haftmann@35050
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1 |
(* Title: HOL/Fields.thy
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wenzelm@32960
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Author: Gertrud Bauer
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wenzelm@32960
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3 |
Author: Steven Obua
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wenzelm@32960
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Author: Tobias Nipkow
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wenzelm@32960
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5 |
Author: Lawrence C Paulson
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wenzelm@32960
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Author: Markus Wenzel
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wenzelm@32960
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Author: Jeremy Avigad
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paulson@14265
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*)
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paulson@14265
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wenzelm@60758
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section \<open>Fields\<close>
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haftmann@25152
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haftmann@35050
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theory Fields
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haftmann@35050
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imports Rings
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haftmann@25186
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begin
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paulson@14421
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wenzelm@60758
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subsection \<open>Division rings\<close>
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huffman@44064
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wenzelm@60758
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text \<open>
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huffman@44064
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A division ring is like a field, but without the commutativity requirement.
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wenzelm@60758
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\<close>
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huffman@44064
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haftmann@60352
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class inverse = divide +
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huffman@44064
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fixes inverse :: "'a \<Rightarrow> 'a"
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hoelzl@58776
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begin
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haftmann@60352
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haftmann@60352
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abbreviation inverse_divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "'/" 70)
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haftmann@60352
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where
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haftmann@60352
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"inverse_divide \<equiv> divide"
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hoelzl@58776
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hoelzl@58776
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end
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hoelzl@58776
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wenzelm@60758
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text\<open>Lemmas @{text divide_simps} move division to the outside and eliminates them on (in)equalities.\<close>
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hoelzl@56481
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wenzelm@57950
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named_theorems divide_simps "rewrite rules to eliminate divisions"
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hoelzl@56481
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huffman@44064
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class division_ring = ring_1 + inverse +
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huffman@44064
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assumes left_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
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huffman@44064
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assumes right_inverse [simp]: "a \<noteq> 0 \<Longrightarrow> a * inverse a = 1"
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huffman@44064
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assumes divide_inverse: "a / b = a * inverse b"
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haftmann@59867
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assumes inverse_zero [simp]: "inverse 0 = 0"
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huffman@44064
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begin
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huffman@44064
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huffman@44064
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subclass ring_1_no_zero_divisors
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huffman@44064
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proof
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huffman@44064
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fix a b :: 'a
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huffman@44064
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assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
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huffman@44064
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show "a * b \<noteq> 0"
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huffman@44064
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proof
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huffman@44064
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assume ab: "a * b = 0"
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huffman@44064
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hence "0 = inverse a * (a * b) * inverse b" by simp
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huffman@44064
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also have "\<dots> = (inverse a * a) * (b * inverse b)"
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haftmann@57512
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by (simp only: mult.assoc)
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huffman@44064
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also have "\<dots> = 1" using a b by simp
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huffman@44064
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finally show False by simp
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huffman@44064
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qed
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huffman@44064
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qed
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huffman@44064
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huffman@44064
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lemma nonzero_imp_inverse_nonzero:
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huffman@44064
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"a \<noteq> 0 \<Longrightarrow> inverse a \<noteq> 0"
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huffman@44064
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proof
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huffman@44064
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assume ianz: "inverse a = 0"
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huffman@44064
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assume "a \<noteq> 0"
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huffman@44064
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hence "1 = a * inverse a" by simp
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huffman@44064
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also have "... = 0" by (simp add: ianz)
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huffman@44064
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finally have "1 = 0" .
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huffman@44064
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thus False by (simp add: eq_commute)
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huffman@44064
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qed
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huffman@44064
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huffman@44064
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lemma inverse_zero_imp_zero:
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huffman@44064
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"inverse a = 0 \<Longrightarrow> a = 0"
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huffman@44064
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apply (rule classical)
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huffman@44064
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apply (drule nonzero_imp_inverse_nonzero)
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huffman@44064
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apply auto
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huffman@44064
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done
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huffman@44064
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lp15@59667
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lemma inverse_unique:
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huffman@44064
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assumes ab: "a * b = 1"
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huffman@44064
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shows "inverse a = b"
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huffman@44064
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proof -
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huffman@44064
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have "a \<noteq> 0" using ab by (cases "a = 0") simp_all
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huffman@44064
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moreover have "inverse a * (a * b) = inverse a" by (simp add: ab)
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haftmann@57512
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ultimately show ?thesis by (simp add: mult.assoc [symmetric])
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huffman@44064
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qed
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huffman@44064
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huffman@44064
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lemma nonzero_inverse_minus_eq:
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huffman@44064
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"a \<noteq> 0 \<Longrightarrow> inverse (- a) = - inverse a"
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huffman@44064
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by (rule inverse_unique) simp
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huffman@44064
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huffman@44064
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lemma nonzero_inverse_inverse_eq:
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huffman@44064
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"a \<noteq> 0 \<Longrightarrow> inverse (inverse a) = a"
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huffman@44064
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by (rule inverse_unique) simp
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huffman@44064
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huffman@44064
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lemma nonzero_inverse_eq_imp_eq:
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huffman@44064
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assumes "inverse a = inverse b" and "a \<noteq> 0" and "b \<noteq> 0"
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huffman@44064
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shows "a = b"
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huffman@44064
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proof -
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wenzelm@60758
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from \<open>inverse a = inverse b\<close>
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huffman@44064
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have "inverse (inverse a) = inverse (inverse b)" by (rule arg_cong)
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wenzelm@60758
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with \<open>a \<noteq> 0\<close> and \<open>b \<noteq> 0\<close> show "a = b"
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huffman@44064
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by (simp add: nonzero_inverse_inverse_eq)
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huffman@44064
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qed
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huffman@44064
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huffman@44064
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lemma inverse_1 [simp]: "inverse 1 = 1"
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huffman@44064
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by (rule inverse_unique) simp
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huffman@44064
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lp15@59667
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lemma nonzero_inverse_mult_distrib:
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huffman@44064
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assumes "a \<noteq> 0" and "b \<noteq> 0"
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huffman@44064
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shows "inverse (a * b) = inverse b * inverse a"
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huffman@44064
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proof -
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huffman@44064
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have "a * (b * inverse b) * inverse a = 1" using assms by simp
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haftmann@57512
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hence "a * b * (inverse b * inverse a) = 1" by (simp only: mult.assoc)
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huffman@44064
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thus ?thesis by (rule inverse_unique)
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huffman@44064
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qed
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huffman@44064
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huffman@44064
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lemma division_ring_inverse_add:
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huffman@44064
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"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = inverse a * (a + b) * inverse b"
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huffman@44064
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by (simp add: algebra_simps)
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huffman@44064
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huffman@44064
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lemma division_ring_inverse_diff:
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huffman@44064
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"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a - inverse b = inverse a * (b - a) * inverse b"
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huffman@44064
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by (simp add: algebra_simps)
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huffman@44064
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huffman@44064
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lemma right_inverse_eq: "b \<noteq> 0 \<Longrightarrow> a / b = 1 \<longleftrightarrow> a = b"
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huffman@44064
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proof
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huffman@44064
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assume neq: "b \<noteq> 0"
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huffman@44064
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{
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haftmann@57512
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hence "a = (a / b) * b" by (simp add: divide_inverse mult.assoc)
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huffman@44064
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also assume "a / b = 1"
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huffman@44064
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finally show "a = b" by simp
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huffman@44064
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next
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huffman@44064
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assume "a = b"
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huffman@44064
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with neq show "a / b = 1" by (simp add: divide_inverse)
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huffman@44064
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}
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huffman@44064
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qed
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huffman@44064
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huffman@44064
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lemma nonzero_inverse_eq_divide: "a \<noteq> 0 \<Longrightarrow> inverse a = 1 / a"
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huffman@44064
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by (simp add: divide_inverse)
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huffman@44064
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huffman@44064
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lemma divide_self [simp]: "a \<noteq> 0 \<Longrightarrow> a / a = 1"
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huffman@44064
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by (simp add: divide_inverse)
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huffman@44064
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hoelzl@56481
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lemma inverse_eq_divide [field_simps, divide_simps]: "inverse a = 1 / a"
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huffman@44064
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by (simp add: divide_inverse)
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huffman@44064
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huffman@44064
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lemma add_divide_distrib: "(a+b) / c = a/c + b/c"
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huffman@44064
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by (simp add: divide_inverse algebra_simps)
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huffman@44064
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huffman@44064
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lemma times_divide_eq_right [simp]: "a * (b / c) = (a * b) / c"
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haftmann@57512
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by (simp add: divide_inverse mult.assoc)
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huffman@44064
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huffman@44064
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lemma minus_divide_left: "- (a / b) = (-a) / b"
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huffman@44064
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by (simp add: divide_inverse)
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huffman@44064
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153 |
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huffman@44064
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lemma nonzero_minus_divide_right: "b \<noteq> 0 ==> - (a / b) = a / (- b)"
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huffman@44064
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by (simp add: divide_inverse nonzero_inverse_minus_eq)
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huffman@44064
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156 |
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huffman@44064
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157 |
lemma nonzero_minus_divide_divide: "b \<noteq> 0 ==> (-a) / (-b) = a / b"
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huffman@44064
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by (simp add: divide_inverse nonzero_inverse_minus_eq)
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huffman@44064
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hoelzl@56479
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lemma divide_minus_left [simp]: "(-a) / b = - (a / b)"
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huffman@44064
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by (simp add: divide_inverse)
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huffman@44064
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162 |
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huffman@44064
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163 |
lemma diff_divide_distrib: "(a - b) / c = a / c - b / c"
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hoelzl@56479
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164 |
using add_divide_distrib [of a "- b" c] by simp
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huffman@44064
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165 |
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huffman@44064
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166 |
lemma nonzero_eq_divide_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> a = b / c \<longleftrightarrow> a * c = b"
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huffman@44064
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167 |
proof -
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huffman@44064
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168 |
assume [simp]: "c \<noteq> 0"
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huffman@44064
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169 |
have "a = b / c \<longleftrightarrow> a * c = (b / c) * c" by simp
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haftmann@57512
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170 |
also have "... \<longleftrightarrow> a * c = b" by (simp add: divide_inverse mult.assoc)
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huffman@44064
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171 |
finally show ?thesis .
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huffman@44064
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172 |
qed
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huffman@44064
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173 |
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huffman@44064
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174 |
lemma nonzero_divide_eq_eq [field_simps]: "c \<noteq> 0 \<Longrightarrow> b / c = a \<longleftrightarrow> b = a * c"
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huffman@44064
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175 |
proof -
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huffman@44064
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176 |
assume [simp]: "c \<noteq> 0"
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huffman@44064
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177 |
have "b / c = a \<longleftrightarrow> (b / c) * c = a * c" by simp
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lp15@59667
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178 |
also have "... \<longleftrightarrow> b = a * c" by (simp add: divide_inverse mult.assoc)
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huffman@44064
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179 |
finally show ?thesis .
