src/HOL/Fields.thy
author haftmann
Fri, 23 Apr 2010 13:58:14 +0200
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child 36304 6984744e6b34
permissions -rw-r--r--
more localization; factored out lemmas for division_ring
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(*  Title:      HOL/Fields.thy
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    Author:     Gertrud Bauer
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    Author:     Steven Obua
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    Author:     Tobias Nipkow
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    Author:     Lawrence C Paulson
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    Author:     Markus Wenzel
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    Author:     Jeremy Avigad
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*)
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header {* Fields *}
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theory Fields
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imports Rings
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begin
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class field = comm_ring_1 + inverse +
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  assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
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  assumes field_divide_inverse: "a / b = a * inverse b"
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begin
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subclass division_ring
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proof
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  fix a :: 'a
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  assume "a \<noteq> 0"
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  thus "inverse a * a = 1" by (rule field_inverse)
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  thus "a * inverse a = 1" by (simp only: mult_commute)
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next
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  fix a b :: 'a
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  show "a / b = a * inverse b" by (rule field_divide_inverse)
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qed
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subclass idom ..
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text{*There is no slick version using division by zero.*}
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lemma inverse_add:
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  "[| a \<noteq> 0;  b \<noteq> 0 |]
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   ==> inverse a + inverse b = (a + b) * inverse a * inverse b"
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by (simp add: division_ring_inverse_add mult_ac)
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lemma nonzero_mult_divide_mult_cancel_left [simp, no_atp]:
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assumes [simp]: "b\<noteq>0" and [simp]: "c\<noteq>0" shows "(c*a)/(c*b) = a/b"
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proof -
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  have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
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    by (simp add: divide_inverse nonzero_inverse_mult_distrib)
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  also have "... =  a * inverse b * (inverse c * c)"
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    by (simp only: mult_ac)
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  also have "... =  a * inverse b" by simp
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    finally show ?thesis by (simp add: divide_inverse)
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qed
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lemma nonzero_mult_divide_mult_cancel_right [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (b * c) = a / b"
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by (simp add: mult_commute [of _ c])
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lemma times_divide_eq_left: "(b / c) * a = (b * a) / c"
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  by (simp add: divide_inverse mult_ac)
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text {* These are later declared as simp rules. *}
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lemmas times_divide_eq [no_atp] = times_divide_eq_right times_divide_eq_left
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lemma add_frac_eq:
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  assumes "y \<noteq> 0" and "z \<noteq> 0"
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  shows "x / y + w / z = (x * z + w * y) / (y * z)"
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proof -
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  have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
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    using assms by simp
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  also have "\<dots> = (x * z + y * w) / (y * z)"
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    by (simp only: add_divide_distrib)
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  finally show ?thesis
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    by (simp only: mult_commute)
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qed
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text{*Special Cancellation Simprules for Division*}
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lemma nonzero_mult_divide_cancel_right [simp, no_atp]:
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  "b \<noteq> 0 \<Longrightarrow> a * b / b = a"
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  using nonzero_mult_divide_mult_cancel_right [of 1 b a] by simp
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lemma nonzero_mult_divide_cancel_left [simp, no_atp]:
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  "a \<noteq> 0 \<Longrightarrow> a * b / a = b"
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using nonzero_mult_divide_mult_cancel_left [of 1 a b] by simp
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lemma nonzero_divide_mult_cancel_right [simp, no_atp]:
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  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> b / (a * b) = 1 / a"
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using nonzero_mult_divide_mult_cancel_right [of a b 1] by simp
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lemma nonzero_divide_mult_cancel_left [simp, no_atp]:
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  "\<lbrakk>a \<noteq> 0; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / (a * b) = 1 / b"
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using nonzero_mult_divide_mult_cancel_left [of b a 1] by simp
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lemma nonzero_mult_divide_mult_cancel_left2 [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (c * a) / (b * c) = a / b"
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using nonzero_mult_divide_mult_cancel_left [of b c a] by (simp add: mult_ac)
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lemma nonzero_mult_divide_mult_cancel_right2 [simp, no_atp]:
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  "\<lbrakk>b \<noteq> 0; c \<noteq> 0\<rbrakk> \<Longrightarrow> (a * c) / (c * b) = a / b"
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using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: mult_ac)
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lemma add_divide_eq_iff:
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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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lemmas field_eq_simps[no_atp] = 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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  times_divide_eq_left times_divide_eq_right
