src/HOL/Fields.thy
author huffman
Thu, 18 Feb 2010 14:21:44 -0800
changeset 35216 7641e8d831d2
parent 35090 88cc65ae046e
child 35579 cc9a5a0ab5ea
permissions -rw-r--r--
get rid of many duplicate simp rule warnings
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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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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lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
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proof
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  assume neq: "b \<noteq> 0"
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  {
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    hence "a = (a / b) * b" by (simp add: divide_inverse mult_ac)
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    also assume "a / b = 1"
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    finally show "a = b" by simp
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  next
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    assume "a = b"
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    with neq show "a / b = 1" by (simp add: divide_inverse)
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  }
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qed
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lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
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by (simp add: divide_inverse)
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lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
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by (simp add: divide_inverse)
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lemma divide_zero_left [simp]: "0 / a = 0"
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by (simp add: divide_inverse)
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lemma inverse_eq_divide: "inverse a = 1 / a"
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by (simp add: divide_inverse)
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lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
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by (simp add: divide_inverse algebra_simps)
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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, noatp]:
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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, noatp]:
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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 divide_1 [simp]: "a / 1 = a"
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by (simp add: divide_inverse)
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lemma times_divide_eq_right: "a * (b / c) = (a * b) / c"
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by (simp add: divide_inverse mult_assoc)
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lemma times_divide_eq_left: "(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 [noatp] = 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, noatp]:
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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, noatp]:
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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, noatp]:
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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, noatp]:
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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, noatp]:
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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, noatp]:
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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 minus_divide_left: "- (a / b) = (-a) / b"
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by (simp add: divide_inverse)
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lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
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by (simp add: divide_inverse nonzero_inverse_minus_eq)
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lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
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by (simp add: divide_inverse nonzero_inverse_minus_eq)
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lemma divide_minus_left [simp, noatp]: "(-a) / b = - (a / b)"
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by (simp add: divide_inverse)
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lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
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by (simp add: diff_minus add_divide_distrib)
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lemma add_divide_eq_iff:
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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:
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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:
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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:
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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 nonzero_eq_divide_eq: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
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proof -
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  assume [simp]: "c \<noteq> 0"
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  have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
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  also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult_assoc)
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  finally show ?thesis .
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qed
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lemma nonzero_divide_eq_eq: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
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proof -
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  assume [simp]: "c \<noteq> 0"
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  have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
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  also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult_assoc) 
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  finally show ?thesis .
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qed
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lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
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by simp
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lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
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by (erule subst, simp)
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lemmas field_eq_simps[noatp] = algebra_simps
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  (* pull / out*)
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  add_divide_eq_iff divide_add_eq_iff
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  diff_divide_eq_iff divide_diff_eq_iff
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  (* multiply eqn *)
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  nonzero_eq_divide_eq nonzero_divide_eq_eq
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(* is added later:
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  times_divide_eq_left times_divide_eq_right
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*)
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text{*An example:*}
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lemma "\<lbrakk>a\<noteq>b; c\<noteq>d; e\<noteq>f\<rbrakk> \<Longrightarrow> ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) = 1"
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apply(subgoal_tac "(c-d)*(e-f)*(a-b) \<noteq> 0")
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 apply(simp add:field_eq_simps)
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apply(simp)
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done
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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_eq_simps times_divide_eq)
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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_eq_simps times_divide_eq)
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022029099a83 continued localization
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end
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class division_by_zero = zero + inverse +
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  assumes inverse_zero [simp]: "inverse 0 = 0"
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lemma divide_zero [simp]:
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  "a / 0 = (0::'a::{field,division_by_zero})"
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by (simp add: divide_inverse)
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lemma divide_self_if [simp]:
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  "a / (a::'a::{field,division_by_zero}) = (if a=0 then 0 else 1)"
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by simp
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class linordered_field = field + linordered_idom
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lemma inverse_nonzero_iff_nonzero [simp]:
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   "(inverse a = 0) = (a = (0::'a::{division_ring,division_by_zero}))"
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by (force dest: inverse_zero_imp_zero) 