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huffman@44064
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180 |
qed
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huffman@44064
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181 |
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hoelzl@56480
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182 |
lemma nonzero_neg_divide_eq_eq [field_simps]: "b \<noteq> 0 \<Longrightarrow> - (a / b) = c \<longleftrightarrow> - a = c * b"
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haftmann@59535
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183 |
using nonzero_divide_eq_eq[of b "-a" c] by simp
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nipkow@56441
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184 |
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hoelzl@56480
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185 |
lemma nonzero_neg_divide_eq_eq2 [field_simps]: "b \<noteq> 0 \<Longrightarrow> c = - (a / b) \<longleftrightarrow> c * b = - a"
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hoelzl@56480
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186 |
using nonzero_neg_divide_eq_eq[of b a c] by auto
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nipkow@56441
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187 |
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huffman@44064
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188 |
lemma divide_eq_imp: "c \<noteq> 0 \<Longrightarrow> b = a * c \<Longrightarrow> b / c = a"
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haftmann@57512
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189 |
by (simp add: divide_inverse mult.assoc)
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huffman@44064
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190 |
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huffman@44064
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191 |
lemma eq_divide_imp: "c \<noteq> 0 \<Longrightarrow> a * c = b \<Longrightarrow> a = b / c"
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haftmann@57512
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192 |
by (drule sym) (simp add: divide_inverse mult.assoc)
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huffman@44064
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193 |
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nipkow@56445
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194 |
lemma add_divide_eq_iff [field_simps]:
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nipkow@56445
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195 |
"z \<noteq> 0 \<Longrightarrow> x + y / z = (x * z + y) / z"
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nipkow@56445
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196 |
by (simp add: add_divide_distrib nonzero_eq_divide_eq)
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nipkow@56445
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197 |
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nipkow@56445
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198 |
lemma divide_add_eq_iff [field_simps]:
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nipkow@56445
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199 |
"z \<noteq> 0 \<Longrightarrow> x / z + y = (x + y * z) / z"
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nipkow@56445
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200 |
by (simp add: add_divide_distrib nonzero_eq_divide_eq)
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nipkow@56445
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201 |
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nipkow@56445
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202 |
lemma diff_divide_eq_iff [field_simps]:
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nipkow@56445
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203 |
"z \<noteq> 0 \<Longrightarrow> x - y / z = (x * z - y) / z"
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nipkow@56445
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204 |
by (simp add: diff_divide_distrib nonzero_eq_divide_eq eq_diff_eq)
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nipkow@56445
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205 |
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hoelzl@56480
|
206 |
lemma minus_divide_add_eq_iff [field_simps]:
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hoelzl@56480
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207 |
"z \<noteq> 0 \<Longrightarrow> - (x / z) + y = (- x + y * z) / z"
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haftmann@59535
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208 |
by (simp add: add_divide_distrib diff_divide_eq_iff)
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hoelzl@56480
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209 |
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nipkow@56445
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210 |
lemma divide_diff_eq_iff [field_simps]:
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nipkow@56445
|
211 |
"z \<noteq> 0 \<Longrightarrow> x / z - y = (x - y * z) / z"
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nipkow@56445
|
212 |
by (simp add: field_simps)
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nipkow@56445
|
213 |
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hoelzl@56480
|
214 |
lemma minus_divide_diff_eq_iff [field_simps]:
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hoelzl@56480
|
215 |
"z \<noteq> 0 \<Longrightarrow> - (x / z) - y = (- x - y * z) / z"
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haftmann@59535
|
216 |
by (simp add: divide_diff_eq_iff[symmetric])
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hoelzl@56480
|
217 |
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haftmann@60353
|
218 |
lemma division_ring_divide_zero [simp]:
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huffman@44064
|
219 |
"a / 0 = 0"
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huffman@44064
|
220 |
by (simp add: divide_inverse)
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huffman@44064
|
221 |
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huffman@44064
|
222 |
lemma divide_self_if [simp]:
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huffman@44064
|
223 |
"a / a = (if a = 0 then 0 else 1)"
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huffman@44064
|
224 |
by simp
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huffman@44064
|
225 |
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huffman@44064
|
226 |
lemma inverse_nonzero_iff_nonzero [simp]:
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huffman@44064
|
227 |
"inverse a = 0 \<longleftrightarrow> a = 0"
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huffman@44064
|
228 |
by rule (fact inverse_zero_imp_zero, simp)
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huffman@44064
|
229 |
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huffman@44064
|
230 |
lemma inverse_minus_eq [simp]:
|
huffman@44064
|
231 |
"inverse (- a) = - inverse a"
|
huffman@44064
|
232 |
proof cases
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huffman@44064
|
233 |
assume "a=0" thus ?thesis by simp
|
huffman@44064
|
234 |
next
|
lp15@59667
|
235 |
assume "a\<noteq>0"
|
huffman@44064
|
236 |
thus ?thesis by (simp add: nonzero_inverse_minus_eq)
|
huffman@44064
|
237 |
qed
|
huffman@44064
|
238 |
|
huffman@44064
|
239 |
lemma inverse_inverse_eq [simp]:
|
huffman@44064
|
240 |
"inverse (inverse a) = a"
|
huffman@44064
|
241 |
proof cases
|
huffman@44064
|
242 |
assume "a=0" thus ?thesis by simp
|
huffman@44064
|
243 |
next
|
lp15@59667
|
244 |
assume "a\<noteq>0"
|
huffman@44064
|
245 |
thus ?thesis by (simp add: nonzero_inverse_inverse_eq)
|
huffman@44064
|
246 |
qed
|
huffman@44064
|
247 |
|
huffman@44680
|
248 |
lemma inverse_eq_imp_eq:
|
huffman@44680
|
249 |
"inverse a = inverse b \<Longrightarrow> a = b"
|
huffman@44680
|
250 |
by (drule arg_cong [where f="inverse"], simp)
|
huffman@44680
|
251 |
|
huffman@44680
|
252 |
lemma inverse_eq_iff_eq [simp]:
|
huffman@44680
|
253 |
"inverse a = inverse b \<longleftrightarrow> a = b"
|
huffman@44680
|
254 |
by (force dest!: inverse_eq_imp_eq)
|
huffman@44680
|
255 |
|
hoelzl@56481
|
256 |
lemma add_divide_eq_if_simps [divide_simps]:
|
hoelzl@56481
|
257 |
"a + b / z = (if z = 0 then a else (a * z + b) / z)"
|
hoelzl@56481
|
258 |
"a / z + b = (if z = 0 then b else (a + b * z) / z)"
|
hoelzl@56481
|
259 |
"- (a / z) + b = (if z = 0 then b else (-a + b * z) / z)"
|
hoelzl@56481
|
260 |
"a - b / z = (if z = 0 then a else (a * z - b) / z)"
|
hoelzl@56481
|
261 |
"a / z - b = (if z = 0 then -b else (a - b * z) / z)"
|
hoelzl@56481
|
262 |
"- (a / z) - b = (if z = 0 then -b else (- a - b * z) / z)"
|
hoelzl@56481
|
263 |
by (simp_all add: add_divide_eq_iff divide_add_eq_iff diff_divide_eq_iff divide_diff_eq_iff
|
hoelzl@56481
|
264 |
minus_divide_diff_eq_iff)
|
hoelzl@56481
|
265 |
|
hoelzl@56481
|
266 |
lemma [divide_simps]:
|
hoelzl@56481
|
267 |
shows divide_eq_eq: "b / c = a \<longleftrightarrow> (if c \<noteq> 0 then b = a * c else a = 0)"
|
hoelzl@56481
|
268 |
and eq_divide_eq: "a = b / c \<longleftrightarrow> (if c \<noteq> 0 then a * c = b else a = 0)"
|
hoelzl@56481
|
269 |
and minus_divide_eq_eq: "- (b / c) = a \<longleftrightarrow> (if c \<noteq> 0 then - b = a * c else a = 0)"
|
hoelzl@56481
|
270 |
and eq_minus_divide_eq: "a = - (b / c) \<longleftrightarrow> (if c \<noteq> 0 then a * c = - b else a = 0)"
|
hoelzl@56481
|
271 |
by (auto simp add: field_simps)
|
hoelzl@56481
|
272 |
|
huffman@44064
|
273 |
end
|
huffman@44064
|
274 |
|
wenzelm@60758
|
275 |
subsection \<open>Fields\<close>
|
huffman@44064
|
276 |
|
huffman@22987