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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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end
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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_ac)
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  next
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    assume "~ (a \<noteq> 0 & b \<noteq> 0)" 
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    thus ?thesis by force
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  qed
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lemma inverse_divide [simp]:
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  "inverse (a/b) = b / (a::'a::{field,division_by_zero})"
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  by (simp add: divide_inverse mult_commute)
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text {* Calculations with fractions *}
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text{* There is a whole bunch of simp-rules just for class @{text
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field} but none for class @{text field} and @{text nonzero_divides}
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because the latter are covered by a simproc. *}
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lemma mult_divide_mult_cancel_left:
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  "c\<noteq>0 ==> (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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lemma divide_divide_eq_right [simp,no_atp]:
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  "a / (b/c) = (a*c) / (b::'a::{field,division_by_zero})"
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by (simp add: divide_inverse mult_ac)
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lemma divide_divide_eq_left [simp,no_atp]:
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  "(a / b) / (c::'a::{field,division_by_zero}) = a / (b*c)"
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by (simp add: divide_inverse mult_assoc)
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text {*Special Cancellation Simprules for Division*}
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lemma mult_divide_mult_cancel_left_if[simp,no_atp]:
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fixes c :: "'a :: {field,division_by_zero}"
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shows "(c*a) / (c*b) = (if c=0 then 0 else a/b)"
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by (simp add: mult_divide_mult_cancel_left)
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text {* Division and Unary Minus *}
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lemma minus_divide_right: "- (a/b) = a / -(b::'a::{field,division_by_zero})"
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by (simp add: divide_inverse)
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lemma divide_minus_right [simp, no_atp]:
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  "a / -(b::'a::{field,division_by_zero}) = -(a / b)"
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by (simp add: divide_inverse)
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lemma minus_divide_divide:
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  "(-a)/(-b) = a / (b::'a::{field,division_by_zero})"
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apply (cases "b=0", simp) 
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apply (simp add: nonzero_minus_divide_divide) 
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done
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lemma eq_divide_eq:
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  "((a::'a::{field,division_by_zero}) = b/c) = (if c\<noteq>0 then a*c = b else a=0)"
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by (simp add: nonzero_eq_divide_eq)
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lemma divide_eq_eq:
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  "(b/c = (a::'a::{field,division_by_zero})) = (if c\<noteq>0 then b = a*c else a=0)"
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by (force simp add: nonzero_divide_eq_eq)
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lemma inverse_eq_1_iff [simp]:
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  "(inverse x = 1) = (x = (1::'a::{field,division_by_zero}))"
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by (insert inverse_eq_iff_eq [of x 1], simp) 
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lemma divide_eq_0_iff [simp,no_atp]:
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     "(a/b = 0) = (a=0 | b=(0::'a::{field,division_by_zero}))"
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by (simp add: divide_inverse)
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lemma divide_cancel_right [simp,no_atp]:
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     "(a/c = b/c) = (c = 0 | a = (b::'a::{field,division_by_zero}))"
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apply (cases "c=0", simp)
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apply (simp add: divide_inverse)
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done
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lemma divide_cancel_left [simp,no_atp]:
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     "(c/a = c/b) = (c = 0 | a = (b::'a::{field,division_by_zero}))" 
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apply (cases "c=0", simp)
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apply (simp add: divide_inverse)
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done
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lemma divide_eq_1_iff [simp,no_atp]:
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     "(a/b = 1) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
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apply (cases "b=0", simp)
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apply (simp add: right_inverse_eq)
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done
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lemma one_eq_divide_iff [simp,no_atp]:
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     "(1 = a/b) = (b \<noteq> 0 & a = (b::'a::{field,division_by_zero}))"
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by (simp add: eq_commute [of 1])
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text {* Ordered Fields *}
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class linordered_field = field + linordered_idom
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begin
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lemma positive_imp_inverse_positive: 
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  assumes a_gt_0: "0 < a" 
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  shows "0 < inverse a"
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proof -
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  have "0 < a * inverse a" 
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    by (simp add: a_gt_0 [THEN less_imp_not_eq2])
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  thus "0 < inverse a" 
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    by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
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qed
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lemma negative_imp_inverse_negative:
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  "a < 0 \<Longrightarrow> inverse a < 0"
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  by (insert positive_imp_inverse_positive [of "-a"], 
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    simp add: nonzero_inverse_minus_eq less_imp_not_eq)
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lemma inverse_le_imp_le:
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  assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
