src/HOL/Fields.thy
author haftmann
Mon, 26 Apr 2010 15:37:50 +0200
changeset 36409 d323e7773aa8
parent 36348 89c54f51f55a
child 36414 a19ba9bbc8dc
permissions -rw-r--r--
use new classes (linordered_)field_inverse_zero
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(*  Title:      HOL/Fields.thy
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    Author:     Gertrud Bauer
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    Author:     Steven Obua
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Jeremy Avigad
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*)
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header {* Fields *}
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theory Fields
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imports Rings
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begin
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class field = comm_ring_1 + inverse +
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  assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
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  assumes field_divide_inverse: "a / b = a * inverse b"
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begin
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subclass division_ring
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proof
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  fix a :: 'a
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  assume "a \<noteq> 0"
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  thus "inverse a * a = 1" by (rule field_inverse)
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  thus "a * inverse a = 1" by (simp only: mult_commute)
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next
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  fix a b :: 'a
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  show "a / b = a * inverse b" by (rule field_divide_inverse)
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qed
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subclass idom ..
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text{*There is no slick version using division by zero.*}
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lemma inverse_add:
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  "[| a \<noteq> 0;  b \<noteq> 0 |]
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   ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
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by (simp add: division_ring_inverse_add mult_ac)
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lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]:
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assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
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proof -
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  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
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    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
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  also have "... =  a * inverse b * (inverse c * c)"
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    by (simp only: mult_ac)
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  also have "... =  a * inverse b" by simp
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    finally show ?thesis by (simp add: divide_inverse)
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qed
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lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
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by (simp add: mult_commute [of _ c])
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lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
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  by (simp add: divide_inverse mult_ac)
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text {* These are later declared as simp rules. *}
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lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left
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lemma add_frac_eq:
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  assumes "y \<noteq> 0" and "z \<noteq> 0"
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  shows "x / y + w / z = (x * z + w * y) / (y * z)"
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proof -
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  have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
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    using assms by simp
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  also have "\<dots> = (x * z + y * w) / (y * z)"
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    by (simp only: add_divide_distrib)
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  finally show ?thesis
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    by (simp only: mult_commute)
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qed
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text{*Special Cancellation Simprules for Division*}
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lemma nonzero_mult_divide_cancel_right [simp, no_atp]:
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  "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
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  using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
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lemma nonzero_mult_divide_cancel_left [simp, no_atp]:
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  "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
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using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
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lemma nonzero_divide_mult_cancel_right [simp, no_atp]:
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  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
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using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
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lemma nonzero_divide_mult_cancel_left [simp, no_atp]:
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  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
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using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
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lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
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using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
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lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
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using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
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lemma add_divide_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x + y / z = (z * x + y) / z"
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  by (simp add: add_divide_distrib)
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lemma divide_add_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x / z + y = (x + z * y) / z"
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  by (simp add: add_divide_distrib)
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lemma diff_divide_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x - y / z = (z * x - y) / z"
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  by (simp add: diff_divide_distrib)
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lemma divide_diff_eq_iff [field_simps]:
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  "z \<noteq> 0 \<Longrightarrow> x / z - y = (x - z * y) / z"
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  by (simp add: diff_divide_distrib)
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lemma diff_frac_eq:
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  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
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  by (simp add: field_simps)
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lemma frac_eq_eq:
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  "y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
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  by (simp add: field_simps)
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end
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class field_inverse_zero = field +
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  assumes field_inverse_zero: "inverse 0 = 0"
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begin
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subclass division_ring_inverse_zero proof
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qed (fact field_inverse_zero)
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text{*This version builds in division by zero while also re-orienting
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      the right-hand side.*}
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lemma inverse_mult_distrib [simp]:
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  "inverse (a * b) = inverse a * inverse b"
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proof cases
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  assume "a \<noteq> 0 & b \<noteq> 0" 
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  thus ?thesis by (simp add: nonzero_inverse_mult_distrib mult_ac)
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next
