r) (simp add: eq_rat)
@@ -187,7 +187,7 @@
by (cases q, cases r) (simp add: eq_rat)
next
fix q r s :: rat show "(q + r) * s = q * s + r * s"
- by (cases q, cases r, cases s) (simp add: eq_rat ring_simps)
+ by (cases q, cases r, cases s) (simp add: eq_rat algebra_simps)
next
show "(0::rat) \<noteq> 1" by (simp add: Zero_rat_def One_rat_def eq_rat)
next
--- a/src/HOL/RealDef.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/RealDef.thy Wed Jan 28 16:29:16 2009 +0100
@@ -202,18 +202,17 @@
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
apply (cases z1, cases z2, cases z3)
-apply (simp add: real_mult right_distrib add_ac mult_ac)
+apply (simp add: real_mult algebra_simps)
done
lemma real_mult_1: "(1::real) * z = z"
apply (cases z)
-apply (simp add: real_mult real_one_def right_distrib
- mult_1_right mult_ac add_ac)
+apply (simp add: real_mult real_one_def algebra_simps)
done
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
apply (cases z1, cases z2, cases w)
-apply (simp add: real_add real_mult right_distrib add_ac mult_ac)
+apply (simp add: real_add real_mult algebra_simps)
done
text{*one and zero are distinct*}
@@ -253,7 +252,7 @@
apply (rule_tac [2]
x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})"
in exI)
-apply (auto simp add: real_mult preal_mult_inverse_right ring_simps)
+apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps)
done
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
@@ -373,7 +372,7 @@
assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
proof -
have "z + x - (z + y) = (z + -z) + (x - y)"
- by (simp add: diff_minus add_ac)
+ by (simp add: algebra_simps)
with le show ?thesis
by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
qed
@@ -391,8 +390,7 @@
real_zero_def real_le real_mult)
--{*Reduce to the (simpler) @{text "\<le>"} relation *}
apply (auto dest!: less_add_left_Ex
- simp add: add_ac mult_ac
- right_distrib preal_self_less_add_left)
+ simp add: algebra_simps preal_self_less_add_left)
done
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
@@ -437,11 +435,11 @@
lemma real_of_preal_add:
"real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
-by (simp add: real_of_preal_def real_add left_distrib add_ac)
+by (simp add: real_of_preal_def real_add algebra_simps)
lemma real_of_preal_mult:
"real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
-by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac)
+by (simp add: real_of_preal_def real_mult algebra_simps)
text{*Gleason prop 9-4.4 p 127*}
@@ -650,7 +648,7 @@
then have "real x / real d = ... / real d"
by simp
then show ?thesis
- by (auto simp add: add_divide_distrib ring_simps prems)
+ by (auto simp add: add_divide_distrib algebra_simps prems)
qed
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
@@ -665,13 +663,13 @@
apply (case_tac "x = 0")
apply simp
apply (case_tac "0 < x")
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (subst real_of_int_div_aux)
apply simp
apply simp
apply (subst zero_le_divide_iff)
apply auto
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (subst real_of_int_div_aux)
apply simp
apply simp
@@ -683,7 +681,7 @@
"real (n::int) / real (x) - real (n div x) <= 1"
apply(case_tac "x = 0")
apply simp
- apply (simp add: compare_rls)
+ apply (simp add: algebra_simps)
apply (subst real_of_int_div_aux)
apply assumption
apply simp
@@ -795,7 +793,7 @@
then have "real x / real d = \<dots> / real d"
by simp
then show ?thesis
- by (auto simp add: add_divide_distrib ring_simps prems)
+ by (auto simp add: add_divide_distrib algebra_simps prems)
qed
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
@@ -810,7 +808,7 @@
"0 <= real (n::nat) / real (x) - real (n div x)"
apply(case_tac "x = 0")
apply (simp)
-apply (simp add: compare_rls)
+apply (simp add: algebra_simps)
apply (subst real_of_nat_div_aux)
apply simp
apply simp
@@ -822,14 +820,14 @@
"real (n::nat) / real (x) - real (n div x) <= 1"
apply(case_tac "x = 0")
apply (simp)
-apply (simp add: compare_rls)
+apply (simp add: algebra_simps)
apply (subst real_of_nat_div_aux)
apply simp
apply simp
done
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
- by (insert real_of_nat_div2 [of n x], simp)
+by (insert real_of_nat_div2 [of n x], simp)
lemma real_of_int_real_of_nat: "real (int n) = real n"
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
--- a/src/HOL/RealPow.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/RealPow.thy Wed Jan 28 16:29:16 2009 +0100
@@ -58,7 +58,7 @@
lemma realpow_two_diff:
"(x::real)^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
apply (unfold real_diff_def)
-apply (simp add: ring_simps)
+apply (simp add: algebra_simps)
done
lemma realpow_two_disj:
--- a/src/HOL/Ring_and_Field.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Ring_and_Field.thy Wed Jan 28 16:29:16 2009 +0100
@@ -24,14 +24,14 @@
*}
class semiring = ab_semigroup_add + semigroup_mult +
- assumes left_distrib: "(a + b) * c = a * c + b * c"
- assumes right_distrib: "a * (b + c) = a * b + a * c"
+ assumes left_distrib[algebra_simps]: "(a + b) * c = a * c + b * c"
+ assumes right_distrib[algebra_simps]: "a * (b + c) = a * b + a * c"
begin
text{*For the @{text combine_numerals} simproc*}
lemma combine_common_factor:
"a * e + (b * e + c) = (a + b) * e + c"
- by (simp add: left_distrib add_ac)
+by (simp add: left_distrib add_ac)
end
@@ -47,16 +47,12 @@
subclass semiring_0
proof
fix a :: 'a
- have "0 * a + 0 * a = 0 * a + 0"
- by (simp add: left_distrib [symmetric])
- thus "0 * a = 0"
- by (simp only: add_left_cancel)
+ have "0 * a + 0 * a = 0 * a + 0" by (simp add: left_distrib [symmetric])
+ thus "0 * a = 0" by (simp only: add_left_cancel)
next
fix a :: 'a
- have "a * 0 + a * 0 = a * 0 + 0"
- by (simp add: right_distrib [symmetric])
- thus "a * 0 = 0"
- by (simp only: add_left_cancel)
+ have "a * 0 + a * 0 = a * 0 + 0" by (simp add: right_distrib [symmetric])
+ thus "a * 0 = 0" by (simp only: add_left_cancel)
qed
end
@@ -98,7 +94,7 @@
begin
lemma one_neq_zero [simp]: "1 \<noteq> 0"
- by (rule not_sym) (rule zero_neq_one)
+by (rule not_sym) (rule zero_neq_one)
end
@@ -142,7 +138,7 @@
qed
lemma dvd_0_left_iff [noatp, simp]: "0 dvd a \<longleftrightarrow> a = 0"
- by (auto intro: dvd_refl elim!: dvdE)
+by (auto intro: dvd_refl elim!: dvdE)
lemma dvd_0_right [iff]: "a dvd 0"
proof
@@ -150,10 +146,10 @@
qed
lemma one_dvd [simp]: "1 dvd a"
- by (auto intro!: dvdI)
+by (auto intro!: dvdI)
lemma dvd_mult: "a dvd c \<Longrightarrow> a dvd (b * c)"
- by (auto intro!: mult_left_commute dvdI elim!: dvdE)
+by (auto intro!: mult_left_commute dvdI elim!: dvdE)
lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)"
apply (subst mult_commute)
@@ -161,10 +157,10 @@
done
lemma dvd_triv_right [simp]: "a dvd b * a"
- by (rule dvd_mult) (rule dvd_refl)
+by (rule dvd_mult) (rule dvd_refl)
lemma dvd_triv_left [simp]: "a dvd a * b"
- by (rule dvd_mult2) (rule dvd_refl)
+by (rule dvd_mult2) (rule dvd_refl)
lemma mult_dvd_mono:
assumes ab: "a dvd b"
@@ -178,13 +174,13 @@
qed
lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
- by (simp add: dvd_def mult_assoc, blast)
+by (simp add: dvd_def mult_assoc, blast)
lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
unfolding mult_ac [of a] by (rule dvd_mult_left)
lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
- by simp
+by simp
lemma dvd_add:
assumes ab: "a dvd b"
@@ -230,43 +226,40 @@
text {* Distribution rules *}
lemma minus_mult_left: "- (a * b) = - a * b"
- by (rule equals_zero_I) (simp add: left_distrib [symmetric])
+by (rule equals_zero_I) (simp add: left_distrib [symmetric])
lemma minus_mult_right: "- (a * b) = a * - b"
- by (rule equals_zero_I) (simp add: right_distrib [symmetric])
+by (rule equals_zero_I) (simp add: right_distrib [symmetric])
text{*Extract signs from products*}
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
lemma minus_mult_minus [simp]: "- a * - b = a * b"
- by simp
+by simp
lemma minus_mult_commute: "- a * b = a * - b"
- by simp
-
-lemma right_diff_distrib: "a * (b - c) = a * b - a * c"
- by (simp add: right_distrib diff_minus)
-
-lemma left_diff_distrib: "(a - b) * c = a * c - b * c"
- by (simp add: left_distrib diff_minus)
+by simp
+
+lemma right_diff_distrib[algebra_simps]: "a * (b - c) = a * b - a * c"
+by (simp add: right_distrib diff_minus)
+
+lemma left_diff_distrib[algebra_simps]: "(a - b) * c = a * c - b * c"
+by (simp add: left_distrib diff_minus)
lemmas ring_distribs =
right_distrib left_distrib left_diff_distrib right_diff_distrib