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  shows "b \<le> a"
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proof (rule classical)
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  assume "~ b \<le> a"
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  hence "a < b"  by (simp add: linorder_not_le)
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  hence bpos: "0 < b"  by (blast intro: apos less_trans)
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  hence "a * inverse a \<le> a * inverse b"
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    by (simp add: apos invle less_imp_le mult_left_mono)
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  hence "(a * inverse a) * b \<le> (a * inverse b) * b"
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    by (simp add: bpos less_imp_le mult_right_mono)
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  thus "b \<le> a"  by (simp add: mult_assoc apos bpos less_imp_not_eq2)
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qed
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lemma inverse_positive_imp_positive:
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  assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
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  shows "0 < a"
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proof -
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   284
  have "0 < inverse (inverse a)"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   285
    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
   286
  thus "0 < a"
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   287
    using nz by (simp add: nonzero_inverse_inverse_eq)
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   288
qed
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   289
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   290
lemma inverse_negative_imp_negative:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   291
  assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   292
  shows "a < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   293
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   294
  have "inverse (inverse a) < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   295
    using inv_less_0 by (rule negative_imp_inverse_negative)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   296
  thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   297
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   298
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   299
lemma linordered_field_no_lb:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   300
  "\<forall>x. \<exists>y. y < x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   301
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   302
  fix x::'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   303
  have m1: "- (1::'a) < 0" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   304
  from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   305
  have "(- 1) + x < x" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   306
  thus "\<exists>y. y < x" by blast
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   307
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   308
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   309
lemma linordered_field_no_ub:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   310
  "\<forall> x. \<exists>y. y > x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   311
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   312
  fix x::'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   313
  have m1: " (1::'a) > 0" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   314
  from add_strict_right_mono[OF m1, where c=x] 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   315
  have "1 + x > x" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   316
  thus "\<exists>y. y > x" by blast
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   317
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   318
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   319
lemma less_imp_inverse_less:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   320
  assumes less: "a < b" and apos:  "0 < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   321
  shows "inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   322
proof (rule ccontr)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   323
  assume "~ inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   324
  hence "inverse a \<le> inverse b" by simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   325
  hence "~ (a < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   326
    by (simp add: not_less inverse_le_imp_le [OF _ apos])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   327
  thus False by (rule notE [OF _ less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   328
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   329
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   330
lemma inverse_less_imp_less:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   331
  "inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   332
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   333
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   334
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   335
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   336
text{*Both premises are essential. Consider -1 and 1.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   337
lemma inverse_less_iff_less [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   338
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   339
  by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   340
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   341
lemma le_imp_inverse_le:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   342
  "a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   343
  by (force simp add: le_less less_imp_inverse_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   344
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   345
lemma inverse_le_iff_le [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   346
  "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   347
  by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   348
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   349
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   350
text{*These results refer to both operands being negative.  The opposite-sign
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   351
case is trivial, since inverse preserves signs.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   352
lemma inverse_le_imp_le_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   353
  "inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   354
apply (rule classical) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   355
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   356
 prefer 2 apply force
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   357
apply (insert inverse_le_imp_le [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   358
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   359
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   360
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   361
lemma less_imp_inverse_less_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   362
   "a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   363
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   364
 prefer 2 apply (blast intro: less_trans) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   365
apply (insert less_imp_inverse_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   366
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   367
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   368
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   369
lemma inverse_less_imp_less_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   370