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  assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
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  thus ?thesis by force
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qed
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lemma inverse_divide [simp]:
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  "inverse (a / b) = b / a"
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  by (simp add: divide_inverse mult_commute)
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text {* Calculations with fractions *}
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text{* There is a whole bunch of simp-rules just for class @{text
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field} but none for class @{text field} and @{text nonzero_divides}
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because the latter are covered by a simproc. *}
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lemma mult_divide_mult_cancel_left:
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  "c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
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apply (cases "b = 0")
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apply simp_all
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done
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lemma mult_divide_mult_cancel_right:
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  "c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
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apply (cases "b = 0")
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apply simp_all
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done
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lemma divide_divide_eq_right [simp, no_atp]:
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  "a / (b / c) = (a * c) / b"
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  by (simp add: divide_inverse mult_ac)
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lemma divide_divide_eq_left [simp, no_atp]:
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  "(a / b) / c = a / (b * c)"
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  by (simp add: divide_inverse mult_assoc)
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text {*Special Cancellation Simprules for Division*}
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lemma mult_divide_mult_cancel_left_if [simp,no_atp]:
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  shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
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  by (simp add: mult_divide_mult_cancel_left)
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text {* Division and Unary Minus *}
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lemma minus_divide_right:
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  "- (a / b) = a / - b"
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  by (simp add: divide_inverse)
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lemma divide_minus_right [simp, no_atp]:
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  "a / - b = - (a / b)"
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  by (simp add: divide_inverse)
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lemma minus_divide_divide:
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  "(- a) / (- b) = a / b"
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apply (cases "b=0", simp) 
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apply (simp add: nonzero_minus_divide_divide) 
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done
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lemma eq_divide_eq:
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  "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
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  by (simp add: nonzero_eq_divide_eq)
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lemma divide_eq_eq:
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  "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
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  by (force simp add: nonzero_divide_eq_eq)
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lemma inverse_eq_1_iff [simp]:
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  "inverse x = 1 \<longleftrightarrow> x = 1"
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  by (insert inverse_eq_iff_eq [of x 1], simp) 
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lemma divide_eq_0_iff [simp, no_atp]:
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  "a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
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  by (simp add: divide_inverse)
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lemma divide_cancel_right [simp, no_atp]:
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  "a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
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  apply (cases "c=0", simp)
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  apply (simp add: divide_inverse)
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  done
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lemma divide_cancel_left [simp, no_atp]:
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  "c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b" 
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  apply (cases "c=0", simp)
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  apply (simp add: divide_inverse)
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  done
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lemma divide_eq_1_iff [simp, no_atp]:
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  "a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
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  apply (cases "b=0", simp)
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  apply (simp add: right_inverse_eq)
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  done
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lemma one_eq_divide_iff [simp, no_atp]:
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  "1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
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  by (simp add: eq_commute [of 1])
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end
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text {* Ordered Fields *}
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class linordered_field = field + linordered_idom
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begin
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lemma positive_imp_inverse_positive: 
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  assumes a_gt_0: "0 < a" 
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  shows "0 < inverse a"
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   248
proof -
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  have "0 < a * inverse a" 
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    by (simp add: a_gt_0 [THEN less_imp_not_eq2])
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  thus "0 < inverse a" 
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    by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
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qed
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lemma negative_imp_inverse_negative:
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  "a < 0 \<Longrightarrow> inverse a < 0"
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  by (insert positive_imp_inverse_positive [of "-a"], 
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    simp add: nonzero_inverse_minus_eq less_imp_not_eq)
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   259
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lemma inverse_le_imp_le:
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  assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
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  shows "b \<le> a"
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   263
proof (rule classical)
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   264
  assume "~ b \<le> a"
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   265
  hence "a < b"  by (simp add: linorder_not_le)
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  hence bpos: "0 < b"  by (blast intro: apos less_trans)
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  hence "a * inverse a \<le> a * inverse b"
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    by (simp add: apos invle less_imp_le mult_left_mono)
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  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
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    by (simp add: bpos less_imp_le mult_right_mono)
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  thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
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qed
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lemma inverse_positive_imp_positive:
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  assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
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  shows "0 < a"
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   277
proof -
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  have "0 < inverse (inverse a)"
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   279