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lemma inverse_minus_eq [simp]:
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   "inverse(-a) = -inverse(a::'a::{division_ring,division_by_zero})"
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proof cases
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  assume "a=0" thus ?thesis by simp
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next
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  assume "a\<noteq>0" 
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  thus ?thesis by (simp add: nonzero_inverse_minus_eq)
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qed
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lemma inverse_eq_imp_eq:
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  "inverse a = inverse b ==> a = (b::'a::{division_ring,division_by_zero})"
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apply (cases "a=0 | b=0") 
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 apply (force dest!: inverse_zero_imp_zero
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              simp add: eq_commute [of "0::'a"])
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apply (force dest!: nonzero_inverse_eq_imp_eq) 
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done
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lemma inverse_eq_iff_eq [simp]:
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  "(inverse a = inverse b) = (a = (b::'a::{division_ring,division_by_zero}))"
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by (force dest!: inverse_eq_imp_eq)
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lemma inverse_inverse_eq [simp]:
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     "inverse(inverse (a::'a::{division_ring,division_by_zero})) = a"
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  proof cases
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    assume "a=0" thus ?thesis by simp
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  next
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    assume "a\<noteq>0" 
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    thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
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  qed
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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::'a::{field,division_by_zero})"
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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_commute)
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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::'a::{field,division_by_zero})"
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by (simp add: divide_inverse mult_commute)
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23389
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c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
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subsection {* 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 ==> (c*a) / (c*b) = a / (b::'a::{field,division_by_zero})"
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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 ==> (a*c) / (b*c) = a / (b::'a::{field,division_by_zero})"
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apply (cases "b = 0")
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apply simp_all
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done
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   294
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lemma divide_divide_eq_right [simp,noatp]:
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  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
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5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
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by (simp add: divide_inverse mult_ac)
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   298
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lemma divide_divide_eq_left [simp,noatp]:
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  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   301
by (simp add: divide_inverse mult_assoc)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   302
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   303
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   304
subsubsection{*Special Cancellation Simprules for Division*}
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   305
24427
bc5cf3b09ff3 revised blacklisting for ATP linkup
paulson
parents: 24422
diff changeset
   306
lemma mult_divide_mult_cancel_left_if[simp,noatp]:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   307
fixes c :: "'a :: {field,division_by_zero}"
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   308
shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
23413
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   309
by (simp add: mult_divide_mult_cancel_left)
5caa2710dd5b tuned laws for cancellation in divisions for fields.
nipkow
parents: 23406
diff changeset
   310
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   311
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   312
subsection {* Division and Unary Minus *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   313
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   314
lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
29407
5ef7e97fd9e4 move lemmas mult_minus{left,right} inside class ring
huffman
parents: 29406
diff changeset
   315
by (simp add: divide_inverse)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   316
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   317
lemma divide_minus_right [simp, noatp]:
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   318
  "a / -(b::'a::{field,division_by_zero}) = -(a / b)"
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   319
by (simp add: divide_inverse)
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   320
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   321
lemma minus_divide_divide:
23477
f4b83f03cac9 tuned and renamed group_eq_simps and ring_eq_simps
nipkow
parents: 23413
diff changeset
   322
  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   323
apply (cases "b=0", simp) 
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   324
apply (simp add: nonzero_minus_divide_divide) 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   325
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   326
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   327
lemma eq_divide_eq:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   328
  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   329
by (simp add: nonzero_eq_divide_eq)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   330
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   331
lemma divide_eq_eq:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   332
  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
30630
4fbe1401bac2 move field lemmas into class locale context
huffman
parents: 30242
diff changeset
   333
by (force simp add: nonzero_divide_eq_eq)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   334
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   335
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   336
subsection {* Ordered Fields *}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   337
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   338
lemma positive_imp_inverse_positive: 
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   339
assumes a_gt_0: "0 < a"  shows "0 < inverse (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   340
proof -
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   341
  have "0 < a * inverse a" 
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35090
diff changeset
   342
    by (simp add: a_gt_0 [THEN order_less_imp_not_eq2])
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   343
  thus "0 < inverse a" 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   344
    by (simp add: a_gt_0 [THEN order_less_not_sym] zero_less_mult_iff)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   345
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   346
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   347
lemma negative_imp_inverse_negative:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   348
  "a < 0 ==> inverse a < (0::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   349
by (insert positive_imp_inverse_positive [of "-a"], 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   350
    simp add: nonzero_inverse_minus_eq order_less_imp_not_eq)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   351
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   352
lemma inverse_le_imp_le:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   353
assumes invle: "inverse a \<le> inverse b" and apos:  "0 < a"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   354
shows "b \<le> (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   355
proof (rule classical)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   356
  assume "~ b \<le> a"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   357
  hence "a < b"  by (simp add: linorder_not_le)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   358
  hence bpos: "0 < b"  by (blast intro: apos order_less_trans)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   359
  hence "a * inverse a \<le> a * inverse b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   360
    by (simp add: apos invle order_less_imp_le mult_left_mono)