   "inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   371
apply (rule classical) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   372
apply (subgoal_tac "a < 0") 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   373
 prefer 2
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   374
 apply force
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   375
apply (insert inverse_less_imp_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   376
apply (simp add: nonzero_inverse_minus_eq) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   377
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   378
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   379
lemma inverse_less_iff_less_neg [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   380
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   381
apply (insert inverse_less_iff_less [of "-b" "-a"])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   382
apply (simp del: inverse_less_iff_less 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   383
            add: nonzero_inverse_minus_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   384
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   385
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   386
lemma le_imp_inverse_le_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   387
  "a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   388
  by (force simp add: le_less less_imp_inverse_less_neg)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   389
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   390
lemma inverse_le_iff_le_neg [simp,no_atp]:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   391
  "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   392
  by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   393
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   394
lemma pos_le_divide_eq: "0 < c ==> (a \<le> b/c) = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   395
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   396
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   397
  hence "(a \<le> b/c) = (a*c \<le> (b/c)*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   398
    by (simp add: mult_le_cancel_right less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   399
  also have "... = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   400
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   401
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   402
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   403
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   404
lemma neg_le_divide_eq: "c < 0 ==> (a \<le> b/c) = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   405
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   406
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   407
  hence "(a \<le> b/c) = ((b/c)*c \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   408
    by (simp add: mult_le_cancel_right less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   409
  also have "... = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   410
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   411
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   412
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   413
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   414
lemma pos_less_divide_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   415
     "0 < c ==> (a < b/c) = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   416
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   417
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   418
  hence "(a < b/c) = (a*c < (b/c)*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   419
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   420
  also have "... = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   421
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   422
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   423
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   424
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   425
lemma neg_less_divide_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   426
 "c < 0 ==> (a < b/c) = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   427
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   428
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   429
  hence "(a < b/c) = ((b/c)*c < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   430
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   431
  also have "... = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   432
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   433
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   434
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   435
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   436
lemma pos_divide_less_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   437
     "0 < c ==> (b/c < a) = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   438
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   439
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   440
  hence "(b/c < a) = ((b/c)*c < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   441
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   442
  also have "... = (b < a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   443
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   444
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   445
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   446
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   447
lemma neg_divide_less_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   448
 "c < 0 ==> (b/c < a) = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   449
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   450
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   451
  hence "(b/c < a) = (a*c < (b/c)*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   452
    by (simp add: mult_less_cancel_right_disj less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   453
  also have "... = (a*c < b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   454
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   455
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   456
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   457
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   458
lemma pos_divide_le_eq: "0 < c ==> (b/c \<le> a) = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   459
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   460
  assume less: "0<c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   461
  hence "(b/c \<le> a) = ((b/c)*c \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   462
    by (simp add: mult_le_cancel_right less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   463
  also have "... = (b \<le> a*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   464
    by (simp add: less_imp_not_eq2 [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   465
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   466
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   467
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   468
lemma neg_divide_le_eq: "c < 0 ==> (b/c \<le> a) = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   469
proof -
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   470
  assume less: "c<0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   471
  hence "(b/c \<le> a) = (a*c \<le> (b/c)*c)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   472
    by (simp add: mult_le_cancel_right less_not_sym [OF less])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   473
  also have "... = (a*c \<le> b)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   474
    by (simp add: less_imp_not_eq [OF less] divide_inverse mult_assoc) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   475
  finally show ?thesis .