    using inv_gt_0 by (rule positive_imp_inverse_positive)
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parents: 14272
diff changeset
   280
  thus "0 < a"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   281
    using nz by (simp add: nonzero_inverse_inverse_eq)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   282
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   283
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   284
lemma inverse_negative_imp_negative:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   285
  assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   286
  shows "a < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   287
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   288
  have "inverse (inverse a) < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   289
    using inv_less_0 by (rule negative_imp_inverse_negative)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   290
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   291
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   292
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   293
lemma linordered_field_no_lb:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   294
  "\<forall>x. \<exists>y. y < x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   295
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   296
  fix x::'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   297
  have m1: "- (1::'a) < 0" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   298
  from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   299
  have "(- 1) + x < x" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   300
  thus "\<exists>y. y < x" by blast
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   301
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   302
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   303
lemma linordered_field_no_ub:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   304
  "\<forall> x. \<exists>y. y > x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   305
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   306
  fix x::'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   307
  have m1: " (1::'a) > 0" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   308
  from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   309
  have "1 + x > x" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   310
  thus "\<exists>y. y > x" by blast
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   311
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   312
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   313
lemma less_imp_inverse_less:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   314
  assumes less: "a < b" and apos:  "0 < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   315
  shows "inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   316
proof (rule ccontr)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   317
  assume "~ inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   318
  hence "inverse a \<le> inverse b" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   319
  hence "~ (a < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   320
    by (simp add: not_less inverse_le_imp_le [OF _ apos])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   321
  thus False by (rule notE [OF _ less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   322
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   323
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   324
lemma inverse_less_imp_less:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   325
  "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   326
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   327
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   328
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   329
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   330
text{*Both premises are essential. Consider -1 and 1.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   331
lemma inverse_less_iff_less [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   332
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   333
  by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   334
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   335
lemma le_imp_inverse_le:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   336
  "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   337
  by (force simp add: le_less less_imp_inverse_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   338
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   339
lemma inverse_le_iff_le [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   340
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   341
  by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   342
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   343
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   344
text{*These results refer to both operands being negative.  The opposite-sign
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   345
case is trivial, since inverse preserves signs.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   346
lemma inverse_le_imp_le_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   347
  "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   348
apply (rule classical) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   349
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   350
 prefer 2 apply force
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   351
apply (insert inverse_le_imp_le [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   352
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   353
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   354
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   355
lemma less_imp_inverse_less_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   356
   "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   357
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   358
 prefer 2 apply (blast intro: less_trans) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   359
apply (insert less_imp_inverse_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   360
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   361
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   362
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   363
lemma inverse_less_imp_less_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   364
   "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   365
apply (rule classical) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   366
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   367
 prefer 2
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   368
 apply force
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   369
apply (insert inverse_less_imp_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   370
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   371
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   372
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   373
lemma inverse_less_iff_less_neg [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   374
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   375
apply (insert inverse_less_iff_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   376
apply (simp del: inverse_less_iff_less 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   377
            add: nonzero_inverse_minus_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   378
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   379
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   380
lemma le_imp_inverse_le_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   381
  "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   382
  by (force simp add: le_less less_imp_inverse_less_neg)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   383
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   384
lemma inverse_le_iff_le_neg [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   385
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   386
  by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   387
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   388
lemma pos_le_divide_eq [field_simps]: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   389
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   390
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   391
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   392
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   393
  also have "... = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   394
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   395
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   396
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   397
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   398
lemma neg_le_divide_eq [field_simps]: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   399
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   400
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   401
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   402
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   403
  also have "... = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   404
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   405
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   406
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   407
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   408
lemma pos_less_divide_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   409
