author | wenzelm |
Thu, 05 Nov 2015 10:39:49 +0100 | |
changeset 61585 | a9599d3d7610 |
parent 61260 | e6f03fae14d5 |
child 63092 | a949b2a5f51d |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Fraction_Field.thy |
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Author: Amine Chaieb, University of Cambridge |
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*) |
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section\<open>A formalization of the fraction field of any integral domain; |
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generalization of theory Rat from int to any integral domain\<close> |
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theory Fraction_Field |
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imports Main |
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begin |
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subsection \<open>General fractions construction\<close> |
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subsubsection \<open>Construction of the type of fractions\<close> |
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context idom begin |
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definition fractrel :: "'a \<times> 'a \<Rightarrow> 'a * 'a \<Rightarrow> bool" where |
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"fractrel = (\<lambda>x y. snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)" |
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lemma fractrel_iff [simp]: |
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"fractrel x y \<longleftrightarrow> snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x" |
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by (simp add: fractrel_def) |
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lemma symp_fractrel: "symp fractrel" |
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by (simp add: symp_def) |
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lemma transp_fractrel: "transp fractrel" |
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proof (rule transpI, unfold split_paired_all) |
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fix a b a' b' a'' b'' :: 'a |
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assume A: "fractrel (a, b) (a', b')" |
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assume B: "fractrel (a', b') (a'', b'')" |
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have "b' * (a * b'') = b'' * (a * b')" by (simp add: ac_simps) |
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also from A have "a * b' = a' * b" by auto |
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also have "b'' * (a' * b) = b * (a' * b'')" by (simp add: ac_simps) |
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also from B have "a' * b'' = a'' * b'" by auto |
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also have "b * (a'' * b') = b' * (a'' * b)" by (simp add: ac_simps) |
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finally have "b' * (a * b'') = b' * (a'' * b)" . |
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moreover from B have "b' \<noteq> 0" by auto |
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ultimately have "a * b'' = a'' * b" by simp |
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with A B show "fractrel (a, b) (a'', b'')" by auto |
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qed |
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lemma part_equivp_fractrel: "part_equivp fractrel" |
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using _ symp_fractrel transp_fractrel |
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by(rule part_equivpI)(rule exI[where x="(0, 1)"]; simp) |
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end |
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quotient_type (overloaded) 'a fract = "'a :: idom \<times> 'a" / partial: "fractrel" |
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by(rule part_equivp_fractrel) |
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subsubsection \<open>Representation and basic operations\<close> |
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lift_definition Fract :: "'a :: idom \<Rightarrow> 'a \<Rightarrow> 'a fract" |
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is "\<lambda>a b. if b = 0 then (0, 1) else (a, b)" |
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by simp |
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lemma Fract_cases [cases type: fract]: |
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obtains (Fract) a b where "q = Fract a b" "b \<noteq> 0" |
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by transfer simp |
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lemma Fract_induct [case_names Fract, induct type: fract]: |
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"(\<And>a b. b \<noteq> 0 \<Longrightarrow> P (Fract a b)) \<Longrightarrow> P q" |
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by (cases q) simp |
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lemma eq_fract: |
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shows "\<And>a b c d. b \<noteq> 0 \<Longrightarrow> d \<noteq> 0 \<Longrightarrow> Fract a b = Fract c d \<longleftrightarrow> a * d = c * b" |
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and "\<And>a. Fract a 0 = Fract 0 1" |
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and "\<And>a c. Fract 0 a = Fract 0 c" |
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by(transfer; simp)+ |
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instantiation fract :: (idom) "{comm_ring_1,power}" |
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begin |
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lift_definition zero_fract :: "'a fract" is "(0, 1)" by simp |
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lemma Zero_fract_def: "0 = Fract 0 1" |
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by transfer simp |
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lift_definition one_fract :: "'a fract" is "(1, 1)" by simp |
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lemma One_fract_def: "1 = Fract 1 1" |
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by transfer simp |
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lift_definition plus_fract :: "'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract" |
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is "\<lambda>q r. (fst q * snd r + fst r * snd q, snd q * snd r)" |
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by(auto simp add: algebra_simps) |
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lemma add_fract [simp]: |
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"\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b + Fract c d = Fract (a * d + c * b) (b * d)" |
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by transfer simp |
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lift_definition uminus_fract :: "'a fract \<Rightarrow> 'a fract" |
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is "\<lambda>x. (- fst x, snd x)" |
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by simp |
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lemma minus_fract [simp]: |
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fixes a b :: "'a::idom" |
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shows "- Fract a b = Fract (- a) b" |
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by transfer simp |
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lemma minus_fract_cancel [simp]: "Fract (- a) (- b) = Fract a b" |
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by (cases "b = 0") (simp_all add: eq_fract) |
