--- a/src/HOL/Fields.thy Mon Apr 26 11:34:19 2010 +0200
+++ b/src/HOL/Fields.thy Mon Apr 26 15:37:50 2010 +0200
@@ -129,22 +129,20 @@
subclass division_ring_inverse_zero proof
qed (fact field_inverse_zero)
-end
-
text{*This version builds in division by zero while also re-orienting
the right-hand side.*}
lemma inverse_mult_distrib [simp]:
- "inverse(a*b) = inverse(a) * inverse(b::'a::{field,division_ring_inverse_zero})"
- proof cases
- assume "a \<noteq> 0 & b \<noteq> 0"
- thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)
- next
- assume "~ (a \<noteq> 0 & b \<noteq> 0)"
- thus ?thesis by force
- qed
+ "inverse (a * b) = inverse a * inverse b"
+proof cases
+ assume "a \<noteq> 0 & b \<noteq> 0"
+ thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)
+next
+ assume "~ (a \<noteq> 0 & b \<noteq> 0)"
+ thus ?thesis by force
+qed
lemma inverse_divide [simp]:
- "inverse (a/b) = b / (a::'a::{field,division_ring_inverse_zero})"
+ "inverse (a / b) = b / a"
by (simp add: divide_inverse mult_commute)
@@ -155,86 +153,88 @@
because the latter are covered by a simproc. *}
lemma mult_divide_mult_cancel_left:
- "c\<noteq>0 ==> (c*a) / (c*b) = a / (b::'a::{field,division_ring_inverse_zero})"
+ "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
apply (cases "b = 0")
apply simp_all
done
lemma mult_divide_mult_cancel_right:
- "c\<noteq>0 ==> (a*c) / (b*c) = a / (b::'a::{field,division_ring_inverse_zero})"
+ "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
apply (cases "b = 0")
apply simp_all
done
-lemma divide_divide_eq_right [simp,no_atp]:
- "a / (b/c) = (a*c) / (b::'a::{field,division_ring_inverse_zero})"
-by (simp add: divide_inverse mult_ac)
+lemma divide_divide_eq_right [simp, no_atp]:
+ "a / (b / c) = (a * c) / b"
+ by (simp add: divide_inverse mult_ac)
-lemma divide_divide_eq_left [simp,no_atp]:
- "(a / b) / (c::'a::{field,division_ring_inverse_zero}) = a / (b*c)"
-by (simp add: divide_inverse mult_assoc)
+lemma divide_divide_eq_left [simp, no_atp]:
+ "(a / b) / c = a / (b * c)"
+ by (simp add: divide_inverse mult_assoc)
text {*Special Cancellation Simprules for Division*}
-lemma mult_divide_mult_cancel_left_if[simp,no_atp]:
-fixes c :: "'a :: {field,division_ring_inverse_zero}"
-shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
-by (simp add: mult_divide_mult_cancel_left)
+lemma mult_divide_mult_cancel_left_if [simp,no_atp]:
+ shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
+ by (simp add: mult_divide_mult_cancel_left)
text {* Division and Unary Minus *}
-lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_ring_inverse_zero})"
-by (simp add: divide_inverse)
+lemma minus_divide_right:
+ "- (a / b) = a / - b"
+ by (simp add: divide_inverse)
lemma divide_minus_right [simp, no_atp]:
- "a / -(b::'a::{field,division_ring_inverse_zero}) = -(a / b)"
-by (simp add: divide_inverse)
+ "a / - b = - (a / b)"
+ by (simp add: divide_inverse)
lemma minus_divide_divide:
- "(-a)/(-b) = a / (b::'a::{field,division_ring_inverse_zero})"
+ "(- a) / (- b) = a / b"
apply (cases "b=0", simp)
apply (simp add: nonzero_minus_divide_divide)
done
lemma eq_divide_eq:
- "((a::'a::{field,division_ring_inverse_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
-by (simp add: nonzero_eq_divide_eq)
+ "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
+ by (simp add: nonzero_eq_divide_eq)
lemma divide_eq_eq:
- "(b/c = (a::'a::{field,division_ring_inverse_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
-by (force simp add: nonzero_divide_eq_eq)
+ "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
+ by (force simp add: nonzero_divide_eq_eq)
lemma inverse_eq_1_iff [simp]:
- "(inverse x = 1) = (x = (1::'a::{field,division_ring_inverse_zero}))"
-by (insert inverse_eq_iff_eq [of x 1], simp)
+ "inverse x = 1 \<longleftrightarrow> x = 1"
+ by (insert inverse_eq_iff_eq [of x 1], simp)
-lemma divide_eq_0_iff [simp,no_atp]:
- "(a/b = 0) = (a=0 | b=(0::'a::{field,division_ring_inverse_zero}))"
-by (simp add: divide_inverse)
+lemma divide_eq_0_iff [simp, no_atp]:
+ "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
+ by (simp add: divide_inverse)
-lemma divide_cancel_right [simp,no_atp]:
- "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_ring_inverse_zero}))"
-apply (cases "c=0", simp)
-apply (simp add: divide_inverse)
-done
+lemma divide_cancel_right [simp, no_atp]:
+ "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
+ apply (cases "c=0", simp)
+ apply (simp add: divide_inverse)
+ done
-lemma divide_cancel_left [simp,no_atp]:
- "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_ring_inverse_zero}))"
-apply (cases "c=0", simp)
-apply (simp add: divide_inverse)
-done
+lemma divide_cancel_left [simp, no_atp]:
+ "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b"
+ apply (cases "c=0", simp)
+ apply (simp add: divide_inverse)
+ done
-lemma divide_eq_1_iff [simp,no_atp]:
- "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_ring_inverse_zero}))"
-apply (cases "b=0", simp)
-apply (simp add: right_inverse_eq)
-done
+lemma divide_eq_1_iff [simp, no_atp]:
+ "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
+ apply (cases "b=0", simp)
+ apply (simp add: right_inverse_eq)
+ done
-lemma one_eq_divide_iff [simp,no_atp]:
- "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_ring_inverse_zero}))"
-by (simp add: eq_commute [of 1])
+lemma one_eq_divide_iff [simp, no_atp]:
+ "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
+ by (simp add: eq_commute [of 1])
+
+end
text {* Ordered Fields *}
@@ -650,39 +650,37 @@
subclass field_inverse_zero proof
qed (fact linordered_field_inverse_zero)
-end
-
lemma le_divide_eq:
"(a \<le> b/c) =
(if 0 < c then a*c \<le> b
else if c < 0 then b \<le> a*c
- else a \<le> (0::'a::{linordered_field,division_ring_inverse_zero}))"
+ else a \<le> 0)"
apply (cases "c=0", simp)
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff)
done
lemma inverse_positive_iff_positive [simp]:
- "(0 < inverse a) = (0 < (a::'a::{linordered_field,division_ring_inverse_zero}))"
+ "(0 < inverse a) = (0 < a)"
apply (cases "a = 0", simp)
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
done
lemma inverse_negative_iff_negative [simp]:
- "(inverse a < 0) = (a < (0::'a::{linordered_field,division_ring_inverse_zero}))"
+ "(inverse a < 0) = (a < 0)"
apply (cases "a = 0", simp)
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
done
lemma inverse_nonnegative_iff_nonnegative [simp]:
- "(0 \<le> inverse a) = (0 \<le> (a::'a::{linordered_field,division_ring_inverse_zero}))"
-by (simp add: linorder_not_less [symmetric])
+ "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
+ by (simp add: not_less [symmetric])
lemma inverse_nonpositive_iff_nonpositive [simp]:
- "(inverse a \<le> 0) = (a \<le> (0::'a::{linordered_field,division_ring_inverse_zero}))"
-by (simp add: linorder_not_less [symmetric])
+ "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
+ by (simp add: not_less [symmetric])
lemma one_less_inverse_iff:
- "(1 < inverse x) = (0 < x & x < (1::'a::{linordered_field,division_ring_inverse_zero}))"
+ "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
proof cases
assume "0 < x"
with inverse_less_iff_less [OF zero_less_one, of x]
@@ -692,7 +690,7 @@
have "~ (1 < inverse x)"
proof
assume "1 < inverse x"
- also with notless have "... \<le> 0" by (simp add: linorder_not_less)
+ also with notless have "... \<le> 0" by simp
also have "... < 1" by (rule zero_less_one)
finally show False by auto
qed
@@ -700,62 +698,69 @@
qed
lemma one_le_inverse_iff:
- "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{linordered_field,division_ring_inverse_zero}))"
-by (force simp add: order_le_less one_less_inverse_iff)
+ "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
+proof (cases "x = 1")