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   361
  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   362
    by (simp add: bpos order_less_imp_le mult_right_mono)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   363
  thus "b \<le> a"  by (simp add: mult_assoc apos bpos order_less_imp_not_eq2)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   364
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   365
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   366
lemma inverse_positive_imp_positive:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   367
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   368
shows "0 < (a::'a::linordered_field)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   369
proof -
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   370
  have "0 < inverse (inverse a)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   371
    using inv_gt_0 by (rule positive_imp_inverse_positive)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   372
  thus "0 < a"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   373
    using nz by (simp add: nonzero_inverse_inverse_eq)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   374
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   375
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   376
lemma inverse_positive_iff_positive [simp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   377
  "(0 < inverse a) = (0 < (a::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   378
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   379
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
   380
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   381
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   382
lemma inverse_negative_imp_negative:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   383
assumes inv_less_0: "inverse a < 0" and nz:  "a \<noteq> 0"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   384
shows "a < (0::'a::linordered_field)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   385
proof -
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   386
  have "inverse (inverse a) < 0"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   387
    using inv_less_0 by (rule negative_imp_inverse_negative)
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   388
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   389
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   390
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   391
lemma inverse_negative_iff_negative [simp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   392
  "(inverse a < 0) = (a < (0::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   393
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   394
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
   395
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   396
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   397
lemma inverse_nonnegative_iff_nonnegative [simp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   398
  "(0 \<le> inverse a) = (0 \<le> (a::'a::{linordered_field,division_by_zero}))"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   399
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   400
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   401
lemma inverse_nonpositive_iff_nonpositive [simp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   402
  "(inverse a \<le> 0) = (a \<le> (0::'a::{linordered_field,division_by_zero}))"
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   403
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   404
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   405
lemma linordered_field_no_lb: "\<forall> x. \<exists>y. y < (x::'a::linordered_field)"
23406
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   406
proof
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   407
  fix x::'a
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   408
  have m1: "- (1::'a) < 0" by simp
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   409
  from add_strict_right_mono[OF m1, where c=x] 
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   410
  have "(- 1) + x < x" by simp
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   411
  thus "\<exists>y. y < x" by blast
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   412
qed
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   413
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35032
diff changeset
   414
lemma linordered_field_no_ub: "\<forall> x. \<exists>y. y > (x::'a::linordered_field)"
23406
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   415
proof
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   416
  fix x::'a
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   417
  have m1: " (1::'a) > 0" by simp
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   418
  from add_strict_right_mono[OF m1, where c=x] 
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   419
  have "1 + x > x" by simp
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   420
  thus "\<exists>y. y > x" by blast
167b53019d6f added theorems nonzero_mult_divide_cancel_right' nonzero_mult_divide_cancel_left' ordered_field_no_lb ordered_field_no_ub
chaieb
parents: 23400
diff changeset
   421
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   422
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   423
subsection{*Anti-Monotonicity of @{term inverse}*}
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   424
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   425
lemma less_imp_inverse_less:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   426
assumes less: "a < b" and apos:  "0 < a"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   427
shows "inverse b < inverse (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   428
proof (rule ccontr)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   429
  assume "~ inverse b < inverse a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   430
  hence "inverse a \<le> inverse b" by (simp add: linorder_not_less)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   431
  hence "~ (a < b)"
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   432
    by (simp add: linorder_not_less inverse_le_imp_le [OF _ apos])
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   433
  thus False by (rule notE [OF _ less])
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   434
qed
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   435
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   436
lemma inverse_less_imp_less:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   437
  "[|inverse a < inverse b; 0 < a|] ==> b < (a::'a::linordered_field)"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   438
apply (simp add: order_less_le [of "inverse a"] order_less_le [of "b"])
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   439
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   440
done
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   441
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   442
text{*Both premises are essential. Consider -1 and 1.*}
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   443
lemma inverse_less_iff_less [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   444
  "[|0 < a; 0 < b|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   445
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   446
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   447
lemma le_imp_inverse_le:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   448
  "[|a \<le> b; 0 < a|] ==> inverse b \<le> inverse (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   449
by (force simp add: order_le_less less_imp_inverse_less)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   450
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   451
lemma inverse_le_iff_le [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   452
 "[|0 < a; 0 < b|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   453
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   454
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   455
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   456
text{*These results refer to both operands being negative.  The opposite-sign
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   457
case is trivial, since inverse preserves signs.*}
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   458
lemma inverse_le_imp_le_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   459
  "[|inverse a \<le> inverse b; b < 0|] ==> b \<le> (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   460
apply (rule classical) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   461
apply (subgoal_tac "a < 0") 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   462
 prefer 2 apply (force simp add: linorder_not_le intro: order_less_trans) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   463
apply (insert inverse_le_imp_le [of "-b" "-a"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   464
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   465
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   466
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   467
lemma less_imp_inverse_less_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   468
   "[|a < b; b < 0|] ==> inverse b < inverse (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   469
apply (subgoal_tac "a < 0") 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   470
 prefer 2 apply (blast intro: order_less_trans) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   471