|
277 |
class field = comm_ring_1 + inverse +
|
haftmann@35084
|
278 |
assumes field_inverse: "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
|
haftmann@35084
|
279 |
assumes field_divide_inverse: "a / b = a * inverse b"
|
haftmann@59867
|
280 |
assumes field_inverse_zero: "inverse 0 = 0"
|
haftmann@25267
|
281 |
begin
|
huffman@20496
|
282 |
|
haftmann@25267
|
283 |
subclass division_ring
|
haftmann@28823
|
284 |
proof
|
huffman@22987
|
285 |
fix a :: 'a
|
huffman@22987
|
286 |
assume "a \<noteq> 0"
|
huffman@22987
|
287 |
thus "inverse a * a = 1" by (rule field_inverse)
|
haftmann@57512
|
288 |
thus "a * inverse a = 1" by (simp only: mult.commute)
|
haftmann@35084
|
289 |
next
|
haftmann@35084
|
290 |
fix a b :: 'a
|
haftmann@35084
|
291 |
show "a / b = a * inverse b" by (rule field_divide_inverse)
|
haftmann@59867
|
292 |
next
|
haftmann@59867
|
293 |
show "inverse 0 = 0"
|
haftmann@59867
|
294 |
by (fact field_inverse_zero)
|
obua@14738
|
295 |
qed
|
haftmann@25230
|
296 |
|
haftmann@60353
|
297 |
subclass idom_divide
|
haftmann@60353
|
298 |
proof
|
haftmann@60353
|
299 |
fix b a
|
haftmann@60353
|
300 |
assume "b \<noteq> 0"
|
haftmann@60353
|
301 |
then show "a * b / b = a"
|
haftmann@60353
|
302 |
by (simp add: divide_inverse ac_simps)
|
haftmann@60353
|
303 |
next
|
haftmann@60353
|
304 |
fix a
|
haftmann@60353
|
305 |
show "a / 0 = 0"
|
haftmann@60353
|
306 |
by (simp add: divide_inverse)
|
haftmann@60353
|
307 |
qed
|
haftmann@25230
|
308 |
|
wenzelm@60758
|
309 |
text\<open>There is no slick version using division by zero.\<close>
|
huffman@30630
|
310 |
lemma inverse_add:
|
haftmann@60353
|
311 |
"a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse a + inverse b = (a + b) * inverse a * inverse b"
|
haftmann@60353
|
312 |
by (simp add: division_ring_inverse_add ac_simps)
|
huffman@30630
|
313 |
|
blanchet@54147
|
314 |
lemma nonzero_mult_divide_mult_cancel_left [simp]:
|
haftmann@60353
|
315 |
assumes [simp]: "c \<noteq> 0"
|
haftmann@60353
|
316 |
shows "(c * a) / (c * b) = a / b"
|
haftmann@60353
|
317 |
proof (cases "b = 0")
|
haftmann@60353
|
318 |
case True then show ?thesis by simp
|
haftmann@60353
|
319 |
next
|
haftmann@60353
|
320 |
case False
|
haftmann@60353
|
321 |
then have "(c*a)/(c*b) = c * a * (inverse b * inverse c)"
|
huffman@30630
|
322 |
by (simp add: divide_inverse nonzero_inverse_mult_distrib)
|
huffman@30630
|
323 |
also have "... = a * inverse b * (inverse c * c)"
|
haftmann@57514
|
324 |
by (simp only: ac_simps)
|
huffman@30630
|
325 |
also have "... = a * inverse b" by simp
|
huffman@30630
|
326 |
finally show ?thesis by (simp add: divide_inverse)
|
huffman@30630
|
327 |
qed
|
huffman@30630
|
328 |
|
blanchet@54147
|
329 |
lemma nonzero_mult_divide_mult_cancel_right [simp]:
|
haftmann@60353
|
330 |
"c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
|
haftmann@60353
|
331 |
using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)
|
huffman@30630
|
332 |
|
haftmann@36304
|
333 |
lemma times_divide_eq_left [simp]: "(b / c) * a = (b * a) / c"
|
haftmann@57514
|
334 |
by (simp add: divide_inverse ac_simps)
|
huffman@30630
|
335 |
|
huffman@30630
|
336 |
lemma add_frac_eq:
|
huffman@30630
|
337 |
assumes "y \<noteq> 0" and "z \<noteq> 0"
|
huffman@30630
|
338 |
shows "x / y + w / z = (x * z + w * y) / (y * z)"
|
huffman@30630
|
339 |
proof -
|
huffman@30630
|
340 |
have "x / y + w / z = (x * z) / (y * z) + (y * w) / (y * z)"
|
huffman@30630
|
341 |
using assms by simp
|
huffman@30630
|
342 |
also have "\<dots> = (x * z + y * w) / (y * z)"
|
huffman@30630
|
343 |
by (simp only: add_divide_distrib)
|
huffman@30630
|
344 |
finally show ?thesis
|
haftmann@57512
|
345 |
by (simp only: mult.commute)
|
huffman@30630
|
346 |
qed
|
huffman@30630
|
347 |
|
wenzelm@60758
|
348 |
text\<open>Special Cancellation Simprules for Division\<close>
|
huffman@30630
|
349 |
|
blanchet@54147
|
350 |
lemma nonzero_divide_mult_cancel_right [simp]:
|
haftmann@60353
|
351 |
"b \<noteq> 0 \<Longrightarrow> b / (a * b) = 1 / a"
|
haftmann@60353
|
352 |
using nonzero_mult_divide_mult_cancel_right [of b 1 a] by simp
|
huffman@30630
|
353 |
|
blanchet@54147
|
354 |
lemma nonzero_divide_mult_cancel_left [simp]:
|
haftmann@60353
|
355 |
"a \<noteq> 0 \<Longrightarrow> a / (a * b) = 1 / b"
|
haftmann@60353
|
356 |
using nonzero_mult_divide_mult_cancel_left [of a 1 b] by simp
|
huffman@30630
|
357 |
|
blanchet@54147
|
358 |
lemma nonzero_mult_divide_mult_cancel_left2 [simp]:
|
haftmann@60353
|
359 |
"c \<noteq> 0 \<Longrightarrow> (c * a) / (b * c) = a / b"
|
haftmann@60353
|
360 |
using nonzero_mult_divide_mult_cancel_left [of c a b] by (simp add: ac_simps)
|
huffman@30630
|
361 |
|
blanchet@54147
|
362 |
lemma nonzero_mult_divide_mult_cancel_right2 [simp]:
|
haftmann@60353
|
363 |
"c \<noteq> 0 \<Longrightarrow> (a * c) / (c * b) = a / b"
|
haftmann@60353
|
364 |
using nonzero_mult_divide_mult_cancel_right [of b c a] by (simp add: ac_simps)
|
huffman@30630
|
365 |
|
huffman@30630
|
366 |
lemma diff_frac_eq:
|
huffman@30630
|
367 |
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y - w / z = (x * z - w * y) / (y * z)"
|
haftmann@36348
|
368 |
by (simp add: field_simps)
|
huffman@30630
|
369 |
|
huffman@30630
|
370 |
lemma frac_eq_eq:
|
huffman@30630
|
371 |
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (x / y = w / z) = (x * z = w * y)"
|
haftmann@36348
|
372 |
by (simp add: field_simps)
|
haftmann@36348
|
373 |
|
haftmann@58512
|
374 |
lemma divide_minus1 [simp]: "x / - 1 = - x"
|
haftmann@58512
|
375 |
using nonzero_minus_divide_right [of "1" x] by simp
|
lp15@59667
|
376 |
|
wenzelm@60758
|
377 |
text\<open>This version builds in division by zero while also re-orienting
|
wenzelm@60758
|
378 |
the right-hand side.\<close>
|
paulson@14270
|
379 |
lemma inverse_mult_distrib [simp]:
|
haftmann@36409
|
380 |
"inverse (a * b) = inverse a * inverse b"
|
haftmann@36409
|
381 |
proof cases
|
lp15@59667
|
382 |
assume "a \<noteq> 0 & b \<noteq> 0"
|
haftmann@57514
|
383 |
thus ?thesis by (simp add: nonzero_inverse_mult_distrib ac_simps)
|
haftmann@36409
|
384 |
next
|
lp15@59667
|
385 |
assume "~ (a \<noteq> 0 & b \<noteq> 0)"
|
haftmann@36409
|
386 |
thus ?thesis by force
|
haftmann@36409
|
387 |
qed
|
paulson@14270
|
388 |
|
paulson@14365
|
389 |
lemma inverse_divide [simp]:
|
haftmann@36409
|
390 |
"inverse (a / b) = b / a"
|
haftmann@57512
|
391 |
by (simp add: divide_inverse mult.commute)
|
paulson@14365
|
392 |
|
wenzelm@23389
|
393 |
|
wenzelm@60758
|
394 |
text \<open>Calculations with fractions\<close>
|
avigad@16775
|
395 |
|
wenzelm@60758
|
396 |
text\<open>There is a whole bunch of simp-rules just for class @{text
|
nipkow@23413
|
397 |
field} but none for class @{text field} and @{text nonzero_divides}
|
wenzelm@60758
|
398 |
because the latter are covered by a simproc.\<close>
|
nipkow@23413
|
399 |
|
nipkow@23413
|
400 |
lemma mult_divide_mult_cancel_left:
|
haftmann@36409
|
401 |
"c \<noteq> 0 \<Longrightarrow> (c * a) / (c * b) = a / b"
|
haftmann@21328
|
402 |
apply (cases "b = 0")
|
huffman@35216
|
403 |
apply simp_all
|
paulson@14277
|
404 |
done
|
paulson@14277
|
405 |
|
nipkow@23413
|
406 |
lemma mult_divide_mult_cancel_right:
|
haftmann@36409
|
407 |
"c \<noteq> 0 \<Longrightarrow> (a * c) / (b * c) = a / b"
|
haftmann@21328
|
408 |
apply (cases "b = 0")
|
huffman@35216
|
409 |
apply simp_all
|
paulson@14321
|
410 |
done
|
nipkow@23413
|
411 |
|
blanchet@54147
|
412 |
lemma divide_divide_eq_right [simp]:
|
haftmann@36409
|
413 |
"a / (b / c) = (a * c) / b"
|
haftmann@57514
|
414 |
by (simp add: divide_inverse ac_simps)
|
paulson@14288
|
415 |
|
blanchet@54147
|
416 |
lemma divide_divide_eq_left [simp]:
|
haftmann@36409
|
417 |
"(a / b) / c = a / (b * c)"
|
haftmann@57512
|
418 |
by (simp add: divide_inverse mult.assoc)
|
paulson@14288
|
419 |
|
lp15@56365
|
420 |
lemma divide_divide_times_eq:
|
lp15@56365
|
421 |
"(x / y) / (z / w) = (x * w) / (y * z)"
|
lp15@56365
|
422 |
by simp
|
wenzelm@23389
|
423 |
|
wenzelm@60758
|
424 |
text \<open>Special Cancellation Simprules for Division\<close>
|
paulson@15234
|
425 |
|
blanchet@54147
|
426 |
lemma mult_divide_mult_cancel_left_if [simp]:
|
haftmann@36409
|
427 |
shows "(c * a) / (c * b) = (if c = 0 then 0 else a / b)"
|
haftmann@60353
|
428 |
by simp
|
nipkow@23413
|
429 |
|
paulson@15234
|
430 |
|
wenzelm@60758
|
431 |
text \<open>Division and Unary Minus\<close>
|
paulson@14293
|
432 |
|
haftmann@36409
|
433 |
lemma minus_divide_right:
|
haftmann@36409
|
434 |
"- (a / b) = a / - b"
|
haftmann@36409
|
435 |
by (simp add: divide_inverse)
|
paulson@14430
|
436 |
|
hoelzl@56479
|
437 |
lemma divide_minus_right [simp]:
|
haftmann@36409
|
438 |
"a / - b = - (a / b)"
|
haftmann@36409
|
439 |
by (simp add: divide_inverse)
|
huffman@30630
|
440 |
|
hoelzl@56479
|
441 |
lemma minus_divide_divide:
|
haftmann@36409
|
442 |
"(- a) / (- b) = a / b"
|
lp15@59667
|
443 |
apply (cases "b=0", simp)
|
lp15@59667
|
444 |
apply (simp add: nonzero_minus_divide_divide)
|
paulson@14293
|
445 |
done
|
paulson@14293
|
446 |
|
haftmann@36301
|
447 |
lemma inverse_eq_1_iff [simp]:
|
haftmann@36409
|
448 |
"inverse x = 1 \<longleftrightarrow> x = 1"
|
lp15@59667
|
449 |
by (insert inverse_eq_iff_eq [of x 1], simp)
|
wenzelm@23389
|
450 |
|
blanchet@54147
|
451 |
lemma divide_eq_0_iff [simp]:
|
haftmann@36409
|
452 |
"a / b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
|
haftmann@36409
|
453 |
by (simp add: divide_inverse)
|
haftmann@36301
|
454 |
|
blanchet@54147
|
455 |
lemma divide_cancel_right [simp]:
|
haftmann@36409
|
456 |
"a / c = b / c \<longleftrightarrow> c = 0 \<or> a = b"
|
haftmann@36409
|
457 |
apply (cases "c=0", simp)
|
haftmann@36409
|
458 |
apply (simp add: divide_inverse)
|
haftmann@36409
|
459 |
done
|
haftmann@36301
|
460 |
|
blanchet@54147
|
461 |
lemma divide_cancel_left [simp]:
|
lp15@59667
|
462 |
"c / a = c / b \<longleftrightarrow> c = 0 \<or> a = b"