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   476
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   477
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   478
text{* Lemmas @{text field_simps} multiply with denominators in in(equations)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   479
if they can be proved to be non-zero (for equations) or positive/negative
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   480
(for inequations). Can be too aggressive and is therefore separate from the
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   481
more benign @{text algebra_simps}. *}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   482
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   483
lemmas field_simps[no_atp] = field_eq_simps
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   484
  (* multiply ineqn *)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   485
  pos_divide_less_eq neg_divide_less_eq
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   486
  pos_less_divide_eq neg_less_divide_eq
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   487
  pos_divide_le_eq neg_divide_le_eq
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   488
  pos_le_divide_eq neg_le_divide_eq
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   489
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   490
thm field_eq_simps
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   491
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   492
text{* Lemmas @{text sign_simps} is a first attempt to automate proofs
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   493
of positivity/negativity needed for @{text field_simps}. Have not added @{text
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   494
sign_simps} to @{text field_simps} because the former can lead to case
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   495
explosions. *}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   496
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   497
lemmas sign_simps[no_atp] = group_simps
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   498
  zero_less_mult_iff mult_less_0_iff
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   499
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   500
(* Only works once linear arithmetic is installed:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   501
text{*An example:*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   502
lemma fixes a b c d e f :: "'a::linordered_field"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   503
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   504
 ((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   505
 ((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   506
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   507
 prefer 2 apply(simp add:sign_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   508
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   509
 prefer 2 apply(simp add:sign_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   510
apply(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   511
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   512
*)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   513
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   514
lemma divide_pos_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   515
  "0 < x ==> 0 < y ==> 0 < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   516
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   517
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   518
lemma divide_nonneg_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   519
  "0 <= x ==> 0 < y ==> 0 <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   520
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   521
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   522
lemma divide_neg_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   523
  "x < 0 ==> 0 < y ==> x / y < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   524
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   525
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   526
lemma divide_nonpos_pos:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   527
  "x <= 0 ==> 0 < y ==> x / y <= 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   528
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   529
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   530
lemma divide_pos_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   531
  "0 < x ==> y < 0 ==> x / y < 0"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   532
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   533
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   534
lemma divide_nonneg_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   535
  "0 <= x ==> y < 0 ==> x / y <= 0" 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   536
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   537
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   538
lemma divide_neg_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   539
  "x < 0 ==> y < 0 ==> 0 < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   540
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   541
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   542
lemma divide_nonpos_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   543
  "x <= 0 ==> y < 0 ==> 0 <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   544
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   545
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   546
lemma divide_strict_right_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   547
     "[|a < b; 0 < c|] ==> a / c < b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   548
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   549
              positive_imp_inverse_positive)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   550
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   551
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   552
lemma divide_strict_right_mono_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   553
     "[|b < a; c < 0|] ==> a / c < b / c"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   554
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   555
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   556
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   557
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   558
text{*The last premise ensures that @{term a} and @{term b} 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   559
      have the same sign*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   560
lemma divide_strict_left_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   561
  "[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   562
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   563
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   564
lemma divide_left_mono:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   565
  "[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   566
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   567
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   568
lemma divide_strict_left_mono_neg:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   569
  "[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   570
by(auto simp: field_simps times_divide_eq zero_less_mult_iff mult_strict_right_mono_neg)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   571
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   572
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   573
    x / y <= z"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   574
by (subst pos_divide_le_eq, assumption+)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   575
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   576
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   577
    z <= x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   578
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   579
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   580
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   581
    x / y < z"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   582
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   583
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   584
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   585
    z < x / y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   586
by(simp add:field_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   587
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   588
lemma frac_le: "0 <= x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   589
    x <= y ==> 0 < w ==> w <= z  ==> x / z <= y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   590
  apply (rule mult_imp_div_pos_le)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   591
  apply simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   592
  apply (subst times_divide_eq_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   593
  apply (rule mult_imp_le_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   594
  apply (rule mult_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   595
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   596
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   597
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   598
lemma frac_less: "0 <= x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   599
    x < y ==> 0 < w ==> w <= z  ==> x / z < y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   600
  apply (rule mult_imp_div_pos_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   601
  apply simp
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   602
  apply (subst times_divide_eq_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   603
  apply (rule mult_imp_less_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   604
  apply (erule mult_less_le_imp_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   605
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   606
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   607
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   608
lemma frac_less2: "0 < x ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   609
    x <= y ==> 0 < w ==> w < z  ==> x / z < y / w"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   610
  apply (rule mult_imp_div_pos_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   611
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   612
  apply (subst times_divide_eq_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   613
  apply (rule mult_imp_less_div_pos, assumption)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   614
  apply (erule mult_le_less_imp_less)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   615
  apply simp_all
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   616
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   617
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   618
text{*It's not obvious whether these should be simprules or not. 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   619
  Their effect is to gather terms into one big fraction, like
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   620
  a*b*c / x*y*z. The rationale for that is unclear, but many proofs 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   621
  seem to need them.*}
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   622
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   623
declare times_divide_eq [simp]
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   624
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   625
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   626
by (simp add: field_simps zero_less_two)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   627
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   628
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   629
by (simp add: field_simps zero_less_two)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   630
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   631
subclass dense_linorder
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   632
proof
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   633
  fix x y :: 'a
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   634
  from less_add_one show "\<exists>y. x < y" .. 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   635
  from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   636
  then have "x - 1 < x + 1 - 1" by (simp only: diff_minus [symmetric])
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   637
  then have "x - 1 < x" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   638
  then show "\<exists>y. y < x" ..