-lemmas ring_simps =
- add_ac
- add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
- diff_eq_eq eq_diff_eq diff_minus [symmetric] uminus_add_conv_diff
- ring_distribs
+text{*Legacy - use @{text algebra_simps} *}
+lemmas ring_simps = algebra_simps
lemma eq_add_iff1:
"a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
- by (simp add: ring_simps)
+by (simp add: algebra_simps)
lemma eq_add_iff2:
"a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
- by (simp add: ring_simps)
+by (simp add: algebra_simps)
end
@@ -320,7 +313,7 @@
qed
lemma dvd_diff: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
- by (simp add: diff_minus dvd_add dvd_minus_iff)
+by (simp add: diff_minus dvd_add dvd_minus_iff)
end
@@ -341,18 +334,16 @@
"a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
proof -
have "(a * c = b * c) = ((a - b) * c = 0)"
- by (simp add: ring_distribs right_minus_eq)
- thus ?thesis
- by (simp add: disj_commute right_minus_eq)
+ by (simp add: algebra_simps right_minus_eq)
+ thus ?thesis by (simp add: disj_commute right_minus_eq)
qed
lemma mult_cancel_left [simp, noatp]:
"c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
proof -
have "(c * a = c * b) = (c * (a - b) = 0)"
- by (simp add: ring_distribs right_minus_eq)
- thus ?thesis
- by (simp add: right_minus_eq)
+ by (simp add: algebra_simps right_minus_eq)
+ thus ?thesis by (simp add: right_minus_eq)
qed
end
@@ -362,19 +353,19 @@
lemma mult_cancel_right1 [simp]:
"c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
- by (insert mult_cancel_right [of 1 c b], force)
+by (insert mult_cancel_right [of 1 c b], force)
lemma mult_cancel_right2 [simp]:
"a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
- by (insert mult_cancel_right [of a c 1], simp)
+by (insert mult_cancel_right [of a c 1], simp)
lemma mult_cancel_left1 [simp]:
"c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
- by (insert mult_cancel_left [of c 1 b], force)
+by (insert mult_cancel_left [of c 1 b], force)
lemma mult_cancel_left2 [simp]:
"c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
- by (insert mult_cancel_left [of c a 1], simp)
+by (insert mult_cancel_left [of c a 1], simp)
end
@@ -397,14 +388,11 @@
show "a * b \<noteq> 0"
proof
assume ab: "a * b = 0"
- hence "0 = inverse a * (a * b) * inverse b"
- by simp
+ hence "0 = inverse a * (a * b) * inverse b" by simp
also have "\<dots> = (inverse a * a) * (b * inverse b)"
by (simp only: mult_assoc)
- also have "\<dots> = 1"
- using a b by simp
- finally show False
- by simp
+ also have "\<dots> = 1" using a b by simp
+ finally show False by simp
qed
qed
@@ -437,45 +425,41 @@
lemma nonzero_inverse_minus_eq:
"a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
- by (rule inverse_unique) simp
+by (rule inverse_unique) simp
lemma nonzero_inverse_inverse_eq:
"a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
- by (rule inverse_unique) simp
+by (rule inverse_unique) simp
lemma nonzero_inverse_eq_imp_eq:
assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
shows "a = b"
proof -
from `inverse a = inverse b`
- have "inverse (inverse a) = inverse (inverse b)"
- by (rule arg_cong)
+ have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
with `a \<noteq> 0` and `b \<noteq> 0` show "a = b"
by (simp add: nonzero_inverse_inverse_eq)
qed
lemma inverse_1 [simp]: "inverse 1 = 1"
- by (rule inverse_unique) simp
+by (rule inverse_unique) simp
lemma nonzero_inverse_mult_distrib:
assumes "a \<noteq> 0" and "b \<noteq> 0"
shows "inverse (a * b) = inverse b * inverse a"
proof -
- have "a * (b * inverse b) * inverse a = 1"
- using assms by simp
- hence "a * b * (inverse b * inverse a) = 1"
- by (simp only: mult_assoc)
- thus ?thesis
- by (rule inverse_unique)
+ have "a * (b * inverse b) * inverse a = 1" using assms by simp
+ hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult_assoc)
+ thus ?thesis by (rule inverse_unique)
qed
lemma division_ring_inverse_add:
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
- by (simp add: ring_simps mult_assoc)
+by (simp add: algebra_simps)
lemma division_ring_inverse_diff:
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
- by (simp add: ring_simps mult_assoc)
+by (simp add: algebra_simps)
end
@@ -508,19 +492,19 @@
qed
lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
- by (simp add: divide_inverse)
+by (simp add: divide_inverse)
lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
- by (simp add: divide_inverse)
+by (simp add: divide_inverse)
lemma divide_zero_left [simp]: "0 / a = 0"
- by (simp add: divide_inverse)
+by (simp add: divide_inverse)
lemma inverse_eq_divide: "inverse a = 1 / a"
- by (simp add: divide_inverse)
+by (simp add: divide_inverse)
lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
- by (simp add: divide_inverse ring_distribs)
+by (simp add: divide_inverse algebra_simps)
end
@@ -529,11 +513,11 @@
lemma divide_zero [simp]:
"a / 0 = (0::'a::{field,division_by_zero})"
- by (simp add: divide_inverse)
+by (simp add: divide_inverse)
lemma divide_self_if [simp]:
"a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
- by simp
+by simp
class mult_mono = times + zero + ord +
assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
@@ -566,16 +550,16 @@
subclass pordered_semiring ..
lemma mult_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
- by (drule mult_left_mono [of zero b], auto)
+by (drule mult_left_mono [of zero b], auto)
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
- by (drule mult_left_mono [of b zero], auto)
+by (drule mult_left_mono [of b zero], auto)
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
- by (drule mult_right_mono [of b zero], auto)
+by (drule mult_right_mono [of b zero], auto)
lemma split_mult_neg_le: "(0 \<le> a & b \<le> 0) | (a \<le> 0 & 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
- by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
+by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
end
@@ -588,11 +572,11 @@
lemma mult_left_less_imp_less:
"c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
- by (force simp add: mult_left_mono not_le [symmetric])
+by (force simp add: mult_left_mono not_le [symmetric])
lemma mult_right_less_imp_less:
"a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
- by (force simp add: mult_right_mono not_le [symmetric])
+by (force simp add: mult_right_mono not_le [symmetric])
end
@@ -617,23 +601,23 @@
lemma mult_left_le_imp_le:
"c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
- by (force simp add: mult_strict_left_mono _not_less [symmetric])
+by (force simp add: mult_strict_left_mono _not_less [symmetric])
lemma mult_right_le_imp_le:
"a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
- by (force simp add: mult_strict_right_mono not_less [symmetric])
+by (force simp add: mult_strict_right_mono not_less [symmetric])
lemma mult_pos_pos:
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
- by (drule mult_strict_left_mono [of zero b], auto)
+by (drule mult_strict_left_mono [of zero b], auto)
lemma mult_pos_neg:
"0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
- by (drule mult_strict_left_mono [of b zero], auto)
+by (drule mult_strict_left_mono [of b zero], auto)
lemma mult_pos_neg2:
"0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
- by (drule mult_strict_right_mono [of b zero], auto)
+by (drule mult_strict_right_mono [of b zero], auto)
lemma zero_less_mult_pos:
"0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
@@ -666,7 +650,7 @@
lemma mult_strict_mono':
assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
shows "a * c < b * d"
- by (rule mult_strict_mono) (insert assms, auto)
+by (rule mult_strict_mono) (insert assms, auto)
lemma mult_less_le_imp_less:
assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
@@ -697,8 +681,7 @@
assume "\<not> a < b"
hence "b \<le> a" by (simp add: linorder_not_less)
hence "c * b \<le> c * a" using nonneg by (rule mult_left_mono)
- with this and less show False
- by (simp add: not_less [symmetric])
+ with this and less show False by (simp add: not_less [symmetric])
qed
lemma mult_less_imp_less_right:
@@ -708,8 +691,7 @@
assume "\<not> a < b"
hence "b \<le> a" by (simp add: linorder_not_less)
hence "b * c \<le> a * c" using nonneg by (rule mult_right_mono)
- with this and less show False
- by (simp add: not_less [symmetric])
+ with this and less show False by (simp add: not_less [symmetric])
qed
end
@@ -768,23 +750,24 @@
subclass pordered_ab_group_add ..