|
haftmann@36409
|
463 |
apply (cases "c=0", simp)
|
haftmann@36409
|
464 |
apply (simp add: divide_inverse)
|
haftmann@36409
|
465 |
done
|
haftmann@36301
|
466 |
|
blanchet@54147
|
467 |
lemma divide_eq_1_iff [simp]:
|
haftmann@36409
|
468 |
"a / b = 1 \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
|
haftmann@36409
|
469 |
apply (cases "b=0", simp)
|
haftmann@36409
|
470 |
apply (simp add: right_inverse_eq)
|
haftmann@36409
|
471 |
done
|
haftmann@36301
|
472 |
|
blanchet@54147
|
473 |
lemma one_eq_divide_iff [simp]:
|
haftmann@36409
|
474 |
"1 = a / b \<longleftrightarrow> b \<noteq> 0 \<and> a = b"
|
haftmann@36409
|
475 |
by (simp add: eq_commute [of 1])
|
haftmann@36409
|
476 |
|
haftmann@36719
|
477 |
lemma times_divide_times_eq:
|
haftmann@36719
|
478 |
"(x / y) * (z / w) = (x * z) / (y * w)"
|
haftmann@36719
|
479 |
by simp
|
haftmann@36719
|
480 |
|
haftmann@36719
|
481 |
lemma add_frac_num:
|
haftmann@36719
|
482 |
"y \<noteq> 0 \<Longrightarrow> x / y + z = (x + z * y) / y"
|
haftmann@36719
|
483 |
by (simp add: add_divide_distrib)
|
haftmann@36719
|
484 |
|
haftmann@36719
|
485 |
lemma add_num_frac:
|
haftmann@36719
|
486 |
"y \<noteq> 0 \<Longrightarrow> z + x / y = (x + z * y) / y"
|
haftmann@36719
|
487 |
by (simp add: add_divide_distrib add.commute)
|
haftmann@36719
|
488 |
|
haftmann@36409
|
489 |
end
|
haftmann@36301
|
490 |
|
haftmann@36301
|
491 |
|
wenzelm@60758
|
492 |
subsection \<open>Ordered fields\<close>
|
haftmann@36301
|
493 |
|
haftmann@36301
|
494 |
class linordered_field = field + linordered_idom
|
haftmann@36301
|
495 |
begin
|
paulson@14268
|
496 |
|
lp15@59667
|
497 |
lemma positive_imp_inverse_positive:
|
lp15@59667
|
498 |
assumes a_gt_0: "0 < a"
|
haftmann@36301
|
499 |
shows "0 < inverse a"
|
nipkow@23482
|
500 |
proof -
|
lp15@59667
|
501 |
have "0 < a * inverse a"
|
haftmann@36301
|
502 |
by (simp add: a_gt_0 [THEN less_imp_not_eq2])
|
lp15@59667
|
503 |
thus "0 < inverse a"
|
haftmann@36301
|
504 |
by (simp add: a_gt_0 [THEN less_not_sym] zero_less_mult_iff)
|
nipkow@23482
|
505 |
qed
|
paulson@14268
|
506 |
|
paulson@14277
|
507 |
lemma negative_imp_inverse_negative:
|
haftmann@36301
|
508 |
"a < 0 \<Longrightarrow> inverse a < 0"
|
lp15@59667
|
509 |
by (insert positive_imp_inverse_positive [of "-a"],
|
haftmann@36301
|
510 |
simp add: nonzero_inverse_minus_eq less_imp_not_eq)
|
paulson@14268
|
511 |
|
paulson@14268
|
512 |
lemma inverse_le_imp_le:
|
haftmann@36301
|
513 |
assumes invle: "inverse a \<le> inverse b" and apos: "0 < a"
|
haftmann@36301
|
514 |
shows "b \<le> a"
|
nipkow@23482
|
515 |
proof (rule classical)
|
paulson@14268
|
516 |
assume "~ b \<le> a"
|
nipkow@23482
|
517 |
hence "a < b" by (simp add: linorder_not_le)
|
haftmann@36301
|
518 |
hence bpos: "0 < b" by (blast intro: apos less_trans)
|
paulson@14268
|
519 |
hence "a * inverse a \<le> a * inverse b"
|
haftmann@36301
|
520 |
by (simp add: apos invle less_imp_le mult_left_mono)
|
paulson@14268
|
521 |
hence "(a * inverse a) * b \<le> (a * inverse b) * b"
|
haftmann@36301
|
522 |
by (simp add: bpos less_imp_le mult_right_mono)
|
haftmann@57512
|
523 |
thus "b \<le> a" by (simp add: mult.assoc apos bpos less_imp_not_eq2)
|
nipkow@23482
|
524 |
qed
|
paulson@14268
|
525 |
|
paulson@14277
|
526 |
lemma inverse_positive_imp_positive:
|
haftmann@36301
|
527 |
assumes inv_gt_0: "0 < inverse a" and nz: "a \<noteq> 0"
|
haftmann@36301
|
528 |
shows "0 < a"
|
wenzelm@23389
|
529 |
proof -
|
paulson@14277
|
530 |
have "0 < inverse (inverse a)"
|
wenzelm@23389
|
531 |
using inv_gt_0 by (rule positive_imp_inverse_positive)
|
paulson@14277
|
532 |
thus "0 < a"
|
wenzelm@23389
|
533 |
using nz by (simp add: nonzero_inverse_inverse_eq)
|
wenzelm@23389
|
534 |
qed
|
paulson@14277
|
535 |
|
haftmann@36301
|
536 |
lemma inverse_negative_imp_negative:
|
haftmann@36301
|
537 |
assumes inv_less_0: "inverse a < 0" and nz: "a \<noteq> 0"
|
haftmann@36301
|
538 |
shows "a < 0"
|
haftmann@36301
|
539 |
proof -
|
haftmann@36301
|
540 |
have "inverse (inverse a) < 0"
|
haftmann@36301
|
541 |
using inv_less_0 by (rule negative_imp_inverse_negative)
|
haftmann@36301
|
542 |
thus "a < 0" using nz by (simp add: nonzero_inverse_inverse_eq)
|
haftmann@36301
|
543 |
qed
|
haftmann@36301
|
544 |
|
haftmann@36301
|
545 |
lemma linordered_field_no_lb:
|
haftmann@36301
|
546 |
"\<forall>x. \<exists>y. y < x"
|
haftmann@36301
|
547 |
proof
|
haftmann@36301
|
548 |
fix x::'a
|
haftmann@36301
|
549 |
have m1: "- (1::'a) < 0" by simp
|
lp15@59667
|
550 |
from add_strict_right_mono[OF m1, where c=x]
|
haftmann@36301
|
551 |
have "(- 1) + x < x" by simp
|
haftmann@36301
|
552 |
thus "\<exists>y. y < x" by blast
|
haftmann@36301
|
553 |
qed
|
haftmann@36301
|
554 |
|
haftmann@36301
|
555 |
lemma linordered_field_no_ub:
|
haftmann@36301
|
556 |
"\<forall> x. \<exists>y. y > x"
|
haftmann@36301
|
557 |
proof
|
haftmann@36301
|
558 |
fix x::'a
|
haftmann@36301
|
559 |
have m1: " (1::'a) > 0" by simp
|
lp15@59667
|
560 |
from add_strict_right_mono[OF m1, where c=x]
|
haftmann@36301
|
561 |
have "1 + x > x" by simp
|
haftmann@36301
|
562 |
thus "\<exists>y. y > x" by blast
|
haftmann@36301
|
563 |
qed
|
haftmann@36301
|
564 |
|
haftmann@36301
|
565 |
lemma less_imp_inverse_less:
|
haftmann@36301
|
566 |
assumes less: "a < b" and apos: "0 < a"
|
haftmann@36301
|
567 |
shows "inverse b < inverse a"
|
haftmann@36301
|
568 |
proof (rule ccontr)
|
haftmann@36301
|
569 |
assume "~ inverse b < inverse a"
|
haftmann@36301
|
570 |
hence "inverse a \<le> inverse b" by simp
|
haftmann@36301
|
571 |
hence "~ (a < b)"
|
haftmann@36301
|
572 |
by (simp add: not_less inverse_le_imp_le [OF _ apos])
|
haftmann@36301
|
573 |
thus False by (rule notE [OF _ less])
|
haftmann@36301
|
574 |
qed
|
haftmann@36301
|
575 |
|
haftmann@36301
|
576 |
lemma inverse_less_imp_less:
|
haftmann@36301
|
577 |
"inverse a < inverse b \<Longrightarrow> 0 < a \<Longrightarrow> b < a"
|
haftmann@36301
|
578 |
apply (simp add: less_le [of "inverse a"] less_le [of "b"])
|
lp15@59667
|
579 |
apply (force dest!: inverse_le_imp_le nonzero_inverse_eq_imp_eq)
|
haftmann@36301
|
580 |
done
|
haftmann@36301
|
581 |
|
wenzelm@60758
|
582 |
text\<open>Both premises are essential. Consider -1 and 1.\<close>
|
blanchet@54147
|
583 |
lemma inverse_less_iff_less [simp]:
|
haftmann@36301
|
584 |
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
|
lp15@59667
|
585 |
by (blast intro: less_imp_inverse_less dest: inverse_less_imp_less)
|
haftmann@36301
|
586 |
|
haftmann@36301
|
587 |
lemma le_imp_inverse_le:
|
haftmann@36301
|
588 |
"a \<le> b \<Longrightarrow> 0 < a \<Longrightarrow> inverse b \<le> inverse a"
|
haftmann@36301
|
589 |
by (force simp add: le_less less_imp_inverse_less)
|
haftmann@36301
|
590 |
|
blanchet@54147
|
591 |
lemma inverse_le_iff_le [simp]:
|
haftmann@36301
|
592 |
"0 < a \<Longrightarrow> 0 < b \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
|
lp15@59667
|
593 |
by (blast intro: le_imp_inverse_le dest: inverse_le_imp_le)
|
haftmann@36301
|
594 |
|
haftmann@36301
|
595 |
|
wenzelm@60758
|
596 |
text\<open>These results refer to both operands being negative. The opposite-sign
|
wenzelm@60758
|
597 |
case is trivial, since inverse preserves signs.\<close>
|
haftmann@36301
|
598 |
lemma inverse_le_imp_le_neg:
|
haftmann@36301
|
599 |
"inverse a \<le> inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b \<le> a"
|
lp15@59667
|
600 |
apply (rule classical)
|
lp15@59667
|
601 |
apply (subgoal_tac "a < 0")
|
haftmann@36301
|
602 |
prefer 2 apply force
|
haftmann@36301
|
603 |
apply (insert inverse_le_imp_le [of "-b" "-a"])
|
lp15@59667
|
604 |
apply (simp add: nonzero_inverse_minus_eq)
|
haftmann@36301
|
605 |
done
|
haftmann@36301
|
606 |
|
haftmann@36301
|
607 |
lemma less_imp_inverse_less_neg:
|
haftmann@36301
|
608 |
"a < b \<Longrightarrow> b < 0 \<Longrightarrow> inverse b < inverse a"
|
lp15@59667
|
609 |
apply (subgoal_tac "a < 0")
|
lp15@59667
|
610 |
prefer 2 apply (blast intro: less_trans)
|
haftmann@36301
|
611 |
apply (insert less_imp_inverse_less [of "-b" "-a"])
|
lp15@59667
|
612 |
apply (simp add: nonzero_inverse_minus_eq)
|
haftmann@36301
|
613 |
done
|
haftmann@36301
|
614 |
|
haftmann@36301
|
615 |
lemma inverse_less_imp_less_neg:
|
haftmann@36301
|
616 |
"inverse a < inverse b \<Longrightarrow> b < 0 \<Longrightarrow> b < a"
|
lp15@59667
|
617 |
apply (rule classical)
|
lp15@59667
|
618 |
apply (subgoal_tac "a < 0")
|
haftmann@36301
|
619 |
prefer 2
|
haftmann@36301
|
620 |
apply force
|
haftmann@36301
|
621 |
apply (insert inverse_less_imp_less [of "-b" "-a"])
|
lp15@59667
|
622 |
apply (simp add: nonzero_inverse_minus_eq)
|
haftmann@36301
|
623 |
done
|
haftmann@36301
|
624 |
|
blanchet@54147
|
625 |
lemma inverse_less_iff_less_neg [simp]:
|
haftmann@36301
|
626 |
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a < inverse b \<longleftrightarrow> b < a"
|
haftmann@36301
|
627 |
apply (insert inverse_less_iff_less [of "-b" "-a"])
|
lp15@59667
|
628 |
apply (simp del: inverse_less_iff_less
|
haftmann@36301
|
629 |
add: nonzero_inverse_minus_eq)
|
haftmann@36301
|
630 |
done
|
haftmann@36301
|
631 |
|
haftmann@36301
|
632 |
lemma le_imp_inverse_le_neg:
|
haftmann@36301
|
633 |
"a \<le> b \<Longrightarrow> b < 0 ==> inverse b \<le> inverse a"
|
haftmann@36301
|
634 |
by (force simp add: le_less less_imp_inverse_less_neg)
|
haftmann@36301
|
635 |
|
blanchet@54147
|
636 |
lemma inverse_le_iff_le_neg [simp]:
|
haftmann@36301
|
637 |
"a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> inverse a \<le> inverse b \<longleftrightarrow> b \<le> a"
|
lp15@59667
|
638 |
by (blast intro: le_imp_inverse_le_neg dest: inverse_le_imp_le_neg)
|
haftmann@36301
|
639 |
|
huffman@36774
|
640 |
lemma one_less_inverse:
|
huffman@36774
|
641 |
"0 < a \<Longrightarrow> a < 1 \<Longrightarrow> 1 < inverse a"
|
huffman@36774
|
642 |
using less_imp_inverse_less [of a 1, unfolded inverse_1] .