apply (insert less_imp_inverse_less [of "-b" "-a"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   472
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   473
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   474
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   475
lemma inverse_less_imp_less_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   476
   "[|inverse a < inverse b; b < 0|] ==> b < (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   477
apply (rule classical) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   478
apply (subgoal_tac "a < 0") 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   479
 prefer 2
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   480
 apply (force simp add: linorder_not_less intro: order_le_less_trans) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   481
apply (insert inverse_less_imp_less [of "-b" "-a"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   482
apply (simp add: order_less_imp_not_eq nonzero_inverse_minus_eq) 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   483
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   484
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   485
lemma inverse_less_iff_less_neg [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   486
  "[|a < 0; b < 0|] ==> (inverse a < inverse b) = (b < (a::'a::linordered_field))"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   487
apply (insert inverse_less_iff_less [of "-b" "-a"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   488
apply (simp del: inverse_less_iff_less 
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   489
            add: order_less_imp_not_eq nonzero_inverse_minus_eq)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   490
done
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   491
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   492
lemma le_imp_inverse_le_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   493
  "[|a \<le> b; b < 0|] ==> inverse b \<le> inverse (a::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   494
by (force simp add: order_le_less less_imp_inverse_less_neg)
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   495
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   496
lemma inverse_le_iff_le_neg [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   497
 "[|a < 0; b < 0|] ==> (inverse a \<le> inverse b) = (b \<le> (a::'a::linordered_field))"
14268
5cf13e80be0e Removal of Hyperreal/ExtraThms2.ML, sending the material to the correct files.
paulson
parents: 14267
diff changeset
   498
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   499
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   500
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   501
subsection{*Inverses and the Number One*}
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   502
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   503
lemma one_less_inverse_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   504
  "(1 < inverse x) = (0 < x & x < (1::'a::{linordered_field,division_by_zero}))"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   505
proof cases
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   506
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   507
    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
   508
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   509
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   510
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   511
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   512
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   513
    assume "1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   514
    also with notless have "... \<le> 0" by (simp add: linorder_not_less)
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   515
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   516
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   517
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   518
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   519
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   520
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   521
lemma inverse_eq_1_iff [simp]:
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   522
  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   523
by (insert inverse_eq_iff_eq [of x 1], simp) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   524
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   525
lemma one_le_inverse_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   526
  "(1 \<le> inverse x) = (0 < x & x \<le> (1::'a::{linordered_field,division_by_zero}))"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35090
diff changeset
   527
by (force simp add: order_le_less one_less_inverse_iff)
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   528
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   529
lemma inverse_less_1_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   530
  "(inverse x < 1) = (x \<le> 0 | 1 < (x::'a::{linordered_field,division_by_zero}))"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   531
by (simp add: linorder_not_le [symmetric] one_le_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   532
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   533
lemma inverse_le_1_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   534
  "(inverse x \<le> 1) = (x \<le> 0 | 1 \<le> (x::'a::{linordered_field,division_by_zero}))"
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   535
by (simp add: linorder_not_less [symmetric] one_less_inverse_iff) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   536
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   537
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   538
subsection{*Simplification of Inequalities Involving Literal Divisors*}
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   539
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   540
lemma pos_le_divide_eq: "0 < (c::'a::linordered_field) ==> (a \<le> b/c) = (a*c \<le> b)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   541
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   542
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   543
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   544
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   545
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   546
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   547
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   548
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   549
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   550
lemma neg_le_divide_eq: "c < (0::'a::linordered_field) ==> (a \<le> b/c) = (b \<le> a*c)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   551
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   552
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   553
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   554
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   555
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   556
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   557
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   558
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   559
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   560
lemma le_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   561
  "(a \<le> b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   562
   (if 0 < c then a*c \<le> b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   563
             else if c < 0 then b \<le> a*c
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   564
             else  a \<le> (0::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   565
apply (cases "c=0", simp) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   566
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   567
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   568
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   569
lemma pos_divide_le_eq: "0 < (c::'a::linordered_field) ==> (b/c \<le> a) = (b \<le> a*c)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   570
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   571
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   572
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   573
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   574
  also have "... = (b \<le> a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   575
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   576
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   577
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   578
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   579
lemma neg_divide_le_eq: "c < (0::'a::linordered_field) ==> (b/c \<le> a) = (a*c \<le> b)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   580
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   581
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   582
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   583
    by (simp add: mult_le_cancel_right order_less_not_sym [OF less])
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   584
  also have "... = (a*c \<le> b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   585
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   586
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   587
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   588
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   589
lemma divide_le_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   590
  "(b/c \<le> a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   591