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   639
  show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   640
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   641
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   642
lemma nonzero_abs_inverse:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   643
     "a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   644
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   645
                      negative_imp_inverse_negative)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   646
apply (blast intro: positive_imp_inverse_positive elim: less_asym) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   647
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   648
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   649
lemma nonzero_abs_divide:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   650
     "b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   651
  by (simp add: divide_inverse abs_mult nonzero_abs_inverse) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   652
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   653
lemma field_le_epsilon:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   654
  assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   655
  shows "x \<le> y"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   656
proof (rule dense_le)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   657
  fix t assume "t < x"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   658
  hence "0 < x - t" by (simp add: less_diff_eq)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   659
  from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   660
  then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   661
  then show "t \<le> y" by (simp add: algebra_simps)
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   662
qed
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   663
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   664
end
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   665
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   666
lemma le_divide_eq:
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   667
  "(a \<le> b/c) = 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   668
   (if 0 < c then a*c \<le> b
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   669
             else if c < 0 then b \<le> a*c
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   670
             else  a \<le> (0::'a::{linordered_field,division_by_zero}))"
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   671
apply (cases "c=0", simp) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   672
apply (force simp add: pos_le_divide_eq neg_le_divide_eq linorder_neq_iff) 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   673
done
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   674
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   675
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
   676
  "(0 < inverse a) = (0 < (a::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   677
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   678
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
   679
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   680
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   681
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
   682
  "(inverse a < 0) = (a < (0::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   683
apply (cases "a = 0", simp)
14277
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   684
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
   685
done
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   686
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   687
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
   688
  "(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
   689
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   690
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   691
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
   692
  "(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
   693
by (simp add: linorder_not_less [symmetric])
ad66687ece6e more field division lemmas transferred from Real to Ring_and_Field
paulson
parents: 14272
diff changeset
   694
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   695
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
   696
  "(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
   697
proof cases
14365
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   698
  assume "0 < x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   699
    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
   700
    show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   701
next
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   702
  assume notless: "~ (0 < x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   703
  have "~ (1 < inverse x)"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   704
  proof
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   705
    assume "1 < inverse x"
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   706
    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
   707
    also have "... < 1" by (rule zero_less_one) 
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   708
    finally show False by auto
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   709
  qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   710
  with notless show ?thesis by simp
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   711
qed
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   712
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   713
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
   714
  "(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
   715
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
   716
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   717
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
   718
  "(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
   719
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
   720
3d4df8c166ae replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents: 14353
diff changeset
   721
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
   722
  "(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
   723
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
   724
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   725
lemma divide_le_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   726
  "(b/c \<le> a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   727
   (if 0 < c then b \<le> a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   728
             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
   729
             else 0 \<le> (a::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   730
apply (cases "c=0", simp) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   731
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
   732
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   733
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   734
lemma less_divide_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   735
  "(a < b/c) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   736
   (if 0 < c then a*c < b
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   737
             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
   738
             else  a < (0::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   739
apply (cases "c=0", simp) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   740
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
   741