|
huffman@36774
|
643 |
|
huffman@36774
|
644 |
lemma one_le_inverse:
|
huffman@36774
|
645 |
"0 < a \<Longrightarrow> a \<le> 1 \<Longrightarrow> 1 \<le> inverse a"
|
huffman@36774
|
646 |
using le_imp_inverse_le [of a 1, unfolded inverse_1] .
|
huffman@36774
|
647 |
|
haftmann@59546
|
648 |
lemma pos_le_divide_eq [field_simps]:
|
haftmann@59546
|
649 |
assumes "0 < c"
|
haftmann@59546
|
650 |
shows "a \<le> b / c \<longleftrightarrow> a * c \<le> b"
|
haftmann@36301
|
651 |
proof -
|
haftmann@59546
|
652 |
from assms have "a \<le> b / c \<longleftrightarrow> a * c \<le> (b / c) * c"
|
haftmann@59546
|
653 |
using mult_le_cancel_right [of a c "b * inverse c"] by (auto simp add: field_simps)
|
haftmann@59546
|
654 |
also have "... \<longleftrightarrow> a * c \<le> b"
|
lp15@59667
|
655 |
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
656 |
finally show ?thesis .
|
haftmann@36301
|
657 |
qed
|
haftmann@36301
|
658 |
|
haftmann@59546
|
659 |
lemma pos_less_divide_eq [field_simps]:
|
haftmann@59546
|
660 |
assumes "0 < c"
|
haftmann@59546
|
661 |
shows "a < b / c \<longleftrightarrow> a * c < b"
|
haftmann@36301
|
662 |
proof -
|
haftmann@59546
|
663 |
from assms have "a < b / c \<longleftrightarrow> a * c < (b / c) * c"
|
haftmann@59546
|
664 |
using mult_less_cancel_right [of a c "b / c"] by auto
|
haftmann@59546
|
665 |
also have "... = (a*c < b)"
|
lp15@59667
|
666 |
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
667 |
finally show ?thesis .
|
haftmann@36301
|
668 |
qed
|
haftmann@36301
|
669 |
|
haftmann@59546
|
670 |
lemma neg_less_divide_eq [field_simps]:
|
haftmann@59546
|
671 |
assumes "c < 0"
|
haftmann@59546
|
672 |
shows "a < b / c \<longleftrightarrow> b < a * c"
|
haftmann@36301
|
673 |
proof -
|
haftmann@59546
|
674 |
from assms have "a < b / c \<longleftrightarrow> (b / c) * c < a * c"
|
haftmann@59546
|
675 |
using mult_less_cancel_right [of "b / c" c a] by auto
|
haftmann@59546
|
676 |
also have "... \<longleftrightarrow> b < a * c"
|
lp15@59667
|
677 |
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
678 |
finally show ?thesis .
|
haftmann@36301
|
679 |
qed
|
haftmann@36301
|
680 |
|
haftmann@59546
|
681 |
lemma neg_le_divide_eq [field_simps]:
|
haftmann@59546
|
682 |
assumes "c < 0"
|
haftmann@59546
|
683 |
shows "a \<le> b / c \<longleftrightarrow> b \<le> a * c"
|
haftmann@36301
|
684 |
proof -
|
haftmann@59546
|
685 |
from assms have "a \<le> b / c \<longleftrightarrow> (b / c) * c \<le> a * c"
|
haftmann@59546
|
686 |
using mult_le_cancel_right [of "b * inverse c" c a] by (auto simp add: field_simps)
|
haftmann@59546
|
687 |
also have "... \<longleftrightarrow> b \<le> a * c"
|
lp15@59667
|
688 |
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
689 |
finally show ?thesis .
|
haftmann@36301
|
690 |
qed
|
haftmann@36301
|
691 |
|
haftmann@59546
|
692 |
lemma pos_divide_le_eq [field_simps]:
|
haftmann@59546
|
693 |
assumes "0 < c"
|
haftmann@59546
|
694 |
shows "b / c \<le> a \<longleftrightarrow> b \<le> a * c"
|
haftmann@36301
|
695 |
proof -
|
haftmann@59546
|
696 |
from assms have "b / c \<le> a \<longleftrightarrow> (b / c) * c \<le> a * c"
|
haftmann@59546
|
697 |
using mult_le_cancel_right [of "b / c" c a] by auto
|
haftmann@59546
|
698 |
also have "... \<longleftrightarrow> b \<le> a * c"
|
lp15@59667
|
699 |
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
700 |
finally show ?thesis .
|
haftmann@36301
|
701 |
qed
|
haftmann@36301
|
702 |
|
haftmann@59546
|
703 |
lemma pos_divide_less_eq [field_simps]:
|
haftmann@59546
|
704 |
assumes "0 < c"
|
haftmann@59546
|
705 |
shows "b / c < a \<longleftrightarrow> b < a * c"
|
haftmann@36301
|
706 |
proof -
|
haftmann@59546
|
707 |
from assms have "b / c < a \<longleftrightarrow> (b / c) * c < a * c"
|
haftmann@59546
|
708 |
using mult_less_cancel_right [of "b / c" c a] by auto
|
haftmann@59546
|
709 |
also have "... \<longleftrightarrow> b < a * c"
|
lp15@59667
|
710 |
by (simp add: less_imp_not_eq2 [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
711 |
finally show ?thesis .
|
haftmann@36301
|
712 |
qed
|
haftmann@36301
|
713 |
|
haftmann@59546
|
714 |
lemma neg_divide_le_eq [field_simps]:
|
haftmann@59546
|
715 |
assumes "c < 0"
|
haftmann@59546
|
716 |
shows "b / c \<le> a \<longleftrightarrow> a * c \<le> b"
|
haftmann@36301
|
717 |
proof -
|
haftmann@59546
|
718 |
from assms have "b / c \<le> a \<longleftrightarrow> a * c \<le> (b / c) * c"
|
lp15@59667
|
719 |
using mult_le_cancel_right [of a c "b / c"] by auto
|
haftmann@59546
|
720 |
also have "... \<longleftrightarrow> a * c \<le> b"
|
lp15@59667
|
721 |
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
722 |
finally show ?thesis .
|
haftmann@36301
|
723 |
qed
|
haftmann@36301
|
724 |
|
haftmann@59546
|
725 |
lemma neg_divide_less_eq [field_simps]:
|
haftmann@59546
|
726 |
assumes "c < 0"
|
haftmann@59546
|
727 |
shows "b / c < a \<longleftrightarrow> a * c < b"
|
haftmann@36301
|
728 |
proof -
|
haftmann@59546
|
729 |
from assms have "b / c < a \<longleftrightarrow> a * c < b / c * c"
|
haftmann@59546
|
730 |
using mult_less_cancel_right [of a c "b / c"] by auto
|
haftmann@59546
|
731 |
also have "... \<longleftrightarrow> a * c < b"
|
lp15@59667
|
732 |
by (simp add: less_imp_not_eq [OF assms] divide_inverse mult.assoc)
|
haftmann@36301
|
733 |
finally show ?thesis .
|
haftmann@36301
|
734 |
qed
|
haftmann@36301
|
735 |
|
wenzelm@60758
|
736 |
text\<open>The following @{text field_simps} rules are necessary, as minus is always moved atop of
|
wenzelm@60758
|
737 |
division but we want to get rid of division.\<close>
|
hoelzl@56480
|
738 |
|
hoelzl@56480
|
739 |
lemma pos_le_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> a * c \<le> - b"
|
hoelzl@56480
|
740 |
unfolding minus_divide_left by (rule pos_le_divide_eq)
|
hoelzl@56480
|
741 |
|
hoelzl@56480
|
742 |
lemma neg_le_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a \<le> - (b / c) \<longleftrightarrow> - b \<le> a * c"
|
hoelzl@56480
|
743 |
unfolding minus_divide_left by (rule neg_le_divide_eq)
|
hoelzl@56480
|
744 |
|
hoelzl@56480
|
745 |
lemma pos_less_minus_divide_eq [field_simps]: "0 < c \<Longrightarrow> a < - (b / c) \<longleftrightarrow> a * c < - b"
|
hoelzl@56480
|
746 |
unfolding minus_divide_left by (rule pos_less_divide_eq)
|
hoelzl@56480
|
747 |
|
hoelzl@56480
|
748 |
lemma neg_less_minus_divide_eq [field_simps]: "c < 0 \<Longrightarrow> a < - (b / c) \<longleftrightarrow> - b < a * c"
|
hoelzl@56480
|
749 |
unfolding minus_divide_left by (rule neg_less_divide_eq)
|
hoelzl@56480
|
750 |
|
hoelzl@56480
|
751 |
lemma pos_minus_divide_less_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) < a \<longleftrightarrow> - b < a * c"
|
hoelzl@56480
|
752 |
unfolding minus_divide_left by (rule pos_divide_less_eq)
|
hoelzl@56480
|
753 |
|
hoelzl@56480
|
754 |
lemma neg_minus_divide_less_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) < a \<longleftrightarrow> a * c < - b"
|
hoelzl@56480
|
755 |
unfolding minus_divide_left by (rule neg_divide_less_eq)
|
hoelzl@56480
|
756 |
|
hoelzl@56480
|
757 |
lemma pos_minus_divide_le_eq [field_simps]: "0 < c \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> - b \<le> a * c"
|
hoelzl@56480
|
758 |
unfolding minus_divide_left by (rule pos_divide_le_eq)
|
hoelzl@56480
|
759 |
|
hoelzl@56480
|
760 |
lemma neg_minus_divide_le_eq [field_simps]: "c < 0 \<Longrightarrow> - (b / c) \<le> a \<longleftrightarrow> a * c \<le> - b"
|
hoelzl@56480
|
761 |
unfolding minus_divide_left by (rule neg_divide_le_eq)
|
hoelzl@56480
|
762 |
|
lp15@56365
|
763 |
lemma frac_less_eq:
|
lp15@56365
|
764 |
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y < w / z \<longleftrightarrow> (x * z - w * y) / (y * z) < 0"
|
lp15@56365
|
765 |
by (subst less_iff_diff_less_0) (simp add: diff_frac_eq )