   (if 0 < c then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   592
             else if c < 0 then a*c \<le> b
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   593
             else 0 \<le> (a::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   594
apply (cases "c=0", simp) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   595
apply (force simp add: pos_divide_le_eq neg_divide_le_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   596
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   597
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   598
lemma pos_less_divide_eq:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   599
     "0 < (c::'a::linordered_field) ==> (a < b/c) = (a*c < b)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   600
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   601
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   602
  hence "(a < b/c) = (a*c < (b/c)*c)"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   603
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   604
  also have "... = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   605
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   606
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   607
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   608
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   609
lemma neg_less_divide_eq:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   610
 "c < (0::'a::linordered_field) ==> (a < b/c) = (b < a*c)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   611
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   612
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   613
  hence "(a < b/c) = ((b/c)*c < a*c)"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   614
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   615
  also have "... = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   616
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   617
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   618
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   619
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   620
lemma less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   621
  "(a < b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   622
   (if 0 < c then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   623
             else if c < 0 then b < a*c
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   624
             else  a < (0::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   625
apply (cases "c=0", simp) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   626
apply (force simp add: pos_less_divide_eq neg_less_divide_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   627
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   628
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   629
lemma pos_divide_less_eq:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   630
     "0 < (c::'a::linordered_field) ==> (b/c < a) = (b < a*c)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   631
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   632
  assume less: "0<c"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   633
  hence "(b/c < a) = ((b/c)*c < a*c)"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   634
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   635
  also have "... = (b < a*c)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   636
    by (simp add: order_less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   637
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   638
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   639
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   640
lemma neg_divide_less_eq:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   641
 "c < (0::'a::linordered_field) ==> (b/c < a) = (a*c < b)"
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   642
proof -
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   643
  assume less: "c<0"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   644
  hence "(b/c < a) = (a*c < (b/c)*c)"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   645
    by (simp add: mult_less_cancel_right_disj order_less_not_sym [OF less])
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   646
  also have "... = (a*c < b)"
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   647
    by (simp add: order_less_imp_not_eq [OF less] divide_inverse mult_assoc) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   648
  finally show ?thesis .
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   649
qed
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   650
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   651
lemma divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   652
  "(b/c < a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   653
   (if 0 < c then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   654
             else if c < 0 then a*c < b
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   655
             else 0 < (a::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   656
apply (cases "c=0", simp) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   657
apply (force simp add: pos_divide_less_eq neg_divide_less_eq linorder_neq_iff) 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   658
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   659
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   660
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   661
subsection{*Field simplification*}
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   662
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   663
text{* Lemmas @{text field_simps} multiply with denominators in in(equations)
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   664
if they can be proved to be non-zero (for equations) or positive/negative
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   665
(for inequations). Can be too aggressive and is therefore separate from the
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29465
diff changeset
   666
more benign @{text algebra_simps}. *}
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   667
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   668
lemmas field_simps[noatp] = field_eq_simps
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   669
  (* multiply ineqn *)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   670
  pos_divide_less_eq neg_divide_less_eq
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   671
  pos_less_divide_eq neg_less_divide_eq
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   672
  pos_divide_le_eq neg_divide_le_eq
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   673
  pos_le_divide_eq neg_le_divide_eq
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   674
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   675
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
23483
a9356b40fbd3 tex problem fixed
nipkow
parents: 23482
diff changeset
   676
of positivity/negativity needed for @{text field_simps}. Have not added @{text
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   677
sign_simps} to @{text field_simps} because the former can lead to case
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   678
explosions. *}
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   679
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   680
lemmas sign_simps[noatp] = group_simps
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   681
  zero_less_mult_iff  mult_less_0_iff
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   682
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   683
(* Only works once linear arithmetic is installed:
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   684
text{*An example:*}
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   685
lemma fixes a b c d e f :: "'a::linordered_field"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   686
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   687
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   688
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   689
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   690
 prefer 2 apply(simp add:sign_simps)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   691
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   692
 prefer 2 apply(simp add:sign_simps)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   693
apply(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   694
done
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   695
*)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   696
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   697
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   698
subsection{*Division and Signs*}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   699
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   700
lemma zero_less_divide_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   701
     "((0::'a::{linordered_field,division_by_zero}) < 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
   702
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
   703
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   704
lemma divide_less_0_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   705
     "(a/b < (0::'a::{linordered_field,division_by_zero})) = 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   706
      (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
   707
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
   708
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   709