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   742
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   743
lemma divide_less_eq:
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   744
  "(b/c < a) = 
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   745
   (if 0 < c then b < a*c
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   746
             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
   747
             else 0 < (a::'a::{linordered_field,division_by_zero}))"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   748
apply (cases "c=0", simp) 
14288
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   749
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
   750
done
d149e3cbdb39 Moving some theorems from Real/RealArith0.ML
paulson
parents: 14284
diff changeset
   751
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   752
text {*Division and Signs*}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   753
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   754
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
   755
     "((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
   756
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
   757
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   758
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
   759
     "(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
   760
      (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
   761
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
   762
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   763
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
   764
     "((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
   765
      (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
   766
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
   767
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   768
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
   769
     "(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
   770
      (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
   771
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
   772
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   773
text {* Division and the Number One *}
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   774
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   775
text{*Simplify expressions equated with 1*}
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   776
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   777
lemma zero_eq_1_divide_iff [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   778
     "((0::'a::{linordered_field,division_by_zero}) = 1/a) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   779
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   780
apply (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
   781
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   782
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   783
lemma one_divide_eq_0_iff [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   784
     "(1/a = (0::'a::{linordered_field,division_by_zero})) = (a = 0)"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   785
apply (cases "a=0", simp)
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   786
apply (insert zero_neq_one [THEN not_sym])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   787
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
   788
done
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   789
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   790
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
   791
lemmas zero_less_divide_1_iff = zero_less_divide_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   792
lemmas divide_less_0_1_iff = divide_less_0_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   793
lemmas zero_le_divide_1_iff = zero_le_divide_iff [of 1, simplified]
9a5419d5ca01 simplified the special-case simprules
paulson
parents: 17085
diff changeset
   794
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
   795
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   796
declare zero_less_divide_1_iff [simp,no_atp]
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   797
declare divide_less_0_1_iff [simp,no_atp]
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   798
declare zero_le_divide_1_iff [simp,no_atp]
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   799
declare divide_le_0_1_iff [simp,no_atp]
14353
79f9fbef9106 Added lemmas to Ring_and_Field with slightly modified simplification rules
paulson
parents: 14348
diff changeset
   800
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   801
lemma divide_right_mono:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   802
     "[|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
   803
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
   804
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   805
lemma divide_right_mono_neg: "(a::'a::{linordered_field,division_by_zero}) <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   806
    ==> c <= 0 ==> b / c <= a / c"
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   807
apply (drule divide_right_mono [of _ _ "- c"])
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   808
apply auto
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   809
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   810
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   811
lemma divide_left_mono_neg: "(a::'a::{linordered_field,division_by_zero}) <= b 
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   812
    ==> 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
   813
  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
   814
  apply (auto simp add: mult_commute)
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   815
done
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   816
23482
2f4be6844f7c tuned and used field_simps
nipkow
parents: 23477
diff changeset
   817
14293
22542982bffd moving some division theorems to Ring_and_Field
paulson
parents: 14288
diff changeset
   818
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   819
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
   820
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   821
lemma le_divide_eq_1 [no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   822
  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
   823
  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
   824
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
   825
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   826
lemma divide_le_eq_1 [no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   827
  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
   828
  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
   829
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
   830
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   831
lemma less_divide_eq_1 [no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   832
  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
   833
  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
   834
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
   835
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   836
lemma divide_less_eq_1 [no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   837
  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
   838
  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
   839
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
   840
23389
aaca6a8e5414 tuned proofs: avoid implicit prems;
wenzelm
parents: 23326
diff changeset
   841
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   842
text {*Conditional Simplification Rules: No Case Splits*}