     "0 < c ==> (a < b/c) = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   410
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   411
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   412
  hence "(a < b/c) = (a*c < (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   413
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   414
  also have "... = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   415
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   416
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   417
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   418
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   419
lemma neg_less_divide_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   420
 "c < 0 ==> (a < b/c) = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   421
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   422
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   423
  hence "(a < b/c) = ((b/c)*c < a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   424
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   425
  also have "... = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   426
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   427
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   428
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   429
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   430
lemma pos_divide_less_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   431
     "0 < c ==> (b/c < a) = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   432
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   433
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   434
  hence "(b/c < a) = ((b/c)*c < a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   435
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   436
  also have "... = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   437
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   438
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   439
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   440
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   441
lemma neg_divide_less_eq [field_simps]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   442
 "c < 0 ==> (b/c < a) = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   443
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   444
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   445
  hence "(b/c < a) = (a*c < (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   446
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   447
  also have "... = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   448
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   449
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   450
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   451
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   452
lemma pos_divide_le_eq [field_simps]: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   453
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   454
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   455
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   456
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   457
  also have "... = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   458
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   459
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   460
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   461
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   462
lemma neg_divide_le_eq [field_simps]: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   463
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   464
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   465
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
36304
6984744e6b34 less special treatment of times_divide_eq [simp]
haftmann
parents: 36301
diff changeset
   466
    by (simp add: mult_le_cancel_right less_not_sym [OF less] del: times_divide_eq)
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   467
  also have "... = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   468
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   469
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   470
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   471
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   472
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   473
of positivity/negativity needed for @{text field_simps}. Have not added @{text
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   474
sign_simps} to @{text field_simps} because the former can lead to case
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   475
explosions. *}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   476
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   477
lemmas sign_simps [no_atp] = algebra_simps
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   478
  zero_less_mult_iff mult_less_0_iff
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   479
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   480
lemmas (in -) sign_simps [no_atp] = algebra_simps
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   481
  zero_less_mult_iff mult_less_0_iff
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   482
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   483
(* Only works once linear arithmetic is installed:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   484
text{*An example:*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   485
lemma fixes a b c d e f :: "'a::linordered_field"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   486
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   487
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   488
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   489
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   490
 prefer 2 apply(simp add:sign_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   491
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   492
 prefer 2 apply(simp add:sign_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   493
apply(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   494
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   495
*)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   496
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   497
lemma divide_pos_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   498
  "0 < x ==> 0 < y ==> 0 < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   499
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   500
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   501
lemma divide_nonneg_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   502
  "0 <= x ==> 0 < y ==> 0 <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   503
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   504
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   505
lemma divide_neg_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   506
  "x < 0 ==> 0 < y ==> x / y < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   507
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   508
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   509
lemma divide_nonpos_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   510
  "x <= 0 ==> 0 < y ==> x / y <= 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   511
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   512
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   513
lemma divide_pos_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   514
  "0 < x ==> y < 0 ==> x / y < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   515
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   516
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   517
lemma divide_nonneg_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   518
  "0 <= x ==> y < 0 ==> x / y <= 0" 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   519
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   520
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   521
lemma divide_neg_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   522
  "x < 0 ==> y < 0 ==> 0 < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   523
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   524
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   525
lemma divide_nonpos_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   526
  "x <= 0 ==> y < 0 ==> 0 <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   527
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   528
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   529
lemma divide_strict_right_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   530
     "[|a < b; 0 < c|] ==> a / c < b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   531
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   532
              positive_imp_inverse_positive)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   533
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   534
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   535
lemma divide_strict_right_mono_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   536
     "[|b < a; c < 0|] ==> a / c < b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   537
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   538
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   539
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   540
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   541