lemma zero_le_divide_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   710
     "((0::'a::{linordered_field,division_by_zero}) \<le> a/b) =
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   711
      (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
   712
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
   713
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   714
lemma divide_le_0_iff:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   715
     "(a/b \<le> (0::'a::{linordered_field,division_by_zero})) =
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   716
      (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
   717
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
   718
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   719
lemma divide_eq_0_iff [simp,noatp]:
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   720
     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   721
by (simp add: divide_inverse)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   722
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   723
lemma divide_pos_pos:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   724
  "0 < (x::'a::linordered_field) ==> 0 < y ==> 0 < x / y"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   725
by(simp add:field_simps)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   726
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   727
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   728
lemma divide_nonneg_pos:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   729
  "0 <= (x::'a::linordered_field) ==> 0 < y ==> 0 <= x / y"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   730
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   731
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   732
lemma divide_neg_pos:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   733
  "(x::'a::linordered_field) < 0 ==> 0 < y ==> x / y < 0"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   734
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   735
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   736
lemma divide_nonpos_pos:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   737
  "(x::'a::linordered_field) <= 0 ==> 0 < y ==> x / y <= 0"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   738
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   739
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   740
lemma divide_pos_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   741
  "0 < (x::'a::linordered_field) ==> y < 0 ==> x / y < 0"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   742
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   743
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   744
lemma divide_nonneg_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   745
  "0 <= (x::'a::linordered_field) ==> y < 0 ==> x / y <= 0" 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   746
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   747
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   748
lemma divide_neg_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   749
  "(x::'a::linordered_field) < 0 ==> y < 0 ==> 0 < x / y"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   750
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   751
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   752
lemma divide_nonpos_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   753
  "(x::'a::linordered_field) <= 0 ==> y < 0 ==> 0 <= x / y"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   754
by(simp add:field_simps)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   755
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   756
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   757
subsection{*Cancellation Laws for Division*}
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   758
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   759
lemma divide_cancel_right [simp,noatp]:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   760
     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   761
apply (cases "c=0", simp)
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
   762
apply (simp add: divide_inverse)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   763
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   764
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   765
lemma divide_cancel_left [simp,noatp]:
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   766
     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   767
apply (cases "c=0", simp)
23496
84e9216a6d0e removed redundant lemmas
nipkow
parents: 23483
diff changeset
   768
apply (simp add: divide_inverse)
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   769
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   770
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   771
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   772
subsection {* Division and the Number One *}
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   773
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   774
text{*Simplify expressions equated with 1*}
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   775
lemma divide_eq_1_iff [simp,noatp]:
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   776
     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   777
apply (cases "b=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   778
apply (simp add: right_inverse_eq)
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   779
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   780
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   781
lemma one_eq_divide_iff [simp,noatp]:
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   782
     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   783
by (simp add: eq_commute [of 1])
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   784
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   785
lemma zero_eq_1_divide_iff [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   786
     "((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   787
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   788
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
   789
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   790
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   791
lemma one_divide_eq_0_iff [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   792
     "(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   793
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   794
apply (insert zero_neq_one [THEN not_sym])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   795
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
   796
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   797
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   798
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
   799
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   800
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   801
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   802
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
   803
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   804
declare zero_less_divide_1_iff [simp,noatp]
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   805
declare divide_less_0_1_iff [simp,noatp]
29833
409138c4de12 added noatps
nipkow
parents: 29700
diff changeset
   806
declare zero_le_divide_1_iff [simp,noatp]
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   807
declare divide_le_0_1_iff [simp,noatp]
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   808
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   809
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   810
subsection {* Ordering Rules for Division *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   811
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   812
lemma divide_strict_right_mono:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   813
     "[|a < b; 0 < c|] ==> a / c < b / (c::'a::linordered_field)"
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   814
by (simp add: order_less_imp_not_eq2 divide_inverse mult_strict_right_mono 
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   815
              positive_imp_inverse_positive)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   816
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   817
lemma divide_right_mono:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   818
     "[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/(c::'a::{linordered_field,division_by_zero})"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   819
by (force simp add: divide_strict_right_mono order_le_less)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   820
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   821
lemma divide_right_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   822
    ==> c <= 0 ==> b / c <= a / c"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   823
apply (drule divide_right_mono [of _ _ "- c"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   824
apply auto
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   825
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   826
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   827
lemma divide_strict_right_mono_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   828
     "[|b < a; c < 0|] ==> a / c < b / (c::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   829
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   830
apply (simp add: order_less_imp_not_eq nonzero_minus_divide_right [symmetric])