-lemmas ring_simps = ring_simps group_simps
+text{*Legacy - use @{text algebra_simps} *}
+lemmas ring_simps = algebra_simps
lemma less_add_iff1:
"a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
- by (simp add: ring_simps)
+by (simp add: algebra_simps)
lemma less_add_iff2:
"a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
- by (simp add: ring_simps)
+by (simp add: algebra_simps)
lemma le_add_iff1:
"a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
- by (simp add: ring_simps)
+by (simp add: algebra_simps)
lemma le_add_iff2:
"a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
- by (simp add: ring_simps)
+by (simp add: algebra_simps)
lemma mult_left_mono_neg:
"b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
@@ -800,11 +783,11 @@
lemma mult_nonpos_nonpos:
"a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
- by (drule mult_right_mono_neg [of a zero b]) auto
+by (drule mult_right_mono_neg [of a zero b]) auto
lemma split_mult_pos_le:
"(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
- by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
+by (auto simp add: mult_nonneg_nonneg mult_nonpos_nonpos)
end
@@ -827,7 +810,7 @@
proof
fix a b
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
- by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
+by (auto simp add: abs_if not_less neg_less_eq_nonneg less_eq_neg_nonpos)
(auto simp del: minus_add_distrib simp add: minus_add_distrib [symmetric]
neg_less_eq_nonneg less_eq_neg_nonpos, auto intro: add_nonneg_nonneg,
auto intro!: less_imp_le add_neg_neg)
@@ -858,7 +841,7 @@
lemma mult_neg_neg:
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
- by (drule mult_strict_right_mono_neg, auto)
+by (drule mult_strict_right_mono_neg, auto)
subclass ring_no_zero_divisors
proof
@@ -903,7 +886,7 @@
lemma zero_le_mult_iff:
"0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
- by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
+by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
lemma mult_less_0_iff:
"a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
@@ -918,10 +901,10 @@
done
lemma zero_le_square [simp]: "0 \<le> a * a"
- by (simp add: zero_le_mult_iff linear)
+by (simp add: zero_le_mult_iff linear)
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
- by (simp add: not_less)
+by (simp add: not_less)
text{*Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
also with the relations @{text "\<le>"} and equality.*}
@@ -968,19 +951,16 @@
lemma mult_le_cancel_right:
"a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
- by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
+by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
lemma mult_le_cancel_left:
"c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
- by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
+by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
end
-text{*This list of rewrites simplifies ring terms by multiplying
-everything out and bringing sums and products into a canonical form
-(by ordered rewriting). As a result it decides ring equalities but
-also helps with inequalities. *}
-lemmas ring_simps = group_simps ring_distribs
+text{*Legacy - use @{text algebra_simps} *}
+lemmas ring_simps = algebra_simps
class pordered_comm_ring = comm_ring + pordered_comm_semiring
@@ -1001,13 +981,13 @@
using add_strict_mono [of zero a b c] by simp
lemma zero_le_one [simp]: "0 \<le> 1"
- by (rule zero_less_one [THEN less_imp_le])
+by (rule zero_less_one [THEN less_imp_le])
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
- by (simp add: not_le)
+by (simp add: not_le)
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
- by (simp add: not_less)
+by (simp add: not_less)
lemma less_1_mult:
assumes "1 < m" and "1 < n"
@@ -1041,35 +1021,35 @@
lemma mult_le_cancel_right1:
"c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
- by (insert mult_le_cancel_right [of 1 c b], simp)
+by (insert mult_le_cancel_right [of 1 c b], simp)
lemma mult_le_cancel_right2:
"a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
- by (insert mult_le_cancel_right [of a c 1], simp)
+by (insert mult_le_cancel_right [of a c 1], simp)
lemma mult_le_cancel_left1:
"c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
- by (insert mult_le_cancel_left [of c 1 b], simp)
+by (insert mult_le_cancel_left [of c 1 b], simp)
lemma mult_le_cancel_left2:
"c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
- by (insert mult_le_cancel_left [of c a 1], simp)
+by (insert mult_le_cancel_left [of c a 1], simp)
lemma mult_less_cancel_right1:
"c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
- by (insert mult_less_cancel_right [of 1 c b], simp)
+by (insert mult_less_cancel_right [of 1 c b], simp)
lemma mult_less_cancel_right2:
"a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
- by (insert mult_less_cancel_right [of a c 1], simp)
+by (insert mult_less_cancel_right [of a c 1], simp)
lemma mult_less_cancel_left1:
"c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
- by (insert mult_less_cancel_left [of c 1 b], simp)
+by (insert mult_less_cancel_left [of c 1 b], simp)
lemma mult_less_cancel_left2:
"c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
- by (insert mult_less_cancel_left [of c a 1], simp)
+by (insert mult_less_cancel_left [of c a 1], simp)
lemma sgn_sgn [simp]:
"sgn (sgn a) = sgn a"
@@ -1089,7 +1069,7 @@
lemma sgn_times:
"sgn (a * b) = sgn a * sgn b"
- by (auto simp add: sgn_if zero_less_mult_iff)
+by (auto simp add: sgn_if zero_less_mult_iff)
end
@@ -1150,12 +1130,10 @@
"inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_by_zero})"
proof cases
assume "a \<noteq> 0 & b \<noteq> 0"
- thus ?thesis
- by (simp add: nonzero_inverse_mult_distrib mult_commute)
+ thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_commute)
next
assume "~ (a \<noteq> 0 & b \<noteq> 0)"
- thus ?thesis
- by force
+ thus ?thesis by force
qed
text{*There is no slick version using division by zero.*}
@@ -1182,10 +1160,8 @@
by (simp add: divide_inverse nonzero_inverse_mult_distrib)
also have "... = a * inverse b * (inverse c * c)"
by (simp only: mult_ac)
- also have "... = a * inverse b"
- by simp
- finally show ?thesis
- by (simp add: divide_inverse)
+ also have "... = a * inverse b" by simp
+ finally show ?thesis by (simp add: divide_inverse)
qed
lemma mult_divide_mult_cancel_left:
@@ -1346,14 +1322,14 @@
lemma divide_eq_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
b = a * c ==> b / c = a"
- by (subst divide_eq_eq, simp)
+by (subst divide_eq_eq, simp)
lemma eq_divide_imp: "(c::'a::{division_by_zero,field}) ~= 0 ==>
a * c = b ==> a = b / c"
- by (subst eq_divide_eq, simp)
-
-
-lemmas field_eq_simps = ring_simps
+by (subst eq_divide_eq, simp)
+
+
+lemmas field_eq_simps = algebra_simps
(* pull / out*)
add_divide_eq_iff divide_add_eq_iff
diff_divide_eq_iff divide_diff_eq_iff
@@ -1475,12 +1451,10 @@
shows "inverse b < inverse (a::'a::ordered_field)"
proof (rule ccontr)
assume "~ inverse b < inverse a"
- hence "inverse a \<le> inverse b"
- by (simp add: linorder_not_less)
+ hence "inverse a \<le> inverse b" by (simp add: linorder_not_less)
hence "~ (a < b)"
by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
- thus False
- by (rule notE [OF _ less])
+ thus False by (rule notE [OF _ less])
qed
lemma inverse_less_imp_less:
@@ -1711,9 +1685,10 @@
subsection{*Field simplification*}
-text{* Lemmas @{text field_simps} multiply with denominators in
-in(equations) if they can be proved to be non-zero (for equations) or
-positive/negative (for inequations). *}
+text{* Lemmas @{text field_simps} multiply with denominators in in(equations)
+if they can be proved to be non-zero (for equations) or positive/negative
+(for inequations). Can be too aggressive and is therefore separate from the
+more benign @{text algebra_simps}. *}
lemmas field_simps = field_eq_simps
(* multiply ineqn *)
@@ -1981,15 +1956,15 @@
lemma mult_right_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
==> x * y <= x"
- by (auto simp add: mult_compare_simps);
+by (auto simp add: mult_compare_simps);
lemma mult_left_le_one_le: "0 <= (x::'a::ordered_idom) ==> 0 <= y ==> y <= 1
==> y * x <= x"
- by (auto simp add: mult_compare_simps);
+by (auto simp add: mult_compare_simps);
lemma mult_imp_div_pos_le: "0 < (y::'a::ordered_field) ==> x <= z * y ==>
x / y <= z";
- by (subst pos_divide_le_eq, assumption+);
+by (subst pos_divide_le_eq, assumption+);
lemma mult_imp_le_div_pos: "0 < (y::'a::ordered_field) ==> z * y <= x ==>
z <= x / y"
@@ -2054,7 +2029,7 @@
qed
lemma zero_less_two: "0 < 1 + 1"
- by (blast intro: less_trans zero_less_one less_add_one)
+by (blast intro: less_trans zero_less_one less_add_one)
end
@@ -2086,7 +2061,7 @@
end
lemma abs_one [simp]: "abs 1 = (1::'a::ordered_idom)"
- by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
+by (simp add: abs_if zero_less_one [THEN order_less_not_sym])
class pordered_ring_abs = pordered_ring + pordered_ab_group_add_abs +
assumes abs_eq_mult:
@@ -2106,14 +2081,14 @@
let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"
let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
have a: "(abs a) * (abs b) = ?x"
- by (simp only: abs_prts[of a] abs_prts[of b] ring_simps)
+ by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
{
fix u v :: 'a
have bh: "\<lbrakk>u = a; v = b\<rbrakk> \<Longrightarrow>
u * v = pprt a * pprt b + pprt a * nprt b +
nprt a * pprt b + nprt a * nprt b"
apply (subst prts[of u], subst prts[of v])
- apply (simp add: ring_simps)
+ apply (simp add: algebra_simps)
done
}
note b = this[OF refl[of a] refl[of b]]
@@ -2166,7 +2141,7 @@
apply (simp_all add: mulprts abs_prts)
apply (insert prems)
apply (auto simp add:
- ring_simps
+ algebra_simps
iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]
iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])
apply(drule (1) mult_nonneg_nonpos[of a b], simp)
@@ -2178,7 +2153,7 @@
then show ?thesis
apply (simp_all add: mulprts abs_prts)
apply (insert prems)
- apply (auto simp add: ring_simps)
+ apply (auto simp add: algebra_simps)
apply(drule (1) mult_nonneg_nonneg[of a b],simp)
apply(drule (1) mult_nonpos_nonpos[of a b],simp)
done
@@ -2191,10 +2166,10 @@
equal_neg_zero neg_equal_zero mult_less_0_iff)
lemma abs_mult: "abs (a * b) = abs a * abs (b::'a::ordered_idom)"
- by (simp add: abs_eq_mult linorder_linear)
+by (simp add: abs_eq_mult linorder_linear)
lemma abs_mult_self: "abs a * abs a = a * (a::'a::ordered_idom)"
- by (simp add: abs_if)
+by (simp add: abs_if)
lemma nonzero_abs_inverse:
"a \<noteq> 0 ==> abs (inverse (a::'a::ordered_field)) = inverse (abs a)"
@@ -2268,7 +2243,7 @@
apply simp
done
then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
moreover have "pprt a * pprt b <= pprt a2 * pprt b2"
by (simp_all add: prems mult_mono)
moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
--- a/src/HOL/SEQ.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/SEQ.thy Wed Jan 28 16:29:16 2009 +0100
@@ -388,7 +388,7 @@
lemma inverse_diff_inverse:
"\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
\<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
-by (simp add: ring_simps)
+by (simp add: algebra_simps)
lemma Bseq_inverse_lemma:
fixes x :: "'a::real_normed_div_algebra"
--- a/src/HOL/SetInterval.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/SetInterval.thy Wed Jan 28 16:29:16 2009 +0100
@@ -532,15 +532,15 @@
done
lemma card_atLeastAtMost_int [simp]: "card {l..u} = nat (u - l + 1)"
- apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
- apply (auto simp add: compare_rls)
- done
+apply (subst atLeastLessThanPlusOne_atLeastAtMost_int [THEN sym])
+apply (auto simp add: algebra_simps)
+done
lemma card_greaterThanAtMost_int [simp]: "card {l<..u} = nat (u - l)"
- by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
+by (subst atLeastPlusOneAtMost_greaterThanAtMost_int [THEN sym], simp)
lemma card_greaterThanLessThan_int [simp]: "card {l<..<u} = nat (u - (l + 1))"
- by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
+by (subst atLeastPlusOneLessThan_greaterThanLessThan_int [THEN sym], simp)
lemma finite_M_bounded_by_nat: "finite {k. P k \<and> k < (i::nat)}"
proof -
@@ -806,7 +806,7 @@
lemma setsum_head_upt_Suc:
"m < n \<Longrightarrow> setsum f {m..<n} = f m + setsum f {Suc m..<n}"
apply(insert setsum_head_Suc[of m "n - 1" f])
-apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] ring_simps)
+apply (simp add: atLeastLessThanSuc_atLeastAtMost[symmetric] algebra_simps)
done
@@ -870,7 +870,7 @@
show ?case by simp
next
case (Suc n)
- then show ?case by (simp add: ring_simps)
+ then show ?case by (simp add: algebra_simps)
qed
theorem arith_series_general:
@@ -894,11 +894,11 @@
have "(1+1)*?n*a + d*(1+1)*(\<Sum>i\<in>{1..n - 1}. ?I i) = ((1+1)*?n*a + d*?I (n - 1)*?I n)"
by (simp only: mult_ac gauss_sum [of "n - 1"])
(simp add: mult_ac trans [OF add_commute of_nat_Suc [symmetric]])
- finally show ?thesis by (simp add: mult_ac add_ac right_distrib)
+ finally show ?thesis by (simp add: algebra_simps)
next
assume "\<not>(n > 1)"
hence "n = 1 \<or> n = 0" by auto
- thus ?thesis by (auto simp: mult_ac right_distrib)
+ thus ?thesis by (auto simp: algebra_simps)
qed
lemma arith_series_nat:
--- a/src/HOL/Tools/ComputeNumeral.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Tools/ComputeNumeral.thy Wed Jan 28 16:29:16 2009 +0100
@@ -55,7 +55,7 @@
lemma multb0x: "(Int.Bit0 x) * y = Int.Bit0 (x * y)" by (rule mult_Bit0)
lemma multxb0: "x * (Int.Bit0 y) = Int.Bit0 (x * y)" unfolding Bit0_def by simp
lemma multb1: "(Int.Bit1 x) * (Int.Bit1 y) = Int.Bit1 (Int.Bit0 (x * y) + x + y)"
- unfolding Bit0_def Bit1_def by (simp add: ring_simps)
+ unfolding Bit0_def Bit1_def by (simp add: algebra_simps)
lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1
lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul
@@ -193,6 +193,3 @@
compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even
end
-
-
-
--- a/src/HOL/Transcendental.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Transcendental.thy Wed Jan 28 16:29:16 2009 +0100
@@ -40,7 +40,7 @@
apply (simp add: right_distrib del: setsum_op_ivl_Suc)
apply (subst mult_left_commute [where a="x - y"])
apply (erule subst)
-apply (simp add: power_Suc ring_simps)
+apply (simp add: power_Suc algebra_simps)
done
lemma lemma_realpow_rev_sumr:
@@ -146,8 +146,7 @@
fixes z :: "'a :: {recpower,comm_ring}" shows
"(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
(\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
-by (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac
- cong: strong_setsum_cong)
+by(auto simp add: algebra_simps power_add [symmetric] cong: strong_setsum_cong)
lemma sumr_diff_mult_const2:
"setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
@@ -406,7 +405,7 @@
apply (rule summable_diff [OF B A])
apply (rule sums_summable [OF diffs_equiv [OF C]])
apply (rule arg_cong [where f="suminf"], rule ext)
- apply (simp add: ring_simps)
+ apply (simp add: algebra_simps)
done
next
show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
@@ -1122,7 +1121,7 @@
apply (safe, rule lemma_DERIV_subst)
apply (best intro!: DERIV_intros intro: DERIV_chain2)
--{*replaces the old @{text DERIV_tac}*}
-apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
+apply (auto simp add: algebra_simps)
done
lemma sin_cos_add [simp]:
@@ -1146,8 +1145,8 @@
lemma lemma_DERIV_sin_cos_minus:
"\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
apply (safe, rule lemma_DERIV_subst)
-apply (best intro!: DERIV_intros intro: DERIV_chain2)
-apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
+apply (best intro!: DERIV_intros intro: DERIV_chain2)
+apply (simp add: algebra_simps)
done
lemma sin_cos_minus:
@@ -1520,9 +1519,8 @@
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
apply (subgoal_tac "0 \<le> x - real n * pi &
(x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
-apply (auto simp add: compare_rls)
- prefer 3 apply (simp add: cos_diff)
- prefer 2 apply (simp add: real_of_nat_Suc left_distrib)
+apply (auto simp add: algebra_simps real_of_nat_Suc)
+ prefer 2 apply (simp add: cos_diff)
apply (simp add: cos_diff)
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
apply (rule_tac [2] cos_total, safe)
@@ -1530,7 +1528,7 @@
apply (drule_tac x = "pi/2" in spec)
apply (simp add: cos_diff)
apply (rule_tac x = "Suc (2 * n)" in exI)
-apply (simp add: real_of_nat_Suc left_distrib, auto)
+apply (simp add: real_of_nat_Suc algebra_simps, auto)
done
lemma sin_zero_lemma:
@@ -1601,7 +1599,7 @@
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
apply (auto simp del: inverse_mult_distrib
simp add: mult_assoc left_diff_distrib cos_add)
-done
+done
lemma add_tan_eq:
"[| cos x \<noteq> 0; cos y \<noteq> 0 |]
@@ -1982,7 +1980,7 @@
also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
by (simp only: cos_add sin_add)
also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
- by (simp add: ring_simps power2_eq_square)
+ by (simp add: algebra_simps power2_eq_square)
finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
using pos_c by (simp add: sin_squared_eq power_divide)
thus ?thesis
@@ -2051,7 +2049,7 @@
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
proof -
have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
- by (auto simp add: right_distrib sin_add left_distrib mult_ac)
+ by (auto simp add: algebra_simps sin_add)
thus ?thesis
by (simp add: real_of_nat_Suc left_distrib add_divide_distrib
mult_commute [of pi])
--- a/src/HOL/Word/BinBoolList.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Word/BinBoolList.thy Wed Jan 28 16:29:16 2009 +0100
@@ -281,7 +281,7 @@
apply clarsimp
apply safe
apply (erule allE, erule xtr8 [rotated],
- simp add: numeral_simps ring_simps cong add : number_of_False_cong)+
+ simp add: numeral_simps algebra_simps cong add : number_of_False_cong)+
done
lemma bl_to_bin_lt2p: "bl_to_bin bs < (2 ^ length bs)"
@@ -299,7 +299,7 @@
apply clarsimp
apply safe
apply (erule allE, erule preorder_class.order_trans [rotated],
- simp add: numeral_simps ring_simps cong add : number_of_False_cong)+
+ simp add: numeral_simps algebra_simps cong add : number_of_False_cong)+
done
lemma bl_to_bin_ge0: "bl_to_bin bs >= 0"
@@ -1138,4 +1138,3 @@
done
end
-
--- a/src/HOL/Word/WordArith.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/Word/WordArith.thy Wed Jan 28 16:29:16 2009 +0100
@@ -1002,7 +1002,7 @@
apply (auto simp add: unat_def uint_sub_if')
apply (rule nat_diff_distrib)
prefer 3
- apply (simp add: group_simps)
+ apply (simp add: algebra_simps)
apply (rule nat_diff_distrib [THEN trans])
prefer 3
apply (subst nat_add_distrib)
--- a/src/HOL/ZF/LProd.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/ZF/LProd.thy Wed Jan 28 16:29:16 2009 +0100
@@ -45,9 +45,9 @@
case (lprod_list ah at bh bt a b)
from prems have transR: "trans R" by auto
have as: "multiset_of (ah @ a # at) = multiset_of (ah @ at) + {#a#}" (is "_ = ?ma + _")
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
have bs: "multiset_of (bh @ b # bt) = multiset_of (bh @ bt) + {#b#}" (is "_ = ?mb + _")
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
from prems have "(?ma, ?mb) \<in> mult R"
by auto
with mult_implies_one_step[OF transR] have
@@ -66,7 +66,7 @@
then show ?thesis
apply (simp only: as bs)
apply (simp only: decomposed True)
- apply (simp add: ring_simps)
+ apply (simp add: algebra_simps)
done
next
case False
@@ -78,7 +78,7 @@
then show ?thesis
apply (simp only: as bs)
apply (simp only: decomposed)
- apply (simp add: ring_simps)
+ apply (simp add: algebra_simps)
done
qed
qed
--- a/src/HOL/ex/Lagrange.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/ex/Lagrange.thy Wed Jan 28 16:29:16 2009 +0100
@@ -35,7 +35,7 @@
sq (x1*y2 + x2*y1 + x3*y4 - x4*y3) +
sq (x1*y3 - x2*y4 + x3*y1 + x4*y2) +
sq (x1*y4 + x2*y3 - x3*y2 + x4*y1)"
-by (simp add: sq_def ring_simps)
+by (simp add: sq_def algebra_simps)
text {* A challenge by John Harrison. Takes about 12s on a 1.6GHz machine. *}
@@ -51,6 +51,6 @@
sq (p1*u2 + q1*t2 - r1*w2 + s1*v2 - t1*q2 + u1*p2 - v1*s2 + w1*r2) +
sq (p1*v2 + q1*w2 + r1*t2 - s1*u2 - t1*r2 + u1*s2 + v1*p2 - w1*q2) +
sq (p1*w2 - q1*v2 + r1*u2 + s1*t2 - t1*s2 - u1*r2 + v1*q2 + w1*p2)"
-by (simp add: sq_def ring_simps)
+by (simp add: sq_def algebra_simps)
end
--- a/src/HOL/ex/MIR.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/ex/MIR.thy Wed Jan 28 16:29:16 2009 +0100
@@ -709,11 +709,11 @@
next
case (2 n c t) hence gd: "g dvd c" by simp
from gp have gnz: "g \<noteq> 0" by simp
- from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_simps)
+ from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
next
case (3 c s t) hence gd: "g dvd c" by simp
from gp have gnz: "g \<noteq> 0" by simp
- from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_simps)
+ from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
qed (auto simp add: numgcd_def gp)
consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
recdef ismaxcoeff "measure size"
@@ -850,12 +850,12 @@
lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
- apply (case_tac "n1 = n2", simp_all add: ring_simps)
+ apply (case_tac "n1 = n2", simp_all add: algebra_simps)
apply (simp only: left_distrib[symmetric])
apply simp
apply (case_tac "lex_bnd t1 t2", simp_all)
apply (case_tac "c1+c2 = 0")
- by (case_tac "t1 = t2", simp_all add: ring_simps left_distrib[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add left_distrib)
+ by (case_tac "t1 = t2", simp_all add: algebra_simps left_distrib[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add left_distrib)
lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
by (induct t s rule: numadd.induct, auto simp add: Let_def)
@@ -868,7 +868,7 @@
"nummul t = (\<lambda> i. Mul i t)"
lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: ring_simps)
+by (induct t rule: nummul.induct, auto simp add: algebra_simps)
lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
by (induct t rule: nummul.induct, auto)
@@ -928,7 +928,7 @@
with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+
from prems(2) have tibi: "ti = CF c a ?bi" by (simp add: Let_def split_def)
from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF)
-qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def ring_simps)
+qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def algebra_simps)
lemma split_int_nb: "numbound0 t \<Longrightarrow> numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t)) "
by (induct t rule: split_int.induct, auto simp add: Let_def split_def)
@@ -1768,7 +1768,7 @@
have th: "(real a + b >0) = (real (-a) + (-b)< 0)" by arith
show ?thesis using myless[rule_format, where b="real (floor b)"]
by (simp only:th split_int_less_real'[where a="-a" and b="-b"])
- (simp add: ring_simps diff_def[symmetric],arith)
+ (simp add: algebra_simps diff_def[symmetric],arith)
qed
lemma split_int_le_real:
@@ -1795,7 +1795,7 @@