|
lp15@56365
|
766 |
|
lp15@56365
|
767 |
lemma frac_le_eq:
|
lp15@56365
|
768 |
"y \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> x / y \<le> w / z \<longleftrightarrow> (x * z - w * y) / (y * z) \<le> 0"
|
lp15@56365
|
769 |
by (subst le_iff_diff_le_0) (simp add: diff_frac_eq )
|
lp15@56365
|
770 |
|
wenzelm@60758
|
771 |
text\<open>Lemmas @{text sign_simps} is a first attempt to automate proofs
|
haftmann@36301
|
772 |
of positivity/negativity needed for @{text field_simps}. Have not added @{text
|
haftmann@36301
|
773 |
sign_simps} to @{text field_simps} because the former can lead to case
|
wenzelm@60758
|
774 |
explosions.\<close>
|
haftmann@36301
|
775 |
|
blanchet@54147
|
776 |
lemmas sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
|
haftmann@36348
|
777 |
|
blanchet@54147
|
778 |
lemmas (in -) sign_simps = algebra_simps zero_less_mult_iff mult_less_0_iff
|
haftmann@36301
|
779 |
|
haftmann@36301
|
780 |
(* Only works once linear arithmetic is installed:
|
haftmann@36301
|
781 |
text{*An example:*}
|
haftmann@36301
|
782 |
lemma fixes a b c d e f :: "'a::linordered_field"
|
haftmann@36301
|
783 |
shows "\<lbrakk>a>b; c<d; e<f; 0 < u \<rbrakk> \<Longrightarrow>
|
haftmann@36301
|
784 |
((a-b)*(c-d)*(e-f))/((c-d)*(e-f)*(a-b)) <
|
haftmann@36301
|
785 |
((e-f)*(a-b)*(c-d))/((e-f)*(a-b)*(c-d)) + u"
|
haftmann@36301
|
786 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b) > 0")
|
haftmann@36301
|
787 |
prefer 2 apply(simp add:sign_simps)
|
haftmann@36301
|
788 |
apply(subgoal_tac "(c-d)*(e-f)*(a-b)*u > 0")
|
haftmann@36301
|
789 |
prefer 2 apply(simp add:sign_simps)
|
haftmann@36301
|
790 |
apply(simp add:field_simps)
|
haftmann@36301
|
791 |
done
|
haftmann@36301
|
792 |
*)
|
haftmann@36301
|
793 |
|
nipkow@56541
|
794 |
lemma divide_pos_pos[simp]:
|
haftmann@36301
|
795 |
"0 < x ==> 0 < y ==> 0 < x / y"
|
haftmann@36301
|
796 |
by(simp add:field_simps)
|
haftmann@36301
|
797 |
|
haftmann@36301
|
798 |
lemma divide_nonneg_pos:
|
haftmann@36301
|
799 |
"0 <= x ==> 0 < y ==> 0 <= x / y"
|
haftmann@36301
|
800 |
by(simp add:field_simps)
|
haftmann@36301
|
801 |
|
haftmann@36301
|
802 |
lemma divide_neg_pos:
|
haftmann@36301
|
803 |
"x < 0 ==> 0 < y ==> x / y < 0"
|
haftmann@36301
|
804 |
by(simp add:field_simps)
|
haftmann@36301
|
805 |
|
haftmann@36301
|
806 |
lemma divide_nonpos_pos:
|
haftmann@36301
|
807 |
"x <= 0 ==> 0 < y ==> x / y <= 0"
|
haftmann@36301
|
808 |
by(simp add:field_simps)
|
haftmann@36301
|
809 |
|
haftmann@36301
|
810 |
lemma divide_pos_neg:
|
haftmann@36301
|
811 |
"0 < x ==> y < 0 ==> x / y < 0"
|
haftmann@36301
|
812 |
by(simp add:field_simps)
|
haftmann@36301
|
813 |
|
haftmann@36301
|
814 |
lemma divide_nonneg_neg:
|
lp15@59667
|
815 |
"0 <= x ==> y < 0 ==> x / y <= 0"
|
haftmann@36301
|
816 |
by(simp add:field_simps)
|
haftmann@36301
|
817 |
|
haftmann@36301
|
818 |
lemma divide_neg_neg:
|
haftmann@36301
|
819 |
"x < 0 ==> y < 0 ==> 0 < x / y"
|
haftmann@36301
|
820 |
by(simp add:field_simps)
|
haftmann@36301
|
821 |
|
haftmann@36301
|
822 |
lemma divide_nonpos_neg:
|
haftmann@36301
|
823 |
"x <= 0 ==> y < 0 ==> 0 <= x / y"
|
haftmann@36301
|
824 |
by(simp add:field_simps)
|
haftmann@36301
|
825 |
|
haftmann@36301
|
826 |
lemma divide_strict_right_mono:
|
haftmann@36301
|
827 |
"[|a < b; 0 < c|] ==> a / c < b / c"
|
lp15@59667
|
828 |
by (simp add: less_imp_not_eq2 divide_inverse mult_strict_right_mono
|
haftmann@36301
|
829 |
positive_imp_inverse_positive)
|
haftmann@36301
|
830 |
|
haftmann@36301
|
831 |
|
haftmann@36301
|
832 |
lemma divide_strict_right_mono_neg:
|
haftmann@36301
|
833 |
"[|b < a; c < 0|] ==> a / c < b / c"
|
haftmann@36301
|
834 |
apply (drule divide_strict_right_mono [of _ _ "-c"], simp)
|
haftmann@36301
|
835 |
apply (simp add: less_imp_not_eq nonzero_minus_divide_right [symmetric])
|
haftmann@36301
|
836 |
done
|
haftmann@36301
|
837 |
|
wenzelm@60758
|
838 |
text\<open>The last premise ensures that @{term a} and @{term b}
|
wenzelm@60758
|
839 |
have the same sign\<close>
|
haftmann@36301
|
840 |
lemma divide_strict_left_mono:
|
haftmann@36301
|
841 |
"[|b < a; 0 < c; 0 < a*b|] ==> c / a < c / b"
|
huffman@44921
|
842 |
by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono)
|
haftmann@36301
|
843 |
|
haftmann@36301
|
844 |
lemma divide_left_mono:
|
haftmann@36301
|
845 |
"[|b \<le> a; 0 \<le> c; 0 < a*b|] ==> c / a \<le> c / b"
|
huffman@44921
|
846 |
by (auto simp: field_simps zero_less_mult_iff mult_right_mono)
|
haftmann@36301
|
847 |
|
haftmann@36301
|
848 |
lemma divide_strict_left_mono_neg:
|
haftmann@36301
|
849 |
"[|a < b; c < 0; 0 < a*b|] ==> c / a < c / b"
|
huffman@44921
|
850 |
by (auto simp: field_simps zero_less_mult_iff mult_strict_right_mono_neg)
|
haftmann@36301
|
851 |
|
haftmann@36301
|
852 |
lemma mult_imp_div_pos_le: "0 < y ==> x <= z * y ==>
|
haftmann@36301
|
853 |
x / y <= z"
|
haftmann@36301
|
854 |
by (subst pos_divide_le_eq, assumption+)
|
haftmann@36301
|
855 |
|
haftmann@36301
|
856 |
lemma mult_imp_le_div_pos: "0 < y ==> z * y <= x ==>
|
haftmann@36301
|
857 |
z <= x / y"
|
haftmann@36301
|
858 |
by(simp add:field_simps)
|
haftmann@36301
|
859 |
|
haftmann@36301
|
860 |
lemma mult_imp_div_pos_less: "0 < y ==> x < z * y ==>
|
haftmann@36301
|
861 |
x / y < z"
|
haftmann@36301
|
862 |
by(simp add:field_simps)
|
haftmann@36301
|
863 |
|
haftmann@36301
|
864 |
lemma mult_imp_less_div_pos: "0 < y ==> z * y < x ==>
|
haftmann@36301
|
865 |
z < x / y"
|
haftmann@36301
|
866 |
by(simp add:field_simps)
|
haftmann@36301
|
867 |
|
lp15@59667
|
868 |
lemma frac_le: "0 <= x ==>
|
haftmann@36301
|
869 |
x <= y ==> 0 < w ==> w <= z ==> x / z <= y / w"
|
haftmann@36301
|
870 |
apply (rule mult_imp_div_pos_le)
|
haftmann@36301
|
871 |
apply simp
|
haftmann@36301
|
872 |
apply (subst times_divide_eq_left)
|
haftmann@36301
|
873 |
apply (rule mult_imp_le_div_pos, assumption)
|
haftmann@36301
|
874 |
apply (rule mult_mono)
|
haftmann@36301
|
875 |
apply simp_all
|
haftmann@36301
|
876 |
done
|
haftmann@36301
|
877 |
|
lp15@59667
|
878 |
lemma frac_less: "0 <= x ==>
|
haftmann@36301
|
879 |
x < y ==> 0 < w ==> w <= z ==> x / z < y / w"
|
haftmann@36301
|
880 |
apply (rule mult_imp_div_pos_less)
|
haftmann@36301
|
881 |
apply simp
|
haftmann@36301
|
882 |
apply (subst times_divide_eq_left)
|
haftmann@36301
|
883 |
apply (rule mult_imp_less_div_pos, assumption)
|
haftmann@36301
|
884 |
apply (erule mult_less_le_imp_less)
|
haftmann@36301
|
885 |
apply simp_all
|
haftmann@36301
|
886 |
done
|
haftmann@36301
|
887 |
|
lp15@59667
|
888 |
lemma frac_less2: "0 < x ==>
|
haftmann@36301
|
889 |
x <= y ==> 0 < w ==> w < z ==> x / z < y / w"
|
haftmann@36301
|
890 |
apply (rule mult_imp_div_pos_less)
|
haftmann@36301
|
891 |
apply simp_all
|
haftmann@36301
|
892 |
apply (rule mult_imp_less_div_pos, assumption)
|
haftmann@36301
|
893 |
apply (erule mult_le_less_imp_less)
|
haftmann@36301
|
894 |
apply simp_all
|
haftmann@36301
|
895 |
done
|
haftmann@36301
|
896 |
|
haftmann@36301
|
897 |
lemma less_half_sum: "a < b ==> a < (a+b) / (1+1)"
|
haftmann@36301
|
898 |
by (simp add: field_simps zero_less_two)
|
haftmann@36301
|
899 |
|
haftmann@36301
|
900 |
lemma gt_half_sum: "a < b ==> (a+b)/(1+1) < b"
|
haftmann@36301
|
901 |
by (simp add: field_simps zero_less_two)
|
haftmann@36301
|
902 |
|
hoelzl@53215
|
903 |
subclass unbounded_dense_linorder
|
haftmann@36301
|
904 |
proof
|
haftmann@36301
|
905 |
fix x y :: 'a
|
lp15@59667
|
906 |
from less_add_one show "\<exists>y. x < y" ..
|
haftmann@36301
|
907 |
from less_add_one have "x + (- 1) < (x + 1) + (- 1)" by (rule add_strict_right_mono)
|
haftmann@54230
|
908 |
then have "x - 1 < x + 1 - 1" by simp
|
haftmann@36301
|
909 |
then have "x - 1 < x" by (simp add: algebra_simps)
|
haftmann@36301
|
910 |
then show "\<exists>y. y < x" ..