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   831
done
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   832
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   833
text{*The last premise ensures that @{term a} and @{term b} 
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   834
      have the same sign*}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   835
lemma divide_strict_left_mono:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   836
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   837
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   838
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   839
lemma divide_left_mono:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   840
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / (b::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   841
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   842
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   843
lemma divide_left_mono_neg: "(a::'a::{division_by_zero,linordered_field}) <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   844
    ==> 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
   845
  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
   846
  apply (auto simp add: mult_commute)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   847
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   848
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   849
lemma divide_strict_left_mono_neg:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   850
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / (b::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   851
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   852
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   853
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   854
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
   855
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   856
lemma le_divide_eq_1 [noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   857
  fixes a :: "'a :: {linordered_field,division_by_zero}"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   858
  shows "(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   859
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
   860
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   861
lemma divide_le_eq_1 [noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   862
  fixes a :: "'a :: {linordered_field,division_by_zero}"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   863
  shows "(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   864
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
   865
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   866
lemma less_divide_eq_1 [noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   867
  fixes a :: "'a :: {linordered_field,division_by_zero}"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   868
  shows "(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   869
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
   870
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   871
lemma divide_less_eq_1 [noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   872
  fixes a :: "'a :: {linordered_field,division_by_zero}"
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   873
  shows "(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   874
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
   875
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   876
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   877
subsection{*Conditional Simplification Rules: No Case Splits*}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   878
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   879
lemma le_divide_eq_1_pos [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   880
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   881
  shows "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
   882
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
   883
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   884
lemma le_divide_eq_1_neg [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   885
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   886
  shows "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
   887
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
   888
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   889
lemma divide_le_eq_1_pos [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   890
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   891
  shows "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
   892
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
   893
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   894
lemma divide_le_eq_1_neg [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   895
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   896
  shows "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
   897
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
   898
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   899
lemma less_divide_eq_1_pos [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   900
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   901
  shows "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
   902
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
   903
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   904
lemma less_divide_eq_1_neg [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   905
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   906
  shows "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
   907
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
   908
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   909
lemma divide_less_eq_1_pos [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   910
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   911
  shows "0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   912
by (auto simp add: divide_less_eq)
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   913
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   914
lemma divide_less_eq_1_neg [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   915
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   916
  shows "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
   917
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
   918
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   919
lemma eq_divide_eq_1 [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   920
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   921
  shows "(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
   922
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
   923
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 23879
diff changeset
   924
lemma divide_eq_eq_1 [simp,noatp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   925
  fixes a :: "'a :: {linordered_field,division_by_zero}"
18649
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   926
  shows "(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
   927
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
   928
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   929
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   930
subsection {* Reasoning about inequalities with division *}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   931
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   932
lemma mult_imp_div_pos_le: "0 < (y::'a::linordered_field) ==> x <= z * y ==>
33319
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
   933
    x / y <= z"
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
   934
by (subst pos_divide_le_eq, assumption+)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   935
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   936
lemma mult_imp_le_div_pos: "0 < (y::'a::linordered_field) ==> z * y <= x ==>
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   937
    z <= x / y"
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   938
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   939
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   940
lemma mult_imp_div_pos_less: "0 < (y::'a::linordered_field) ==> x < z * y ==>
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   941
    x / y < z"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   942
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   943
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   944
lemma mult_imp_less_div_pos: "0 < (y::'a::linordered_field) ==> z * y < x ==>
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   945
    z < x / y"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   946
by(simp add:field_simps)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   947
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   948
lemma frac_le: "(0::'a::linordered_field) <= x ==> 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   949
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   950
  apply (rule mult_imp_div_pos_le)
25230
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   951
  apply simp
022029099a83 continued localization
haftmann
parents: 25193
diff changeset
   952
  apply (subst times_divide_eq_left)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   953
  apply (rule mult_imp_le_div_pos, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   954
  apply (rule mult_mono)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   955
  apply simp_all
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   956
done
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   957