proof-
have th: "(real a + b \<ge>0) = (real (-a) + (-b) \<le> 0)" by arith
show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"])
- (simp add: ring_simps diff_def[symmetric],arith)
+ (simp add: algebra_simps diff_def[symmetric],arith)
qed
lemma split_int_eq_real: "(real (a::int) = b) = ( a = floor b \<and> b = real (floor b))" (is "?l = ?r")
@@ -2281,9 +2281,9 @@
(is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))"
- by (simp add: ring_simps di_def)
+ by (simp add: algebra_simps di_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
next
@@ -2292,7 +2292,7 @@
hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def)
- hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: ring_simps)
+ hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
by blast
thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp
@@ -2308,9 +2308,9 @@
(is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))"
- by (simp add: ring_simps di_def)
+ by (simp add: algebra_simps di_def)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast
thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp
next
@@ -2319,7 +2319,7 @@
hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def)
- hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: ring_simps)
+ hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)"
by blast
thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp
@@ -2450,16 +2450,16 @@
have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
(real j rdvd - (real c * real x - Inum (real x # bs) e))"
by (simp only: rdvd_minus[symmetric])
- from prems show ?case
- by (simp add: ring_simps th[simplified ring_simps]
+ from prems th show ?case
+ by (simp add: algebra_simps
numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
next
case (10 j c e)
have th: "(real j rdvd real c * real x - Inum (real x # bs) e) =
(real j rdvd - (real c * real x - Inum (real x # bs) e))"
by (simp only: rdvd_minus[symmetric])
- from prems show ?case
- by (simp add: ring_simps th[simplified ring_simps]
+ from prems th show ?case
+ by (simp add: algebra_simps
numbound0_I[where bs="bs" and b'="real x" and b="- real x"])
qed (auto simp add: numbound0_I[where bs="bs" and b="real x" and b'="- real x"] nth_pos2)
@@ -2541,7 +2541,7 @@
hence "(real l * real x + real (l div c) * Inum (real x # bs) e < (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e < 0)"
by simp
- also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) < (real (l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) < (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e < 0)"
using mult_less_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
@@ -2559,7 +2559,7 @@
hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<le> (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<le> 0)"
by simp
- also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<le> (real (l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<le> (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e \<le> 0)"
using mult_le_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
@@ -2577,7 +2577,7 @@
hence "(real l * real x + real (l div c) * Inum (real x # bs) e > (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e > 0)"
by simp
- also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) > (real (l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) > (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e > 0)"
using zero_less_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
@@ -2595,7 +2595,7 @@
hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<ge> (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<ge> 0)"
by simp
- also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<ge> (real (l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<ge> (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e \<ge> 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
@@ -2613,7 +2613,7 @@
hence "(real l * real x + real (l div c) * Inum (real x # bs) e = (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = 0)"
by simp
- also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) = (real (l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) = (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e = 0)"
using mult_eq_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
@@ -2631,7 +2631,7 @@
hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<noteq> (0\<Colon>real)) =
(real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<noteq> 0)"
by simp
- also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<noteq> (real (l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<noteq> (real (l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (real c * real x + Inum (real x # bs) e \<noteq> 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp
@@ -2647,7 +2647,7 @@
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp
- also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: ring_simps)
+ also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
@@ -2664,7 +2664,7 @@
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp
- also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: ring_simps)
+ also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)"
using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp
@@ -2719,7 +2719,7 @@
hence "x + floor ?e \<ge> 1 \<and> x + floor ?e \<le> d" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. real x = real (- floor ?e + j)"
- by (simp only: real_of_int_inject) (simp add: ring_simps)
+ by (simp only: real_of_int_inject) (simp add: algebra_simps)
hence "\<exists> (j::int) \<in> {1 .. d}. real x = - ?e + real j"
by (simp add: ie[simplified isint_iff])
with nob have ?case by auto}
@@ -2744,7 +2744,7 @@
using ie by simp
hence "x + floor ?e +1 \<ge> 1 \<and> x + floor ?e + 1 \<le> d" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e + 1" by simp
- hence "\<exists> (j::int) \<in> {1 .. d}. x= - floor ?e - 1 + j" by (simp add: ring_simps)
+ hence "\<exists> (j::int) \<in> {1 .. d}. x= - floor ?e - 1 + j" by (simp add: algebra_simps)
hence "\<exists> (j::int) \<in> {1 .. d}. real x= real (- floor ?e - 1 + j)"
by (simp only: real_of_int_inject)
hence "\<exists> (j::int) \<in> {1 .. d}. real x= - ?e - 1 + real j"
@@ -2759,7 +2759,7 @@
have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
from p have "real x= - ?e" by (simp add: c1) with prems(11) show ?case using dp
by simp (erule ballE[where x="1"],
- simp_all add:ring_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"])
+ simp_all add:algebra_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"])
next
case (4 c e)hence p: "Ifm (real x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1"
and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
@@ -2983,7 +2983,7 @@
from prems have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3005,7 +3005,7 @@
from prems have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<noteq> 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3027,7 +3027,7 @@
from prems have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))" using pos_less_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3049,7 +3049,7 @@
from prems have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<le> 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Le (CN 0 c e)))" using pos_le_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3071,7 +3071,7 @@
from prems have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))" using pos_divide_less_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3093,7 +3093,7 @@
from prems have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<ge> 0)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))" using pos_divide_le_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3114,7 +3114,7 @@
from prems have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3135,7 +3135,7 @@
from prems have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\<not> (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))"
using real_of_int_div[OF knz kdt]
numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
- numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_simps)
+ numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti algebra_simps)
also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"]
numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"]
by (simp add: ti)
@@ -3154,7 +3154,7 @@
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
- {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_simps)}
+ {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (4 c e)
@@ -3162,7 +3162,7 @@
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
- {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_simps)}
+ {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (5 c e)
@@ -3170,7 +3170,7 @@
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
- {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_less_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_simps)}
+ {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_less_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (6 c e)
@@ -3178,7 +3178,7 @@
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
- {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_le_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_simps)}
+ {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_le_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (7 c e)
@@ -3186,7 +3186,7 @@
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
- {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_less_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_simps)}
+ {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_less_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (8 c e)
@@ -3194,7 +3194,7 @@
from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp
{assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp }
moreover
- {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_le_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_simps)}
+ {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_le_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: algebra_simps)}
ultimately show ?case by blast
next
case (9 i c e)
@@ -3206,7 +3206,7 @@
hence "Ifm (real (x*k)#bs) (a\<rho> (Dvd i (CN 0 c e)) k) =
(real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k)"
using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"]
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
also have "\<dots> = (Ifm (real x#bs) (Dvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"])
finally have ?case . }
ultimately show ?case by blast
@@ -3220,7 +3220,7 @@
hence "Ifm (real (x*k)#bs) (a\<rho> (NDvd i (CN 0 c e)) k) =
(\<not> (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k))"
using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"]
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
also have "\<dots> = (Ifm (real x#bs) (NDvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"])
finally have ?case . }
ultimately show ?case by blast
@@ -3233,7 +3233,7 @@
proof-
have "(\<exists> x. ?D x \<and> ?P' x) = (\<exists> x. k dvd x \<and> ?P' x)" using int_rdvd_iff by simp
also have "\<dots> = (\<exists>x. ?P' (x*k))" using unity_coeff_ex[where P="?P'" and l="k", simplified]
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
also have "\<dots> = (\<exists> x. ?P x)" using a\<rho> iszlfm_gen[OF lp] kp by auto
finally show ?thesis .