+ case True then show ?thesis by simp
+next
+ case False then have "inverse x \<noteq> 1" by simp
+ then have "1 \<noteq> inverse x" by blast
+ then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
+ with False show ?thesis by (auto simp add: one_less_inverse_iff)
+qed
lemma inverse_less_1_iff:
- "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{linordered_field,division_ring_inverse_zero}))"
-by (simp add: linorder_not_le [symmetric] one_le_inverse_iff)
+ "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
+ by (simp add: not_le [symmetric] one_le_inverse_iff)
lemma inverse_le_1_iff:
- "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{linordered_field,division_ring_inverse_zero}))"
-by (simp add: linorder_not_less [symmetric] one_less_inverse_iff)
+ "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
+ by (simp add: not_less [symmetric] one_less_inverse_iff)
lemma divide_le_eq:
"(b/c \<le> a) =
(if 0 < c then b \<le> a*c
else if c < 0 then a*c \<le> b
- else 0 \<le> (a::'a::{linordered_field,division_ring_inverse_zero}))"
+ else 0 \<le> a)"
apply (cases "c=0", simp)
-apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff)
+apply (force simp add: pos_divide_le_eq neg_divide_le_eq)
done
lemma less_divide_eq:
"(a < b/c) =
(if 0 < c then a*c < b
else if c < 0 then b < a*c
- else a < (0::'a::{linordered_field,division_ring_inverse_zero}))"
+ else a < 0)"
apply (cases "c=0", simp)
-apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff)
+apply (force simp add: pos_less_divide_eq neg_less_divide_eq)
done
lemma divide_less_eq:
"(b/c < a) =
(if 0 < c then b < a*c
else if c < 0 then a*c < b
- else 0 < (a::'a::{linordered_field,division_ring_inverse_zero}))"
+ else 0 < a)"
apply (cases "c=0", simp)
-apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff)
+apply (force simp add: pos_divide_less_eq neg_divide_less_eq)
done
text {*Division and Signs*}
lemma zero_less_divide_iff:
- "((0::'a::{linordered_field,division_ring_inverse_zero}) < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
+ "(0 < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
by (simp add: divide_inverse zero_less_mult_iff)
lemma divide_less_0_iff:
- "(a/b < (0::'a::{linordered_field,division_ring_inverse_zero})) =
+ "(a/b < 0) =
(0 < a & b < 0 | a < 0 & 0 < b)"
by (simp add: divide_inverse mult_less_0_iff)
lemma zero_le_divide_iff:
- "((0::'a::{linordered_field,division_ring_inverse_zero}) \<le> a/b) =
+ "(0 \<le> a/b) =
(0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
by (simp add: divide_inverse zero_le_mult_iff)
lemma divide_le_0_iff:
- "(a/b \<le> (0::'a::{linordered_field,division_ring_inverse_zero})) =
+ "(a/b \<le> 0) =
(0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
by (simp add: divide_inverse mult_le_0_iff)
@@ -764,13 +769,13 @@
text{*Simplify expressions equated with 1*}
lemma zero_eq_1_divide_iff [simp,no_atp]:
- "((0::'a::{linordered_field,division_ring_inverse_zero}) = 1/a) = (a = 0)"
+ "(0 = 1/a) = (a = 0)"
apply (cases "a=0", simp)
apply (auto simp add: nonzero_eq_divide_eq)
done
lemma one_divide_eq_0_iff [simp,no_atp]:
- "(1/a = (0::'a::{linordered_field,division_ring_inverse_zero})) = (a = 0)"
+ "(1/a = 0) = (a = 0)"
apply (cases "a=0", simp)
apply (insert zero_neq_one [THEN not_sym])
apply (auto simp add: nonzero_divide_eq_eq)
@@ -788,16 +793,16 @@
declare divide_le_0_1_iff [simp,no_atp]
lemma divide_right_mono:
- "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{linordered_field,division_ring_inverse_zero})"
-by (force simp add: divide_strict_right_mono order_le_less)
+ "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
+by (force simp add: divide_strict_right_mono le_less)
-lemma divide_right_mono_neg: "(a::'a::{linordered_field,division_ring_inverse_zero}) <= b
+lemma divide_right_mono_neg: "a <= b
==> c <= 0 ==> b / c <= a / c"
apply (drule divide_right_mono [of _ _ "- c"])
apply auto