text{*The last premise ensures that @{term a} and @{term b} 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   542
      have the same sign*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   543
lemma divide_strict_left_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   544
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   545
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   546
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   547
lemma divide_left_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   548
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   549
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   550
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   551
lemma divide_strict_left_mono_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   552
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   553
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   554
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   555
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   556
    x / y <= z"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   557
by (subst pos_divide_le_eq, assumption+)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   558
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   559
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   560
    z <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   561
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   562
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   563
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   564
    x / y < z"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   565
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   566
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   567
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   568
    z < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   569
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   570
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   571
lemma frac_le: "0 <= x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   572
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   573
  apply (rule mult_imp_div_pos_le)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   574
  apply simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   575
  apply (subst times_divide_eq_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   576
  apply (rule mult_imp_le_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   577
  apply (rule mult_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   578
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   579
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   580
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   581
lemma frac_less: "0 <= x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   582
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   583
  apply (rule mult_imp_div_pos_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   584
  apply simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   585
  apply (subst times_divide_eq_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   586
  apply (rule mult_imp_less_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   587
  apply (erule mult_less_le_imp_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   588
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   589
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   590
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   591
lemma frac_less2: "0 < x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   592
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   593
  apply (rule mult_imp_div_pos_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   594
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   595
  apply (rule mult_imp_less_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   596
  apply (erule mult_le_less_imp_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   597
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   598
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   599
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   600
text{*It's not obvious whether these should be simprules or not. 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   601
  Their effect is to gather terms into one big fraction, like
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   602
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   603
  seem to need them.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   604
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   605
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   606
by (simp add: field_simps zero_less_two)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   607
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   608
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   609
by (simp add: field_simps zero_less_two)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   610
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   611
subclass dense_linorder
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   612
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   613
  fix x y :: 'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   614
  from less_add_one show "\<exists>y. x < y" .. 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   615
  from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   616
  then have "x - 1 < x + 1 - 1" by (simp only: diff_minus [symmetric])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   617
  then have "x - 1 < x" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   618
  then show "\<exists>y. y < x" ..
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   619
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   620
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   621
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   622
lemma nonzero_abs_inverse:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   623
     "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   624
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   625
                      negative_imp_inverse_negative)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   626
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   627
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   628
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   629
lemma nonzero_abs_divide:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   630
     "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   631
  by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   632
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   633
lemma field_le_epsilon:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   634
  assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   635
  shows "x \<le> y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   636
proof (rule dense_le)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   637
  fix t assume "t < x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   638
  hence "0 < x - t" by (simp add: less_diff_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   639
  from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   640
  then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   641
  then show "t \<le> y" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   642
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   643
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   644
end
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   645
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   646
class linordered_field_inverse_zero = linordered_field +
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   647
  assumes linordered_field_inverse_zero: "inverse 0 = 0"
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   648
begin
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   649
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   650
subclass field_inverse_zero proof
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   651
qed (fact linordered_field_inverse_zero)
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
   652
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   653
lemma le_divide_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   654
  "(a \<le> b/c) = 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   655
   (if 0 < c then a*c \<le> b
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   656
             else if c < 0 then b \<le> a*c
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   657
             else  a \<le> 0)"
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   658
apply (cases "c=0", simp) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   659
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   660
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   661
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   662
lemma inverse_positive_iff_positive [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   663
  "(0 < inverse a) = (0 < a)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   664
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   665
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   666
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   667
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   668
lemma inverse_negative_iff_negative [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   669
  "(inverse a < 0) = (a < 0)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   670
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   671