|
haftmann@36301
|
911 |
show "x < y \<Longrightarrow> \<exists>z>x. z < y" by (blast intro!: less_half_sum gt_half_sum)
|
haftmann@36301
|
912 |
qed
|
haftmann@36301
|
913 |
|
haftmann@36301
|
914 |
lemma nonzero_abs_inverse:
|
haftmann@36301
|
915 |
"a \<noteq> 0 ==> \<bar>inverse a\<bar> = inverse \<bar>a\<bar>"
|
lp15@59667
|
916 |
apply (auto simp add: neq_iff abs_if nonzero_inverse_minus_eq
|
haftmann@36301
|
917 |
negative_imp_inverse_negative)
|
lp15@59667
|
918 |
apply (blast intro: positive_imp_inverse_positive elim: less_asym)
|
haftmann@36301
|
919 |
done
|
haftmann@36301
|
920 |
|
haftmann@36301
|
921 |
lemma nonzero_abs_divide:
|
haftmann@36301
|
922 |
"b \<noteq> 0 ==> \<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
|
lp15@59667
|
923 |
by (simp add: divide_inverse abs_mult nonzero_abs_inverse)
|
haftmann@36301
|
924 |
|
haftmann@36301
|
925 |
lemma field_le_epsilon:
|
haftmann@36301
|
926 |
assumes e: "\<And>e. 0 < e \<Longrightarrow> x \<le> y + e"
|
haftmann@36301
|
927 |
shows "x \<le> y"
|
haftmann@36301
|
928 |
proof (rule dense_le)
|
haftmann@36301
|
929 |
fix t assume "t < x"
|
haftmann@36301
|
930 |
hence "0 < x - t" by (simp add: less_diff_eq)
|
haftmann@36301
|
931 |
from e [OF this] have "x + 0 \<le> x + (y - t)" by (simp add: algebra_simps)
|
haftmann@36301
|
932 |
then have "0 \<le> y - t" by (simp only: add_le_cancel_left)
|
haftmann@36301
|
933 |
then show "t \<le> y" by (simp add: algebra_simps)
|
haftmann@36301
|
934 |
qed
|
haftmann@36301
|
935 |
|
paulson@14277
|
936 |
lemma inverse_positive_iff_positive [simp]:
|
haftmann@36409
|
937 |
"(0 < inverse a) = (0 < a)"
|
haftmann@21328
|
938 |
apply (cases "a = 0", simp)
|
paulson@14277
|
939 |
apply (blast intro: inverse_positive_imp_positive positive_imp_inverse_positive)
|
paulson@14277
|
940 |
done
|
paulson@14277
|
941 |
|
paulson@14277
|
942 |
lemma inverse_negative_iff_negative [simp]:
|
haftmann@36409
|
943 |
"(inverse a < 0) = (a < 0)"
|
haftmann@21328
|
944 |
apply (cases "a = 0", simp)
|
paulson@14277
|
945 |
apply (blast intro: inverse_negative_imp_negative negative_imp_inverse_negative)
|
paulson@14277
|
946 |
done
|
paulson@14277
|
947 |
|
paulson@14277
|
948 |
lemma inverse_nonnegative_iff_nonnegative [simp]:
|
haftmann@36409
|
949 |
"0 \<le> inverse a \<longleftrightarrow> 0 \<le> a"
|
haftmann@36409
|
950 |
by (simp add: not_less [symmetric])
|
paulson@14277
|
951 |
|
paulson@14277
|
952 |
lemma inverse_nonpositive_iff_nonpositive [simp]:
|
haftmann@36409
|
953 |
"inverse a \<le> 0 \<longleftrightarrow> a \<le> 0"
|
haftmann@36409
|
954 |
by (simp add: not_less [symmetric])
|
paulson@14277
|
955 |
|
hoelzl@56480
|
956 |
lemma one_less_inverse_iff: "1 < inverse x \<longleftrightarrow> 0 < x \<and> x < 1"
|
hoelzl@56480
|
957 |
using less_trans[of 1 x 0 for x]
|
hoelzl@56480
|
958 |
by (cases x 0 rule: linorder_cases) (auto simp add: field_simps)
|
paulson@14365
|
959 |
|
hoelzl@56480
|
960 |
lemma one_le_inverse_iff: "1 \<le> inverse x \<longleftrightarrow> 0 < x \<and> x \<le> 1"
|
haftmann@36409
|
961 |
proof (cases "x = 1")
|
haftmann@36409
|
962 |
case True then show ?thesis by simp
|
haftmann@36409
|
963 |
next
|
haftmann@36409
|
964 |
case False then have "inverse x \<noteq> 1" by simp
|
haftmann@36409
|
965 |
then have "1 \<noteq> inverse x" by blast
|
haftmann@36409
|
966 |
then have "1 \<le> inverse x \<longleftrightarrow> 1 < inverse x" by (simp add: le_less)
|
haftmann@36409
|
967 |
with False show ?thesis by (auto simp add: one_less_inverse_iff)
|
haftmann@36409
|
968 |
qed
|
paulson@14365
|
969 |
|
hoelzl@56480
|
970 |
lemma inverse_less_1_iff: "inverse x < 1 \<longleftrightarrow> x \<le> 0 \<or> 1 < x"
|
lp15@59667
|
971 |
by (simp add: not_le [symmetric] one_le_inverse_iff)
|
paulson@14365
|
972 |
|
hoelzl@56480
|
973 |
lemma inverse_le_1_iff: "inverse x \<le> 1 \<longleftrightarrow> x \<le> 0 \<or> 1 \<le> x"
|
lp15@59667
|
974 |
by (simp add: not_less [symmetric] one_less_inverse_iff)
|
paulson@14365
|
975 |
|
hoelzl@56481
|
976 |
lemma [divide_simps]:
|
hoelzl@56480
|
977 |
shows le_divide_eq: "a \<le> b / c \<longleftrightarrow> (if 0 < c then a * c \<le> b else if c < 0 then b \<le> a * c else a \<le> 0)"
|
hoelzl@56480
|
978 |
and divide_le_eq: "b / c \<le> a \<longleftrightarrow> (if 0 < c then b \<le> a * c else if c < 0 then a * c \<le> b else 0 \<le> a)"
|
hoelzl@56480
|
979 |
and less_divide_eq: "a < b / c \<longleftrightarrow> (if 0 < c then a * c < b else if c < 0 then b < a * c else a < 0)"
|
hoelzl@56480
|
980 |
and divide_less_eq: "b / c < a \<longleftrightarrow> (if 0 < c then b < a * c else if c < 0 then a * c < b else 0 < a)"
|
hoelzl@56481
|
981 |
and le_minus_divide_eq: "a \<le> - (b / c) \<longleftrightarrow> (if 0 < c then a * c \<le> - b else if c < 0 then - b \<le> a * c else a \<le> 0)"
|
hoelzl@56481
|
982 |
and minus_divide_le_eq: "- (b / c) \<le> a \<longleftrightarrow> (if 0 < c then - b \<le> a * c else if c < 0 then a * c \<le> - b else 0 \<le> a)"
|
hoelzl@56481
|
983 |
and less_minus_divide_eq: "a < - (b / c) \<longleftrightarrow> (if 0 < c then a * c < - b else if c < 0 then - b < a * c else a < 0)"
|
hoelzl@56481
|
984 |
and minus_divide_less_eq: "- (b / c) < a \<longleftrightarrow> (if 0 < c then - b < a * c else if c < 0 then a * c < - b else 0 < a)"
|
hoelzl@56480
|
985 |
by (auto simp: field_simps not_less dest: antisym)
|
paulson@14288
|
986 |
|
wenzelm@60758
|
987 |
text \<open>Division and Signs\<close>
|
avigad@16775
|
988 |
|
hoelzl@56480
|
989 |
lemma
|
hoelzl@56480
|
990 |
shows zero_less_divide_iff: "0 < a / b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
|
hoelzl@56480
|
991 |
and divide_less_0_iff: "a / b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
|
hoelzl@56480
|
992 |
and zero_le_divide_iff: "0 \<le> a / b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
|
hoelzl@56480
|
993 |
and divide_le_0_iff: "a / b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
|
hoelzl@56481
|
994 |
by (auto simp add: divide_simps)
|
avigad@16775
|
995 |
|
wenzelm@60758
|
996 |
text \<open>Division and the Number One\<close>
|
paulson@14353
|
997 |
|
wenzelm@60758
|
998 |
text\<open>Simplify expressions equated with 1\<close>
|
paulson@14353
|
999 |
|
hoelzl@56480
|
1000 |
lemma zero_eq_1_divide_iff [simp]: "0 = 1 / a \<longleftrightarrow> a = 0"
|
hoelzl@56480
|
1001 |
by (cases "a = 0") (auto simp: field_simps)
|
paulson@14353
|
1002 |
|
hoelzl@56480
|
1003 |
lemma one_divide_eq_0_iff [simp]: "1 / a = 0 \<longleftrightarrow> a = 0"
|
hoelzl@56480
|
1004 |
using zero_eq_1_divide_iff[of a] by simp
|
paulson@14353
|
1005 |
|
wenzelm@60758
|
1006 |
text\<open>Simplify expressions such as @{text "0 < 1/x"} to @{text "0 < x"}\<close>
|
haftmann@36423
|
1007 |
|
blanchet@54147
|
1008 |
lemma zero_le_divide_1_iff [simp]:
|
haftmann@36423
|
1009 |
"0 \<le> 1 / a \<longleftrightarrow> 0 \<le> a"
|
haftmann@36423
|
1010 |
by (simp add: zero_le_divide_iff)
|
paulson@17085
|
1011 |
|
blanchet@54147
|
1012 |
lemma zero_less_divide_1_iff [simp]:
|
haftmann@36423
|
1013 |
"0 < 1 / a \<longleftrightarrow> 0 < a"
|
haftmann@36423
|
1014 |
by (simp add: zero_less_divide_iff)
|
haftmann@36423
|
1015 |
|
blanchet@54147
|
1016 |
lemma divide_le_0_1_iff [simp]:
|
haftmann@36423
|
1017 |
"1 / a \<le> 0 \<longleftrightarrow> a \<le> 0"
|
haftmann@36423
|
1018 |
by (simp add: divide_le_0_iff)
|
haftmann@36423
|
1019 |
|
blanchet@54147
|
1020 |
lemma divide_less_0_1_iff [simp]:
|
haftmann@36423
|
1021 |
"1 / a < 0 \<longleftrightarrow> a < 0"
|
haftmann@36423
|
1022 |
by (simp add: divide_less_0_iff)
|
paulson@14353
|
1023 |
|
paulson@14293
|
1024 |
lemma divide_right_mono:
|
haftmann@36409
|
1025 |
"[|a \<le> b; 0 \<le> c|] ==> a/c \<le> b/c"
|
haftmann@36409
|
1026 |
by (force simp add: divide_strict_right_mono le_less)
|
paulson@14293
|
1027 |
|
lp15@59667
|
1028 |
lemma divide_right_mono_neg: "a <= b
|
avigad@16775
|
1029 |
==> c <= 0 ==> b / c <= a / c"
|
nipkow@23482
|
1030 |
apply (drule divide_right_mono [of _ _ "- c"])
|
hoelzl@56479
|
1031 |
apply auto
|
avigad@16775
|
1032 |
done
|
avigad@16775
|
1033 |
|
lp15@59667
|
1034 |
lemma divide_left_mono_neg: "a <= b