qed
@@ -3297,7 +3297,7 @@
by simp+
{assume "real (c*i) \<noteq> - ?N i e + real (c*d)"
with numbound0_I[OF nb, where bs="bs" and b="real i - real d" and b'="real i"]
- have ?case by (simp add: ring_simps)}
+ have ?case by (simp add: algebra_simps)}
moreover
{assume pi: "real (c*i) = - ?N i e + real (c*d)"
from mult_strict_left_mono[OF dp cp] have d: "(c*d) \<in> {1 .. c*d}" by simp
@@ -3309,27 +3309,27 @@
real_of_int_mult]
show ?case using prems dp
by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"]
- ring_simps)
+ algebra_simps)
next
case (6 c e) hence cp: "c > 0" by simp
from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric]
real_of_int_mult]
show ?case using prems dp
by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"]
- ring_simps)
+ algebra_simps)
next
case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)"
and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j"
and pi: "real (c*i) + ?N i e > 0" and cp': "real c >0"
by simp+
let ?fe = "floor (?N i e)"
- from pi cp have th:"(real i +?N i e / real c)*real c > 0" by (simp add: ring_simps)
+ from pi cp have th:"(real i +?N i e / real c)*real c > 0" by (simp add: algebra_simps)
from pi ei[simplified isint_iff] have "real (c*i + ?fe) > real (0::int)" by simp
hence pi': "c*i + ?fe > 0" by (simp only: real_of_int_less_iff[symmetric])
have "real (c*i) + ?N i e > real (c*d) \<or> real (c*i) + ?N i e \<le> real (c*d)" by auto
moreover
{assume "real (c*i) + ?N i e > real (c*d)" hence ?case
- by (simp add: ring_simps
+ by (simp add: algebra_simps
numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])}
moreover
{assume H:"real (c*i) + ?N i e \<le> real (c*d)"
@@ -3337,7 +3337,7 @@
hence pid: "c*i + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + ?fe = j1" by auto
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - ?N i e + real j1"
- by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] ring_simps)
+ by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps)
with nob have ?case by blast }
ultimately show ?case by blast
next
@@ -3346,13 +3346,13 @@
and pi: "real (c*i) + ?N i e \<ge> 0" and cp': "real c >0"
by simp+
let ?fe = "floor (?N i e)"
- from pi cp have th:"(real i +?N i e / real c)*real c \<ge> 0" by (simp add: ring_simps)
+ from pi cp have th:"(real i +?N i e / real c)*real c \<ge> 0" by (simp add: algebra_simps)
from pi ei[simplified isint_iff] have "real (c*i + ?fe) \<ge> real (0::int)" by simp
hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: real_of_int_le_iff[symmetric])
have "real (c*i) + ?N i e \<ge> real (c*d) \<or> real (c*i) + ?N i e < real (c*d)" by auto
moreover
{assume "real (c*i) + ?N i e \<ge> real (c*d)" hence ?case
- by (simp add: ring_simps
+ by (simp add: algebra_simps
numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])}
moreover
{assume H:"real (c*i) + ?N i e < real (c*d)"
@@ -3360,9 +3360,9 @@
hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: real_of_int_le_iff)
with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + 1+ ?fe = j1" by auto
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) + 1= - ?N i e + real j1"
- by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] ring_simps real_of_one)
+ by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] algebra_simps real_of_one)
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = (- ?N i e + real j1) - 1"
- by (simp only: ring_simps diff_def[symmetric])
+ by (simp only: algebra_simps diff_def[symmetric])
hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - 1 - ?N i e + real j1"
by (simp only: add_ac diff_def)
with nob have ?case by blast }
@@ -3383,10 +3383,10 @@
using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
ie by simp
also have "\<dots> = (real j rdvd real (c*(i - d)) + ?e)"
- using ie by (simp add:ring_simps)
+ using ie by (simp add:algebra_simps)
finally show ?case
using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
next
case (10 j c e) hence p: "\<not> (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+
let ?e = "Inum (real i # bs) e"
@@ -3403,10 +3403,10 @@
using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
ie by simp
also have "\<dots> = Not (real j rdvd real (c*(i - d)) + ?e)"
- using ie by (simp add:ring_simps)
+ using ie by (simp add:algebra_simps)
finally show ?case
using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
qed(auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] nth_pos2)
lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t"
@@ -3649,7 +3649,7 @@
from H
have "?I (?p (p',n',s') j) \<longrightarrow>
(((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))"
- by (simp add: fp_def np ring_simps numsub numadd numfloor)
+ by (simp add: fp_def np algebra_simps numsub numadd numfloor)
also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
moreover
@@ -3675,7 +3675,7 @@
from H
have "?I (?p (p',n',s') j) \<longrightarrow>
(((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))"
- by (simp add: np fp_def ring_simps numneg numfloor numadd numsub)
+ by (simp add: np fp_def algebra_simps numneg numfloor numadd numsub)
also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp
moreover
@@ -3691,7 +3691,7 @@
apply (erule_tac x = "(aa, aaa, ba)" in ballE) apply simp_all
apply (erule_tac x = "(ab, ac, baa)" in ballE) apply simp_all
done
-qed (auto simp add: Let_def split_def ring_simps conj_rl)
+qed (auto simp add: Let_def split_def algebra_simps conj_rl)
lemma real_in_int_intervals:
assumes xb: "real m \<le> x \<and> x < real ((n::int) + 1)"
@@ -3795,7 +3795,7 @@
hence "\<exists> j\<in> {0 .. n}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0"
by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"])
hence "\<exists> j\<in> {0.. n}. ?I (?p (p,n,s) j)"
- using pns by (simp add: fp_def np ring_simps numsub numadd)
+ using pns by (simp add: fp_def np algebra_simps numsub numadd)
then obtain "j" where j_def: "j\<in> {0 .. n} \<and> ?I (?p (p,n,s) j)" by blast
hence "\<exists>x \<in> {?p (p,n,s) j |j. 0\<le> j \<and> j \<le> n }. ?I x" by auto
hence ?case using pns
@@ -3813,7 +3813,7 @@
have "real n *x + ?N s \<ge> real n + ?N s" by simp
moreover from real_of_int_floor_le[where r="?N s"] have "real n + ?N s \<ge> real n + ?N (Floor s)" by simp
ultimately have "real n *x + ?N s \<ge> ?N (Floor s) + real n"
- by (simp only: ring_simps)}
+ by (simp only: algebra_simps)}
ultimately have "?N (Floor s) + real n \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (1::int)" by simp
hence th: "real n \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (1::int)" by simp
have th1: "\<forall> (a::real). (- a > 0) = (a < 0)" by auto
@@ -3919,7 +3919,7 @@
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (lt n s) = ?I (Lt a)" using H by (cases "n=0", (simp add: lt_def))
- (cases "n > 0", simp_all add: lt_def ring_simps myless[rule_format, where b="0"])
+ (cases "n > 0", simp_all add: lt_def algebra_simps myless[rule_format, where b="0"])
qed
lemma lt_l: "isrlfm (rsplit lt a)"
@@ -3931,7 +3931,7 @@
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (le n s) = ?I (Le a)" using H by (cases "n=0", (simp add: le_def))
- (cases "n > 0", simp_all add: le_def ring_simps myl[rule_format, where b="0"])
+ (cases "n > 0", simp_all add: le_def algebra_simps myl[rule_format, where b="0"])
qed
lemma le_l: "isrlfm (rsplit le a)"
@@ -3943,7 +3943,7 @@
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (gt n s) = ?I (Gt a)" using H by (cases "n=0", (simp add: gt_def))
- (cases "n > 0", simp_all add: gt_def ring_simps myless[rule_format, where b="0"])
+ (cases "n > 0", simp_all add: gt_def algebra_simps myless[rule_format, where b="0"])
qed
lemma gt_l: "isrlfm (rsplit gt a)"
by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def)
@@ -3954,7 +3954,7 @@
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
show "?I (ge n s) = ?I (Ge a)" using H by (cases "n=0", (simp add: ge_def))
- (cases "n > 0", simp_all add: ge_def ring_simps myl[rule_format, where b="0"])
+ (cases "n > 0", simp_all add: ge_def algebra_simps myl[rule_format, where b="0"])
qed
lemma ge_l: "isrlfm (rsplit ge a)"
by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def)
@@ -3964,7 +3964,7 @@
proof(clarify)
fix a n s
assume H: "?N a = ?N (CN 0 n s)"
- show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def ring_simps)
+ show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def algebra_simps)
qed
lemma eq_l: "isrlfm (rsplit eq a)"
by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def)
@@ -3974,7 +3974,7 @@
proof(clarify)
fix a n s bs
assume H: "?N a = ?N (CN 0 n s)"
- show "?I (neq n s) = ?I (NEq a)" using H by (auto simp add: neq_def ring_simps)
+ show "?I (neq n s) = ?I (NEq a)" using H by (auto simp add: neq_def algebra_simps)
qed
lemma neq_l: "isrlfm (rsplit neq a)"
@@ -4012,10 +4012,10 @@
also have "\<dots> = (?DE \<and> ?a \<ge> 0 \<and> ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1])
also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. ?a = j))" by simp
also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. real (\<lfloor>real n * u - s\<rfloor>) = real j - real \<lfloor>s\<rfloor> ))"
- by (simp only: ring_simps real_of_int_diff[symmetric] real_of_int_inject del: real_of_int_diff)
+ by (simp only: algebra_simps real_of_int_diff[symmetric] real_of_int_inject del: real_of_int_diff)
also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real n * u - s = real j - real \<lfloor>s\<rfloor> \<and> real i rdvd real n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real n * u - s\<rfloor>"]
by (auto cong: conj_cong)
- also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: ring_simps )
+ also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: algebra_simps )
finally show ?thesis .