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   843
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   844
lemma le_divide_eq_1_pos [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   845
  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
   846
  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
   847
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
   848
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   849
lemma le_divide_eq_1_neg [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   850
  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
   851
  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
   852
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
   853
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   854
lemma divide_le_eq_1_pos [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   855
  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
   856
  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
   857
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
   858
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   859
lemma divide_le_eq_1_neg [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   860
  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
   861
  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
   862
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
   863
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   864
lemma less_divide_eq_1_pos [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   865
  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
   866
  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
   867
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
   868
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   869
lemma less_divide_eq_1_neg [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   870
  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
   871
  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
   872
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
   873
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   874
lemma divide_less_eq_1_pos [simp,no_atp]:
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34146
diff changeset
   875
  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
   876
  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
   877
by (auto simp add: divide_less_eq)
bb99c2e705ca tidied, and added missing thm divide_less_eq_1_neg
paulson
parents: 18623
diff changeset
   878
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   879
lemma divide_less_eq_1_neg [simp,no_atp]:
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 "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
   882
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
   883
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   884
lemma eq_divide_eq_1 [simp,no_atp]:
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 "(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
   887
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
   888
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35579
diff changeset
   889
lemma divide_eq_eq_1 [simp,no_atp]:
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 "(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
   892
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
   893
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   894
lemma abs_inverse [simp]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   895
     "\<bar>inverse (a::'a::{linordered_field,division_by_zero})\<bar> = 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   896
      inverse \<bar>a\<bar>"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   897
apply (cases "a=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   898
apply (simp add: nonzero_abs_inverse) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   899
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   900
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   901
lemma abs_divide [simp]:
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   902
     "\<bar>a / (b::'a::{linordered_field,division_by_zero})\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
21328
73bb86d0f483 dropped Inductive dependency
haftmann
parents: 21258
diff changeset
   903
apply (cases "b=0", simp) 
14294
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   904
apply (simp add: nonzero_abs_divide) 
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   905
done
f4d806fd72ce absolute value theorems moved to HOL/Ring_and_Field
paulson
parents: 14293
diff changeset
   906
36301
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   907
lemma abs_div_pos: "(0::'a::{linordered_field,division_by_zero}) < y ==> 
72f4d079ebf8 more localization; factored out lemmas for division_ring
haftmann
parents: 35828
diff changeset
   908
    \<bar>x\<bar> / y = \<bar>x / y\<bar>"
25304
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   909
  apply (subst abs_divide)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   910
  apply (simp add: order_less_imp_le)
7491c00f0915 removed subclass edge ordered_ring < lordered_ring
haftmann
parents: 25267
diff changeset
   911
done
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16568
diff changeset
   912
35579
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   913
lemma field_le_mult_one_interval:
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   914
  fixes x :: "'a\<Colon>{linordered_field,division_by_zero}"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   915
  assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   916
  shows "x \<le> y"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   917
proof (cases "0 < x")
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   918
  assume "0 < x"
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   919
  thus ?thesis
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   920
    using dense_le_bounded[of 0 1 "y/x"] *
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   921
    unfolding le_divide_eq if_P[OF `0 < x`] by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   922
next
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   923
  assume "\<not>0 < x" hence "x \<le> 0" by simp
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   924
  obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   925
  hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] `x \<le> 0` by auto
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   926
  also note *[OF s]
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   927
  finally show ?thesis .
cc9a5a0ab5ea Add dense_le, dense_le_bounded, field_le_mult_one_interval.
hoelzl
parents: 35216
diff changeset
   928
qed
35090
88cc65ae046e moved lemma field_le_epsilon from Real.thy to Fields.thy
haftmann
parents: 35084
diff changeset
   929
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   930
code_modulename SML
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
   931
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   932
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   933
code_modulename OCaml
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
   934
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   935
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   936
code_modulename Haskell
35050
9f841f20dca6 renamed OrderedGroup to Groups; split theory Ring_and_Field into Rings Fields
haftmann
parents: 35043
diff changeset
   937
  Fields Arith
33364
2bd12592c5e8 tuned code setup
haftmann
parents: 33319
diff changeset
   938
14265
95b42e69436c HOL: installation of Ring_and_Field as the basis for Naturals and Reals
paulson
parents:
diff changeset
   939
end