|
avigad@16775
|
1035 |
==> c <= 0 ==> 0 < a * b ==> c / a <= c / b"
|
avigad@16775
|
1036 |
apply (drule divide_left_mono [of _ _ "- c"])
|
haftmann@57512
|
1037 |
apply (auto simp add: mult.commute)
|
avigad@16775
|
1038 |
done
|
avigad@16775
|
1039 |
|
hoelzl@56480
|
1040 |
lemma inverse_le_iff: "inverse a \<le> inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b \<le> a) \<and> (a * b \<le> 0 \<longrightarrow> a \<le> b)"
|
hoelzl@56480
|
1041 |
by (cases a 0 b 0 rule: linorder_cases[case_product linorder_cases])
|
hoelzl@56480
|
1042 |
(auto simp add: field_simps zero_less_mult_iff mult_le_0_iff)
|
hoelzl@42904
|
1043 |
|
hoelzl@56480
|
1044 |
lemma inverse_less_iff: "inverse a < inverse b \<longleftrightarrow> (0 < a * b \<longrightarrow> b < a) \<and> (a * b \<le> 0 \<longrightarrow> a < b)"
|
hoelzl@42904
|
1045 |
by (subst less_le) (auto simp: inverse_le_iff)
|
hoelzl@42904
|
1046 |
|
hoelzl@56480
|
1047 |
lemma divide_le_cancel: "a / c \<le> b / c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
|
hoelzl@42904
|
1048 |
by (simp add: divide_inverse mult_le_cancel_right)
|
hoelzl@42904
|
1049 |
|
hoelzl@56480
|
1050 |
lemma divide_less_cancel: "a / c < b / c \<longleftrightarrow> (0 < c \<longrightarrow> a < b) \<and> (c < 0 \<longrightarrow> b < a) \<and> c \<noteq> 0"
|
hoelzl@42904
|
1051 |
by (auto simp add: divide_inverse mult_less_cancel_right)
|
hoelzl@42904
|
1052 |
|
wenzelm@60758
|
1053 |
text\<open>Simplify quotients that are compared with the value 1.\<close>
|
avigad@16775
|
1054 |
|
blanchet@54147
|
1055 |
lemma le_divide_eq_1:
|
haftmann@36409
|
1056 |
"(1 \<le> b / a) = ((0 < a & a \<le> b) | (a < 0 & b \<le> a))"
|
avigad@16775
|
1057 |
by (auto simp add: le_divide_eq)
|
avigad@16775
|
1058 |
|
blanchet@54147
|
1059 |
lemma divide_le_eq_1:
|
haftmann@36409
|
1060 |
"(b / a \<le> 1) = ((0 < a & b \<le> a) | (a < 0 & a \<le> b) | a=0)"
|
avigad@16775
|
1061 |
by (auto simp add: divide_le_eq)
|
avigad@16775
|
1062 |
|
blanchet@54147
|
1063 |
lemma less_divide_eq_1:
|
haftmann@36409
|
1064 |
"(1 < b / a) = ((0 < a & a < b) | (a < 0 & b < a))"
|
avigad@16775
|
1065 |
by (auto simp add: less_divide_eq)
|
avigad@16775
|
1066 |
|
blanchet@54147
|
1067 |
lemma divide_less_eq_1:
|
haftmann@36409
|
1068 |
"(b / a < 1) = ((0 < a & b < a) | (a < 0 & a < b) | a=0)"
|
avigad@16775
|
1069 |
by (auto simp add: divide_less_eq)
|
avigad@16775
|
1070 |
|
hoelzl@56571
|
1071 |
lemma divide_nonneg_nonneg [simp]:
|
hoelzl@56571
|
1072 |
"0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x / y"
|
hoelzl@56571
|
1073 |
by (auto simp add: divide_simps)
|
hoelzl@56571
|
1074 |
|
hoelzl@56571
|
1075 |
lemma divide_nonpos_nonpos:
|
hoelzl@56571
|
1076 |
"x \<le> 0 \<Longrightarrow> y \<le> 0 \<Longrightarrow> 0 \<le> x / y"
|
hoelzl@56571
|
1077 |
by (auto simp add: divide_simps)
|
hoelzl@56571
|
1078 |
|
hoelzl@56571
|
1079 |
lemma divide_nonneg_nonpos:
|
hoelzl@56571
|
1080 |
"0 \<le> x \<Longrightarrow> y \<le> 0 \<Longrightarrow> x / y \<le> 0"
|
hoelzl@56571
|
1081 |
by (auto simp add: divide_simps)
|
hoelzl@56571
|
1082 |
|
hoelzl@56571
|
1083 |
lemma divide_nonpos_nonneg:
|
hoelzl@56571
|
1084 |
"x \<le> 0 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x / y \<le> 0"
|
hoelzl@56571
|
1085 |
by (auto simp add: divide_simps)
|
wenzelm@23389
|
1086 |
|
wenzelm@60758
|
1087 |
text \<open>Conditional Simplification Rules: No Case Splits\<close>
|
avigad@16775
|
1088 |
|
blanchet@54147
|
1089 |
lemma le_divide_eq_1_pos [simp]:
|
haftmann@36409
|
1090 |
"0 < a \<Longrightarrow> (1 \<le> b/a) = (a \<le> b)"
|
avigad@16775
|
1091 |
by (auto simp add: le_divide_eq)
|
avigad@16775
|
1092 |
|
blanchet@54147
|
1093 |
lemma le_divide_eq_1_neg [simp]:
|
haftmann@36409
|
1094 |
"a < 0 \<Longrightarrow> (1 \<le> b/a) = (b \<le> a)"
|
avigad@16775
|
1095 |
by (auto simp add: le_divide_eq)
|
avigad@16775
|
1096 |
|
blanchet@54147
|
1097 |
lemma divide_le_eq_1_pos [simp]:
|
haftmann@36409
|
1098 |
"0 < a \<Longrightarrow> (b/a \<le> 1) = (b \<le> a)"
|
avigad@16775
|
1099 |
by (auto simp add: divide_le_eq)
|
avigad@16775
|
1100 |
|
blanchet@54147
|
1101 |
lemma divide_le_eq_1_neg [simp]:
|
haftmann@36409
|
1102 |
"a < 0 \<Longrightarrow> (b/a \<le> 1) = (a \<le> b)"
|
avigad@16775
|
1103 |
by (auto simp add: divide_le_eq)
|
avigad@16775
|
1104 |
|
blanchet@54147
|
1105 |
lemma less_divide_eq_1_pos [simp]:
|
haftmann@36409
|
1106 |
"0 < a \<Longrightarrow> (1 < b/a) = (a < b)"
|
avigad@16775
|
1107 |
by (auto simp add: less_divide_eq)
|
avigad@16775
|
1108 |
|
blanchet@54147
|
1109 |
lemma less_divide_eq_1_neg [simp]:
|
haftmann@36409
|
1110 |
"a < 0 \<Longrightarrow> (1 < b/a) = (b < a)"
|
avigad@16775
|
1111 |
by (auto simp add: less_divide_eq)
|
avigad@16775
|
1112 |
|
blanchet@54147
|
1113 |
lemma divide_less_eq_1_pos [simp]:
|
haftmann@36409
|
1114 |
"0 < a \<Longrightarrow> (b/a < 1) = (b < a)"
|
paulson@18649
|
1115 |
by (auto simp add: divide_less_eq)
|
paulson@18649
|
1116 |
|
blanchet@54147
|
1117 |
lemma divide_less_eq_1_neg [simp]:
|
haftmann@36409
|
1118 |
"a < 0 \<Longrightarrow> b/a < 1 <-> a < b"
|
avigad@16775
|
1119 |
by (auto simp add: divide_less_eq)
|
avigad@16775
|
1120 |
|
blanchet@54147
|
1121 |
lemma eq_divide_eq_1 [simp]:
|
haftmann@36409
|
1122 |
"(1 = b/a) = ((a \<noteq> 0 & a = b))"
|
avigad@16775
|
1123 |
by (auto simp add: eq_divide_eq)
|
avigad@16775
|
1124 |
|
blanchet@54147
|
1125 |
lemma divide_eq_eq_1 [simp]:
|
haftmann@36409
|
1126 |
"(b/a = 1) = ((a \<noteq> 0 & a = b))"
|
avigad@16775
|
1127 |
by (auto simp add: divide_eq_eq)
|
avigad@16775
|
1128 |
|
paulson@14294
|
1129 |
lemma abs_inverse [simp]:
|
lp15@59667
|
1130 |
"\<bar>inverse a\<bar> =
|
haftmann@36301
|
1131 |
inverse \<bar>a\<bar>"
|
lp15@59667
|
1132 |
apply (cases "a=0", simp)
|
lp15@59667
|
1133 |
apply (simp add: nonzero_abs_inverse)
|
paulson@14294
|
1134 |
done
|
paulson@14294
|
1135 |
|
paulson@15234
|
1136 |
lemma abs_divide [simp]:
|
haftmann@36409
|
1137 |
"\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
|
lp15@59667
|
1138 |
apply (cases "b=0", simp)
|
lp15@59667
|
1139 |
apply (simp add: nonzero_abs_divide)
|
paulson@14294
|
1140 |
done
|
paulson@14294
|
1141 |
|
lp15@59667
|
1142 |
lemma abs_div_pos: "0 < y ==>
|
haftmann@36301
|
1143 |
\<bar>x\<bar> / y = \<bar>x / y\<bar>"
|
haftmann@25304
|
1144 |
apply (subst abs_divide)
|
haftmann@25304
|
1145 |
apply (simp add: order_less_imp_le)
|
haftmann@25304
|
1146 |
done
|
avigad@16775
|
1147 |
|
lp15@59667
|
1148 |
lemma zero_le_divide_abs_iff [simp]: "(0 \<le> a / abs b) = (0 \<le> a | b = 0)"
|
lp15@55718
|
1149 |
by (auto simp: zero_le_divide_iff)
|
lp15@55718
|
1150 |
|
lp15@59667
|
1151 |
lemma divide_le_0_abs_iff [simp]: "(a / abs b \<le> 0) = (a \<le> 0 | b = 0)"
|
lp15@55718
|
1152 |
by (auto simp: divide_le_0_iff)
|
lp15@55718
|
1153 |
|
hoelzl@35579
|
1154 |
lemma field_le_mult_one_interval:
|
hoelzl@35579
|
1155 |
assumes *: "\<And>z. \<lbrakk> 0 < z ; z < 1 \<rbrakk> \<Longrightarrow> z * x \<le> y"
|
hoelzl@35579
|
1156 |
shows "x \<le> y"
|
hoelzl@35579
|
1157 |
proof (cases "0 < x")
|
hoelzl@35579
|
1158 |
assume "0 < x"
|
hoelzl@35579
|
1159 |
thus ?thesis
|
hoelzl@35579
|
1160 |
using dense_le_bounded[of 0 1 "y/x"] *
|
wenzelm@60758
|
1161 |
unfolding le_divide_eq if_P[OF \<open>0 < x\<close>] by simp
|
hoelzl@35579
|
1162 |
next
|
hoelzl@35579
|
1163 |
assume "\<not>0 < x" hence "x \<le> 0" by simp
|
hoelzl@35579
|
1164 |
obtain s::'a where s: "0 < s" "s < 1" using dense[of 0 "1\<Colon>'a"] by auto
|
wenzelm@60758
|
1165 |
hence "x \<le> s * x" using mult_le_cancel_right[of 1 x s] \<open>x \<le> 0\<close> by auto
|
hoelzl@35579
|
1166 |
also note *[OF s]
|
hoelzl@35579
|
1167 |
finally show ?thesis .
|
hoelzl@35579
|
1168 |
qed
|
haftmann@35090
|
1169 |
|
haftmann@36409
|
1170 |
end
|
haftmann@36409
|
1171 |
|
haftmann@59557
|
1172 |
hide_fact (open) field_inverse field_divide_inverse field_inverse_zero
|
haftmann@59557
|
1173 |
|
haftmann@52435
|
1174 |
code_identifier
|
haftmann@52435
|
1175 |
code_module Fields \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
|
lp15@59667
|
1176 |
|
paulson@14265
|
1177 |
end
|