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   958
lemma frac_less: "(0::'a::linordered_field) <= x ==> 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   959
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   960
  apply (rule mult_imp_div_pos_less)
33319
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
   961
  apply simp
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
   962
  apply (subst times_divide_eq_left)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   963
  apply (rule mult_imp_less_div_pos, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   964
  apply (erule mult_less_le_imp_less)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   965
  apply simp_all
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   966
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   967
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   968
lemma frac_less2: "(0::'a::linordered_field) < x ==> 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   969
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   970
  apply (rule mult_imp_div_pos_less)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   971
  apply simp_all
33319
74f0dcc0b5fb moved algebraic classes to Ring_and_Field
haftmann
parents: 32960
diff changeset
   972
  apply (subst times_divide_eq_left)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   973
  apply (rule mult_imp_less_div_pos, assumption)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   974
  apply (erule mult_le_less_imp_less)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   975
  apply simp_all
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   976
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   977
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   978
text{*It's not obvious whether these should be simprules or not. 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   979
  Their effect is to gather terms into one big fraction, like
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   980
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   981
  seem to need them.*}
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   982
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   983
declare times_divide_eq [simp]
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   984
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   985
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   986
subsection {* Ordered Fields are Dense *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   987
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   988
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1::'a::linordered_field)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   989
by (simp add: field_simps zero_less_two)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   990
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   991
lemma gt_half_sum: "a < b ==> (a+b)/(1+1::'a::linordered_field) < b"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   992
by (simp add: field_simps zero_less_two)
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   993
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   994
instance linordered_field < dense_linorder
24422
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   995
proof
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   996
  fix x y :: 'a
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   997
  have "x < x + 1" by simp
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   998
  then show "\<exists>y. x < y" .. 
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
   999
  have "x - 1 < x" by simp
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
  1000
  then show "\<exists>y. y < x" ..
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
  1001
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
c0b5ff9e9e4d moved class dense_linear_order to Orderings.thy
haftmann
parents: 24286
diff changeset
  1002
qed
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1003
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1004
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1005
subsection {* Absolute Value *}
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
  1006
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1007
lemma nonzero_abs_inverse:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1008
     "a \<noteq> 0 ==> abs (inverse (a::'a::linordered_field)) = inverse (abs a)"
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1009
apply (auto simp add: linorder_neq_iff abs_if nonzero_inverse_minus_eq 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1010
                      negative_imp_inverse_negative)
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1011
apply (blast intro: positive_imp_inverse_positive elim: order_less_asym) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1012
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1013
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1014
lemma abs_inverse [simp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1015
     "abs (inverse (a::'a::{linordered_field,division_by_zero})) = 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1016
      inverse (abs a)"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1017
apply (cases "a=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1018
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1019
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1020
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1021
lemma nonzero_abs_divide:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1022
     "b \<noteq> 0 ==> abs (a / (b::'a::linordered_field)) = abs a / abs b"
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1023
by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1024
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1025
lemma abs_divide [simp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1026
     "abs (a / (b::'a::{linordered_field,division_by_zero})) = abs a / abs b"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
  1027
apply (cases "b=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1028
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1029
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
  1030
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
  1031
lemma abs_div_pos: "(0::'a::{division_by_zero,linordered_field}) < y ==> 
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1032
    abs x / y = abs (x / y)"
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1033
  apply (subst abs_divide)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1034
  apply (simp add: order_less_imp_le)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
  1035
done
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
  1036
35090
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1037
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1038
lemma field_le_epsilon:
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1039
  fixes x y :: "'a :: {division_by_zero,linordered_field}"
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1040
  assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1041
  shows "x \<le> y"
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1042
proof (rule ccontr)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1043
  obtain two :: 'a where two: "two = 1 + 1" by simp
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1044
  assume "\<not> x \<le> y"
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1045
  then have yx: "y < x" by simp
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1046
  then have "y + - y < x + - y" by (rule add_strict_right_mono)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1047
  then have "x - y > 0" by (simp add: diff_minus)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1048
  then have "(x - y) / two > 0"
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1049
    by (rule divide_pos_pos) (simp add: two)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1050
  then have "x \<le> y + (x - y) / two" by (rule e)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1051
  also have "... = (x - y + two * y) / two"
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1052
    by (simp add: add_divide_distrib two)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1053
  also have "... = (x + y) / two" 
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1054
    by (simp add: two algebra_simps)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1055
  also have "... < x" using yx
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1056
    by (simp add: two pos_divide_less_eq algebra_simps)
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1057
  finally have "x < x" .
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1058
  then show False ..
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1059
qed
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1060
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
  1061
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1062
code_modulename SML
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
  1063
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1064
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1065
code_modulename OCaml
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
  1066
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1067
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1068
code_modulename Haskell
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
  1069
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
  1070
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
  1071
end