qed
@@ -4038,7 +4038,7 @@
from foldr_disj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"]
have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))"
by (simp add: iupt_set np DVDJ_def del: iupt.simps)
- also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s)))" by (simp add: ring_simps diff_def[symmetric])
+ also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s)))" by (simp add: algebra_simps diff_def[symmetric])
also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"]
have "\<dots> = (real i rdvd real n * x - (-?s))" by simp
finally show ?thesis by simp
@@ -4053,7 +4053,7 @@
from foldr_conj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"]
have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))"
by (simp add: iupt_set np NDVDJ_def del: iupt.simps)
- also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s))))" by (simp add: ring_simps diff_def[symmetric])
+ also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s))))" by (simp add: algebra_simps diff_def[symmetric])
also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"]
have "\<dots> = (\<not> (real i rdvd real n * x - (-?s)))" by simp
finally show ?thesis by simp
@@ -4652,40 +4652,40 @@
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
from np have np: "real n \<noteq> 0" by simp
@@ -4693,10 +4693,10 @@
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
from np have np: "real n \<noteq> 0" by simp
@@ -4704,10 +4704,10 @@
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
lemma \<Upsilon>_l:
@@ -4758,7 +4758,7 @@
using lp px noS
proof (induct p rule: isrlfm.induct)
case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e < 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
hence pxc: "x < (- ?N x e) / real c"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -4767,7 +4767,7 @@
moreover {assume y: "y < (-?N x e)/ real c"
hence "y * real c < - ?N x e"
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y > (- ?N x e) / real c"
with yu have eu: "u > (- ?N x e) / real c" by auto
@@ -4777,7 +4777,7 @@
ultimately show ?case by blast
next
case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
- from prems have "x * real c + ?N x e \<le> 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
hence pxc: "x \<le> (- ?N x e) / real c"
by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -4786,7 +4786,7 @@
moreover {assume y: "y < (-?N x e)/ real c"
hence "y * real c < - ?N x e"
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y > (- ?N x e) / real c"
with yu have eu: "u > (- ?N x e) / real c" by auto
@@ -4796,7 +4796,7 @@
ultimately show ?case by blast
next
case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e > 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
hence pxc: "x > (- ?N x e) / real c"
by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -4805,7 +4805,7 @@
moreover {assume y: "y > (-?N x e)/ real c"
hence "y * real c > - ?N x e"
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y < (- ?N x e) / real c"
with ly have eu: "l < (- ?N x e) / real c" by auto
@@ -4815,7 +4815,7 @@
ultimately show ?case by blast
next
case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e \<ge> 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
hence pxc: "x \<ge> (- ?N x e) / real c"
by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -4824,7 +4824,7 @@
moreover {assume y: "y > (-?N x e)/ real c"
hence "y * real c > - ?N x e"
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y < (- ?N x e) / real c"
with ly have eu: "l < (- ?N x e) / real c" by auto
@@ -4835,7 +4835,7 @@
next
case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
from cp have cnz: "real c \<noteq> 0" by simp
- from prems have "x * real c + ?N x e = 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
hence pxc: "x = (- ?N x e) / real c"
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -4848,9 +4848,9 @@
with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
hence "y* real c \<noteq> -?N x e"
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
- hence "y* real c + ?N x e \<noteq> 0" by (simp add: ring_simps)
+ hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
lemma finite_set_intervals:
@@ -5013,7 +5013,7 @@
by (simp add: mult_commute)
from tnb snb have st_nb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnp mp np by (simp add: ring_simps add_divide_distrib)
+ using mnp mp np by (simp add: algebra_simps add_divide_distrib)
from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"]
have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
with rinf_\<Upsilon>[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
@@ -5082,7 +5082,7 @@
lemma exsplitnum:
"Inum (x#y#bs) (exsplitnum t) = Inum ((x+y) #bs) t"
- by(induct t rule: exsplitnum.induct) (simp_all add: ring_simps)
+ by(induct t rule: exsplitnum.induct) (simp_all add: algebra_simps)
lemma exsplit:
assumes qfp: "qfree p"
@@ -5173,14 +5173,14 @@
from Ul th have mnz: "m \<noteq> 0" by auto
from Ul th have nnz: "n \<noteq> 0" by auto
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnz nnz by (simp add: ring_simps add_divide_distrib)
+ using mnz nnz by (simp add: algebra_simps add_divide_distrib)
thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) /
(2 * real n * real m)
\<in> (\<lambda>((t, n), s, m).
(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
(set U \<times> set U)"using mnz nnz th
- apply (auto simp add: th add_divide_distrib ring_simps split_def image_def)
+ apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
by (rule_tac x="(s,m)" in bexI,simp_all)
(rule_tac x="(t,n)" in bexI,simp_all)
next
@@ -5191,7 +5191,7 @@
from Ul smU have mnz: "m \<noteq> 0" by auto
from Ul tnU have nnz: "n \<noteq> 0" by auto
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnz nnz by (simp add: ring_simps add_divide_distrib)
+ using mnz nnz by (simp add: algebra_simps add_divide_distrib)
let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
have Pc:"\<forall> a b. ?P a b = ?P b a"
by auto
@@ -5204,7 +5204,7 @@
from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
let ?st' = "Add (Mul m' t') (Mul n' s')"
have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
- using mnz' nnz' by (simp add: ring_simps add_divide_distrib)
+ using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
from Pts' have
"(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
@@ -5234,7 +5234,7 @@
by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
from tnb snb have stnb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mp np by (simp add: ring_simps add_divide_distrib)
+ using mp np by (simp add: algebra_simps add_divide_distrib)
from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
by auto (rule_tac x="(a,b)" in bexI, auto)
@@ -5262,7 +5262,7 @@
by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
from tnb snb have stnb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mp np by (simp add: ring_simps add_divide_distrib)
+ using mp np by (simp add: algebra_simps add_divide_distrib)
from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
from \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
--- a/src/HOL/ex/Numeral.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/ex/Numeral.thy Wed Jan 28 16:29:16 2009 +0100
@@ -699,7 +699,7 @@
begin
subclass semiring_1_minus
- proof qed (simp_all add: ring_simps)
+ proof qed (simp_all add: algebra_simps)
lemma Dig_zero_minus_of_num [numeral]:
"0 - of_num n = - of_num n"
@@ -783,7 +783,7 @@
"sub (Dig1 m) (Dig1 n) = dup (sub m n)"
"sub (Dig1 m) (Dig0 n) = dup (sub m n) + 1"
"sub (Dig0 m) (Dig1 n) = dup (sub m n) - 1"
- apply (simp_all add: dup_def ring_simps)
+ apply (simp_all add: dup_def algebra_simps)
apply (simp_all add: of_num_plus Dig_one_plus_dec)[4]
apply (simp_all add: of_num.simps)
done
--- a/src/HOL/ex/ReflectedFerrack.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/ex/ReflectedFerrack.thy Wed Jan 28 16:29:16 2009 +0100
@@ -532,7 +532,7 @@
next
case (2 n c t) hence gd: "g dvd c" by simp
from gp have gnz: "g \<noteq> 0" by simp
- from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_simps)
+ from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
qed (auto simp add: numgcd_def gp)
consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
recdef ismaxcoeff "measure size"
@@ -636,7 +636,7 @@
lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
-apply (case_tac "n1 = n2", simp_all add: ring_simps)
+apply (case_tac "n1 = n2", simp_all add: algebra_simps)
by (simp only: left_distrib[symmetric],simp)
lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
@@ -648,7 +648,7 @@
"nummul t = (\<lambda> i. Mul i t)"
lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: ring_simps)
+by (induct t rule: nummul.induct, auto simp add: algebra_simps)
lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
by (induct t rule: nummul.induct, auto )
@@ -1001,10 +1001,10 @@
by(cases "rsplit0 a",auto simp add: Let_def split_def)
have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) =
Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
- by (simp add: Let_def split_def ring_simps)
+ by (simp add: Let_def split_def algebra_simps)
also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
finally show ?case using nb by simp
-qed(auto simp add: Let_def split_def ring_simps , simp add: right_distrib[symmetric])
+qed(auto simp add: Let_def split_def algebra_simps , simp add: right_distrib[symmetric])
(* Linearize a formula*)
definition
@@ -1396,40 +1396,40 @@
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
from np have np: "real n \<noteq> 0" by simp
@@ -1437,10 +1437,10 @@
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
from np have np: "real n \<noteq> 0" by simp
@@ -1448,10 +1448,10 @@
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
- and b="0", simplified divide_zero_left]) (simp only: ring_simps)
+ and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
using np by simp
- finally show ?case using nbt nb by (simp add: ring_simps)
+ finally show ?case using nbt nb by (simp add: algebra_simps)
qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
lemma uset_l:
@@ -1502,7 +1502,7 @@
using lp px noS
proof (induct p rule: isrlfm.induct)
case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e < 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
hence pxc: "x < (- ?N x e) / real c"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -1511,7 +1511,7 @@
moreover {assume y: "y < (-?N x e)/ real c"
hence "y * real c < - ?N x e"
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y > (- ?N x e) / real c"
with yu have eu: "u > (- ?N x e) / real c" by auto
@@ -1521,7 +1521,7 @@
ultimately show ?case by blast
next
case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
- from prems have "x * real c + ?N x e \<le> 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
hence pxc: "x \<le> (- ?N x e) / real c"
by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -1530,7 +1530,7 @@
moreover {assume y: "y < (-?N x e)/ real c"
hence "y * real c < - ?N x e"
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y > (- ?N x e) / real c"
with yu have eu: "u > (- ?N x e) / real c" by auto
@@ -1540,7 +1540,7 @@
ultimately show ?case by blast
next
case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e > 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
hence pxc: "x > (- ?N x e) / real c"
by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -1549,7 +1549,7 @@
moreover {assume y: "y > (-?N x e)/ real c"
hence "y * real c > - ?N x e"
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y < (- ?N x e) / real c"
with ly have eu: "l < (- ?N x e) / real c" by auto
@@ -1559,7 +1559,7 @@
ultimately show ?case by blast
next
case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
- from prems have "x * real c + ?N x e \<ge> 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
hence pxc: "x \<ge> (- ?N x e) / real c"
by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -1568,7 +1568,7 @@
moreover {assume y: "y > (-?N x e)/ real c"
hence "y * real c > - ?N x e"
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
- hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
+ hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y < (- ?N x e) / real c"
with ly have eu: "l < (- ?N x e) / real c" by auto
@@ -1579,7 +1579,7 @@
next
case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
from cp have cnz: "real c \<noteq> 0" by simp
- from prems have "x * real c + ?N x e = 0" by (simp add: ring_simps)
+ from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
hence pxc: "x = (- ?N x e) / real c"
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
@@ -1592,9 +1592,9 @@
with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
hence "y* real c \<noteq> -?N x e"
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
- hence "y* real c + ?N x e \<noteq> 0" by (simp add: ring_simps)
+ hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
lemma finite_set_intervals:
@@ -1757,7 +1757,7 @@
by (simp add: mult_commute)
from tnb snb have st_nb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnp mp np by (simp add: ring_simps add_divide_distrib)
+ using mnp mp np by (simp add: algebra_simps add_divide_distrib)
from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"]
have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
@@ -1801,14 +1801,14 @@
from Ul th have mnz: "m \<noteq> 0" by auto
from Ul th have nnz: "n \<noteq> 0" by auto
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnz nnz by (simp add: ring_simps add_divide_distrib)
+ using mnz nnz by (simp add: algebra_simps add_divide_distrib)
thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) /
(2 * real n * real m)
\<in> (\<lambda>((t, n), s, m).