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   672
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   673
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   674
lemma inverse_nonnegative_iff_nonnegative [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   675
  "0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   676
  by (simp add: not_less [symmetric])
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   677
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   678
lemma inverse_nonpositive_iff_nonpositive [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   679
  "inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   680
  by (simp add: not_less [symmetric])
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   681
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   682
lemma one_less_inverse_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   683
  "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   684
proof cases
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   685
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   686
    with inverse_less_iff_less [OF zero_less_one, of x]
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   687
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   688
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   689
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   690
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   691
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   692
    assume "1 < inverse x"
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   693
    also with notless have "... \<le> 0" by simp
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   694
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   695
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   696
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   697
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   698
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   699
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   700
lemma one_le_inverse_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   701
  "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   702
proof (cases "x = 1")
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   703
  case True then show ?thesis by simp
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   704
next
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   705
  case False then have "inverse x \<noteq> 1" by simp
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   706
  then have "1 \<noteq> inverse x" by blast
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   707
  then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   708
  with False show ?thesis by (auto simp add: one_less_inverse_iff)
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   709
qed
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   710
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   711
lemma inverse_less_1_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   712
  "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   713
  by (simp add: not_le [symmetric] one_le_inverse_iff) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   714
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   715
lemma inverse_le_1_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   716
  "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   717
  by (simp add: not_less [symmetric] one_less_inverse_iff) 
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   718
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   719
lemma divide_le_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   720
  "(b/c \<le> a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   721
   (if 0 < c then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   722
             else if c < 0 then a*c \<le> b
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   723
             else 0 \<le> a)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   724
apply (cases "c=0", simp) 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   725
apply (force simp add: pos_divide_le_eq neg_divide_le_eq) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   726
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   727
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   728
lemma less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   729
  "(a < b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   730
   (if 0 < c then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   731
             else if c < 0 then b < a*c
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   732
             else  a < 0)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   733
apply (cases "c=0", simp) 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   734
apply (force simp add: pos_less_divide_eq neg_less_divide_eq) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   735
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   736
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   737
lemma divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   738
  "(b/c < a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   739
   (if 0 < c then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   740
             else if c < 0 then a*c < b
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   741
             else 0 < a)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   742
apply (cases "c=0", simp) 
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   743
apply (force simp add: pos_divide_less_eq neg_divide_less_eq)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   744
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   745
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   746
text {*Division and Signs*}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   747
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   748
lemma zero_less_divide_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   749
     "(0 < a/b) = (0 < a & 0 < b | a < 0 & b < 0)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   750
by (simp add: divide_inverse zero_less_mult_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   751
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   752
lemma divide_less_0_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   753
     "(a/b < 0) = 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   754
      (0 < a & b < 0 | a < 0 & 0 < b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   755
by (simp add: divide_inverse mult_less_0_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   756
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   757
lemma zero_le_divide_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   758
     "(0 \<le> a/b) =
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   759
      (0 \<le> a & 0 \<le> b | a \<le> 0 & b \<le> 0)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   760
by (simp add: divide_inverse zero_le_mult_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   761
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   762
lemma divide_le_0_iff:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   763
     "(a/b \<le> 0) =
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   764
      (0 \<le> a & b \<le> 0 | a \<le> 0 & 0 \<le> b)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   765
by (simp add: divide_inverse mult_le_0_iff)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   766
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   767
text {* Division and the Number One *}
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   768
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   769
text{*Simplify expressions equated with 1*}
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   770
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   771
lemma zero_eq_1_divide_iff [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   772
     "(0 = 1/a) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   773
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   774
apply (auto simp add: nonzero_eq_divide_eq)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   775
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   776
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   777
lemma one_divide_eq_0_iff [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   778
     "(1/a = 0) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   779
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   780
apply (insert zero_neq_one [THEN not_sym])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   781
apply (auto simp add: nonzero_divide_eq_eq)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   782
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   783
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   784
text{*Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}*}
18623
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   785
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   786
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   787
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   788
lemmas divide_le_0_1_iff = divide_le_0_iff [of 1, simplified]
17085
5b57f995a179 more simprules now have names
paulson
parents: 16775
diff changeset
   789
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   790
declare zero_less_divide_1_iff [simp,no_atp]
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   791
declare divide_less_0_1_iff [simp,no_atp]
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   792