done
-lemma divide_left_mono_neg: "(a::'a::{linordered_field,division_ring_inverse_zero}) <= b
+lemma divide_left_mono_neg: "a <= b
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
apply (drule divide_left_mono [of _ _ "- c"])
apply (auto simp add: mult_commute)
@@ -808,99 +813,84 @@
text{*Simplify quotients that are compared with the value 1.*}
lemma le_divide_eq_1 [no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
+ "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
by (auto simp add: le_divide_eq)
lemma divide_le_eq_1 [no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
+ "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
by (auto simp add: divide_le_eq)
lemma less_divide_eq_1 [no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
+ "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
by (auto simp add: less_divide_eq)
lemma divide_less_eq_1 [no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
+ "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
by (auto simp add: divide_less_eq)
text {*Conditional Simplification Rules: No Case Splits*}
lemma le_divide_eq_1_pos [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
+ "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
by (auto simp add: le_divide_eq)
lemma le_divide_eq_1_neg [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
+ "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
by (auto simp add: le_divide_eq)
lemma divide_le_eq_1_pos [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
+ "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
by (auto simp add: divide_le_eq)
lemma divide_le_eq_1_neg [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
+ "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
by (auto simp add: divide_le_eq)
lemma less_divide_eq_1_pos [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
+ "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
by (auto simp add: less_divide_eq)
lemma less_divide_eq_1_neg [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
+ "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
by (auto simp add: less_divide_eq)
lemma divide_less_eq_1_pos [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
+ "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
by (auto simp add: divide_less_eq)
lemma divide_less_eq_1_neg [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
+ "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
by (auto simp add: divide_less_eq)
lemma eq_divide_eq_1 [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "(1 = b/a) = ((a \<noteq> 0 & a = b))"
+ "(1 = b/a) = ((a \<noteq> 0 & a = b))"
by (auto simp add: eq_divide_eq)
lemma divide_eq_eq_1 [simp,no_atp]:
- fixes a :: "'a :: {linordered_field,division_ring_inverse_zero}"
- shows "(b/a = 1) = ((a \<noteq> 0 & a = b))"
+ "(b/a = 1) = ((a \<noteq> 0 & a = b))"
by (auto simp add: divide_eq_eq)
lemma abs_inverse [simp]:
- "\<bar>inverse (a::'a::{linordered_field,division_ring_inverse_zero})\<bar> =
+ "\<bar>inverse a\<bar> =
inverse \<bar>a\<bar>"
apply (cases "a=0", simp)
apply (simp add: nonzero_abs_inverse)
done
lemma abs_divide [simp]:
- "\<bar>a / (b::'a::{linordered_field,division_ring_inverse_zero})\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
+ "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
apply (cases "b=0", simp)
apply (simp add: nonzero_abs_divide)
done
-lemma abs_div_pos: "(0::'a::{linordered_field,division_ring_inverse_zero}) < y ==>
+lemma abs_div_pos: "0 < y ==>
\<bar>x\<bar> / y = \<bar>x / y\<bar>"
apply (subst abs_divide)
apply (simp add: order_less_imp_le)
done
lemma field_le_mult_one_interval:
- fixes x :: "'a\<Colon>{linordered_field,division_ring_inverse_zero}"
assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
shows "x \<le> y"
proof (cases "0 < x")
@@ -916,6 +906,8 @@
finally show ?thesis .
qed
+end
+
code_modulename SML
Fields Arith