(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
(set U \<times> set U)"using mnz nnz th
- apply (auto simp add: th add_divide_distrib ring_simps split_def image_def)
+ apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
by (rule_tac x="(s,m)" in bexI,simp_all)
(rule_tac x="(t,n)" in bexI,simp_all)
next
@@ -1819,7 +1819,7 @@
from Ul smU have mnz: "m \<noteq> 0" by auto
from Ul tnU have nnz: "n \<noteq> 0" by auto
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mnz nnz by (simp add: ring_simps add_divide_distrib)
+ using mnz nnz by (simp add: algebra_simps add_divide_distrib)
let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
have Pc:"\<forall> a b. ?P a b = ?P b a"
by auto
@@ -1832,7 +1832,7 @@
from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
let ?st' = "Add (Mul m' t') (Mul n' s')"
have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
- using mnz' nnz' by (simp add: ring_simps add_divide_distrib)
+ using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
from Pts' have
"(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
@@ -1862,7 +1862,7 @@
by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
from tnb snb have stnb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mp np by (simp add: ring_simps add_divide_distrib)
+ using mp np by (simp add: algebra_simps add_divide_distrib)
from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
by auto (rule_tac x="(a,b)" in bexI, auto)
@@ -1890,7 +1890,7 @@
by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
from tnb snb have stnb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
- using mp np by (simp add: ring_simps add_divide_distrib)
+ using mp np by (simp add: algebra_simps add_divide_distrib)
from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
--- a/src/HOL/ex/Reflected_Presburger.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/ex/Reflected_Presburger.thy Wed Jan 28 16:29:16 2009 +0100
@@ -437,7 +437,7 @@
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
apply (case_tac "n1 = n2")
- apply(simp_all add: ring_simps)
+ apply(simp_all add: algebra_simps)
apply(simp add: left_distrib[symmetric])
done
@@ -452,7 +452,7 @@
| "nummul i t = Mul i t"
lemma nummul: "\<And> i. Inum bs (nummul i t) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: ring_simps numadd)
+by (induct t rule: nummul.induct, auto simp add: algebra_simps numadd)
lemma nummul_nb: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul i t)"
by (induct t rule: nummul.induct, auto simp add: numadd_nb)
@@ -912,7 +912,7 @@
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (i#bs) t"
from prems Ia nb show ?case
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done
@@ -925,7 +925,7 @@
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (i#bs) t"
from prems Ia nb show ?case
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done
@@ -938,7 +938,7 @@
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (i#bs) t"
from prems Ia nb show ?case
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done
@@ -951,7 +951,7 @@
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (i#bs) t"
from prems Ia nb show ?case
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done
@@ -964,7 +964,7 @@
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (i#bs) t"
from prems Ia nb show ?case
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done
@@ -977,7 +977,7 @@
have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
let ?N = "\<lambda> t. Inum (i#bs) t"
from prems Ia nb show ?case
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done
@@ -996,7 +996,7 @@
moreover
{assume "?c=0" and "j\<noteq>0" hence ?case
using zsplit0_I[OF spl, where x="i" and bs="bs"] zdvd_abs1[where i="j"]
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done}
@@ -1027,7 +1027,7 @@
moreover
{assume "?c=0" and "j\<noteq>0" hence ?case
using zsplit0_I[OF spl, where x="i" and bs="bs"] zdvd_abs1[where i="j"]
- apply (auto simp add: Let_def split_def ring_simps)
+ apply (auto simp add: Let_def split_def algebra_simps)
apply (cases "?r",auto)
apply (case_tac nat, auto)
done}
@@ -1301,9 +1301,9 @@
(is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
hence "\<exists> (l::int). ?rt = i * l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x+ ?I x e = i*l+c*(k * i*di)"
- by (simp add: ring_simps di_def)
+ by (simp add: algebra_simps di_def)
hence "\<exists> (l::int). c*x+ ?I x e = i*(l + c*k*di)"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
hence "\<exists> (l::int). c*x+ ?I x e = i*l" by blast
thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def)
next
@@ -1312,7 +1312,7 @@
hence "\<exists> (l::int). c*x+?e = i*l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
- hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: ring_simps)
+ hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). c*x - c * (k*d) +?e = i*l"
by blast
thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
@@ -1328,9 +1328,9 @@
(is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
hence "\<exists> (l::int). ?rt = i * l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x+ ?I x e = i*l+c*(k * i*di)"
- by (simp add: ring_simps di_def)
+ by (simp add: algebra_simps di_def)
hence "\<exists> (l::int). c*x+ ?I x e = i*(l + c*k*di)"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
hence "\<exists> (l::int). c*x+ ?I x e = i*l" by blast
thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def)
next
@@ -1339,7 +1339,7 @@
hence "\<exists> (l::int). c*x+?e = i*l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
- hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: ring_simps)
+ hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). c*x - c * (k*d) +?e = i*l"
by blast
thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
@@ -1363,7 +1363,7 @@
by (simp only: zdvd_zminus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_def zadd_ac zminus_zadd_distrib)
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
by simp
@@ -1375,7 +1375,7 @@
by (simp only: zdvd_zminus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
apply (simp only: minus_mult_right[symmetric] minus_mult_left[symmetric] diff_def zadd_ac zminus_zadd_distrib)
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
by simp
@@ -1443,7 +1443,7 @@
hence "(l*x + (l div c) * Inum (x # bs) e < 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e < 0)"
by simp
- also have "\<dots> = ((l div c) * (c*x + Inum (x # bs) e) < (l div c) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = ((l div c) * (c*x + Inum (x # bs) e) < (l div c) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e < 0)"
using mult_less_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be by simp
@@ -1461,7 +1461,7 @@
hence "(l*x + (l div c) * Inum (x# bs) e \<le> 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<le> 0)"
by simp
- also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<le> ((l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<le> ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e \<le> 0)"
using mult_le_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be by simp
@@ -1479,7 +1479,7 @@
hence "(l*x + (l div c)* Inum (x # bs) e > 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e > 0)"
by simp
- also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) > ((l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) > ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e > 0)"
using zero_less_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp
@@ -1498,7 +1498,7 @@
((c*(l div c))*x + (l div c)* Inum (x # bs) e \<ge> 0)"
by simp
also have "\<dots> = ((l div c)*(c*x + Inum (x # bs) e) \<ge> ((l div c)) * 0)"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e \<ge> 0)" using ldcp
zero_le_mult_iff [where a="l div c" and b="c*x + Inum (x # bs) e"] by simp
finally show ?case using be numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"]
@@ -1517,7 +1517,7 @@
hence "(l * x + (l div c) * Inum (x # bs) e = 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e = 0)"
by simp
- also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) = ((l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) = ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e = 0)"
using mult_eq_0_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp
@@ -1535,7 +1535,7 @@
hence "(l * x + (l div c) * Inum (x # bs) e \<noteq> 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<noteq> 0)"
by simp
- also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<noteq> ((l div c)) * 0)" by (simp add: ring_simps)
+ also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<noteq> ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e \<noteq> 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be by simp
@@ -1551,7 +1551,7 @@
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\<exists> (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" by simp
- also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: ring_simps)
+ also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e - j * k = 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e = j * k)" by simp
@@ -1568,7 +1568,7 @@
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\<exists> (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)" by simp
- also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: ring_simps)
+ also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e - j * k = 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e = j * k)" by simp
@@ -1617,7 +1617,7 @@
hence "x + ?e \<ge> 1 \<and> x + ?e \<le> d" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + ?e" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. x = (- ?e + j)"
- by (simp add: ring_simps)
+ by (simp add: algebra_simps)
with nob have ?case by auto}
ultimately show ?case by blast
next
@@ -1636,7 +1636,7 @@
from H p have "x + ?e \<ge> 0 \<and> x + ?e < d" by (simp add: c1)
hence "x + ?e +1 \<ge> 1 \<and> x + ?e + 1 \<le> d" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + ?e + 1" by simp
- hence "\<exists> (j::int) \<in> {1 .. d}. x= - ?e - 1 + j" by (simp add: ring_simps)
+ hence "\<exists> (j::int) \<in> {1 .. d}. x= - ?e - 1 + j" by (simp add: algebra_simps)
with nob have ?case by simp }
ultimately show ?case by blast
next
@@ -1646,7 +1646,7 @@
have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
from p have "x= - ?e" by (simp add: c1) with prems(11) show ?case using dp
by simp (erule ballE[where x="1"],
- simp_all add:ring_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"])
+ simp_all add:algebra_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"])
next
case (4 c e)hence p: "Ifm bbs (x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
let ?e = "Inum (x # bs) e"
--- a/src/HOL/ex/ReflectionEx.thy Sun Jan 18 13:58:17 2009 +0100
+++ b/src/HOL/ex/ReflectionEx.thy Wed Jan 28 16:29:16 2009 +0100
@@ -183,7 +183,7 @@
lemma lin_add: "Inum (lin_add (t,s)) bs = Inum (Add t s) bs"
apply (induct t s rule: lin_add.induct, simp_all add: Let_def)
apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
-by (case_tac "n1 = n2", simp_all add: ring_simps)
+by (case_tac "n1 = n2", simp_all add: algebra_simps)
consts lin_mul :: "num \<Rightarrow> nat \<Rightarrow> num"
recdef lin_mul "measure size "
@@ -192,7 +192,7 @@
"lin_mul t = (\<lambda> i. Mul i t)"
lemma lin_mul: "Inum (lin_mul t i) bs = Inum (Mul i t) bs"
-by (induct t arbitrary: i rule: lin_mul.induct, auto simp add: ring_simps)
+by (induct t arbitrary: i rule: lin_mul.induct, auto simp add: algebra_simps)
consts linum:: "num \<Rightarrow> num"
recdef linum "measure num_size"