declare zero_le_divide_1_iff [simp,no_atp]
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   793
declare divide_le_0_1_iff [simp,no_atp]
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   794
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   795
lemma divide_right_mono:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   796
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   797
by (force simp add: divide_strict_right_mono le_less)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   798
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   799
lemma divide_right_mono_neg: "a <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   800
    ==> c <= 0 ==> b / c <= a / c"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   801
apply (drule divide_right_mono [of _ _ "- c"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   802
apply auto
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   803
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   804
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   805
lemma divide_left_mono_neg: "a <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   806
    ==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   807
  apply (drule divide_left_mono [of _ _ "- c"])
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   808
  apply (auto simp add: mult_commute)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   809
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   810
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   811
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   812
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   813
text{*Simplify quotients that are compared with the value 1.*}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   814
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   815
lemma le_divide_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   816
  "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   817
by (auto simp add: le_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   818
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   819
lemma divide_le_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   820
  "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   821
by (auto simp add: divide_le_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   822
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   823
lemma less_divide_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   824
  "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   825
by (auto simp add: less_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   826
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   827
lemma divide_less_eq_1 [no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   828
  "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   829
by (auto simp add: divide_less_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   830
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   831
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   832
text {*Conditional Simplification Rules: No Case Splits*}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   833
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   834
lemma le_divide_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   835
  "0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   836
by (auto simp add: le_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   837
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   838
lemma le_divide_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   839
  "a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   840
by (auto simp add: le_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   841
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   842
lemma divide_le_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   843
  "0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   844
by (auto simp add: divide_le_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   845
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   846
lemma divide_le_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   847
  "a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   848
by (auto simp add: divide_le_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   849
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   850
lemma less_divide_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   851
  "0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   852
by (auto simp add: less_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   853
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   854
lemma less_divide_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   855
  "a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   856
by (auto simp add: less_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   857
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   858
lemma divide_less_eq_1_pos [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   859
  "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   860
by (auto simp add: divide_less_eq)
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   861
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   862
lemma divide_less_eq_1_neg [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   863
  "a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   864
by (auto simp add: divide_less_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   865
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   866
lemma eq_divide_eq_1 [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   867
  "(1 = b/a) = ((a \<noteq> 0 & a = b))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   868
by (auto simp add: eq_divide_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   869
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   870
lemma divide_eq_eq_1 [simp,no_atp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   871
  "(b/a = 1) = ((a \<noteq> 0 & a = b))"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   872
by (auto simp add: divide_eq_eq)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   873
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   874
lemma abs_inverse [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   875
     "\<bar>inverse a\<bar> = 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   876
      inverse \<bar>a\<bar>"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   877
apply (cases "a=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   878
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   879
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   880
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   881
lemma abs_divide [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   882
     "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   883
apply (cases "b=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   884
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   885
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   886
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   887
lemma abs_div_pos: "0 < y ==> 
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   888
    \<bar>x\<bar> / y = \<bar>x / y\<bar>"
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   889
  apply (subst abs_divide)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   890
  apply (simp add: order_less_imp_le)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   891
done
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   892
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   893
lemma field_le_mult_one_interval:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   894
  assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   895
  shows "x \<le> y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   896
proof (cases "0 < x")
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   897
  assume "0 < x"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   898
  thus ?thesis
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   899
    using dense_le_bounded[of 0 1 "y/x"] *
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   900
    unfolding le_divide_eq if_P[OF `0 < x`] by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   901
next
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   902
  assume "\<not>0 < x" hence "x \<le> 0" by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   903
  obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   904
  hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   905
  also note *[OF s]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   906
  finally show ?thesis .
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   907
qed
35090
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
   908
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   909
end
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36348
diff changeset
   910
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   911
code_modulename SML
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
   912
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   913
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   914
code_modulename OCaml
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
   915
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   916
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   917
code_modulename Haskell
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
   918
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   919
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   920
end