--- a/src/HOL/Ring_and_Field.thy Wed Jan 28 06:03:46 2009 -0800
+++ b/src/HOL/Ring_and_Field.thy Wed Jan 28 16:57:12 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)
lemma abs_sgn: "abs k = k * sgn k"
unfolding sgn_if abs_if by auto
@@ -1153,12 +1133,10 @@
"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.*}
@@ -1185,10 +1163,8 @@
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:
@@ -1349,14 +1325,14 @@
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
@@ -1478,12 +1454,10 @@
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:
@@ -1714,9 +1688,10 @@
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 *)
@@ -1984,15 +1959,15 @@
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"
@@ -2057,7 +2032,7 @@
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
@@ -2089,7 +2064,7 @@
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:
@@ -2109,14 +2084,14 @@
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]]
@@ -2169,7 +2144,7 @@
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)
@@ -2181,7 +2156,7 @@
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
@@ -2194,10 +2169,10 @@
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)"
@@ -2271,7 +2246,7 @@
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"