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definition diff_fract_def: "q - r = q + - (r::'a fract)" |
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lemma diff_fract [simp]: |
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"\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b - Fract c d = Fract (a * d - c * b) (b * d)" |
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by (simp add: diff_fract_def) |
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lift_definition times_fract :: "'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract" |
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is "\<lambda>q r. (fst q * fst r, snd q * snd r)" |
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by(simp add: algebra_simps) |
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lemma mult_fract [simp]: "Fract (a::'a::idom) b * Fract c d = Fract (a * c) (b * d)" |
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by transfer simp |
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lemma mult_fract_cancel: |
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"c \<noteq> 0 \<Longrightarrow> Fract (c * a) (c * b) = Fract a b" |
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by transfer simp |
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instance |
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proof |
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fix q r s :: "'a fract" |
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show "(q * r) * s = q * (r * s)" |
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by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps) |
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show "q * r = r * q" |
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by (cases q, cases r) (simp add: eq_fract algebra_simps) |
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show "1 * q = q" |
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by (cases q) (simp add: One_fract_def eq_fract) |
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show "(q + r) + s = q + (r + s)" |
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by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps) |
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show "q + r = r + q" |
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by (cases q, cases r) (simp add: eq_fract algebra_simps) |
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show "0 + q = q" |
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by (cases q) (simp add: Zero_fract_def eq_fract) |
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show "- q + q = 0" |
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by (cases q) (simp add: Zero_fract_def eq_fract) |
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show "q - r = q + - r" |
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by (cases q, cases r) (simp add: eq_fract) |
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show "(q + r) * s = q * s + r * s" |
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by (cases q, cases r, cases s) (simp add: eq_fract algebra_simps) |
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show "(0::'a fract) \<noteq> 1" |
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by (simp add: Zero_fract_def One_fract_def eq_fract) |
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qed |
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end |
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lemma of_nat_fract: "of_nat k = Fract (of_nat k) 1" |
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by (induct k) (simp_all add: Zero_fract_def One_fract_def) |
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lemma Fract_of_nat_eq: "Fract (of_nat k) 1 = of_nat k" |
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by (rule of_nat_fract [symmetric]) |
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lemma fract_collapse: |
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"Fract 0 k = 0" |
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"Fract 1 1 = 1" |
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"Fract k 0 = 0" |
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by(transfer; simp)+ |
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lemma fract_expand: |
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"0 = Fract 0 1" |
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"1 = Fract 1 1" |
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by (simp_all add: fract_collapse) |
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lemma Fract_cases_nonzero: |
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obtains (Fract) a b where "q = Fract a b" and "b \<noteq> 0" and "a \<noteq> 0" |
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| (0) "q = 0" |
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proof (cases "q = 0") |
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case True |
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then show thesis using 0 by auto |
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next |
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case False |
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then obtain a b where "q = Fract a b" and "b \<noteq> 0" by (cases q) auto |
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with False have "0 \<noteq> Fract a b" by simp |
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with \<open>b \<noteq> 0\<close> have "a \<noteq> 0" by (simp add: Zero_fract_def eq_fract) |
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with Fract \<open>q = Fract a b\<close> \<open>b \<noteq> 0\<close> show thesis by auto |
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qed |
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subsubsection \<open>The field of rational numbers\<close> |
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Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
183 |
|
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
184 |
context idom |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
185 |
begin |
53196 | 186 |
|
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
187 |
subclass ring_no_zero_divisors .. |
53196 | 188 |
|
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
189 |
end |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
190 |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58881
diff
changeset
|
191 |
instantiation fract :: (idom) field |
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
192 |
begin |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
193 |
|
61106 | 194 |
lift_definition inverse_fract :: "'a fract \<Rightarrow> 'a fract" |
195 |
is "\<lambda>x. if fst x = 0 then (0, 1) else (snd x, fst x)" |
|
196 |
by(auto simp add: algebra_simps) |
|
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
197 |
|
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
198 |
lemma inverse_fract [simp]: "inverse (Fract a b) = Fract (b::'a::idom) a" |
61106 | 199 |
by transfer simp |
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
200 |
|
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
201 |
definition divide_fract_def: "q div r = q * inverse (r:: 'a fract)" |
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
202 |
|
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
203 |
lemma divide_fract [simp]: "Fract a b div Fract c d = Fract (a * d) (b * c)" |
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
204 |
by (simp add: divide_fract_def) |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
205 |
|
47252 | 206 |
instance |
207 |
proof |
|
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
208 |
fix q :: "'a fract" |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
209 |
assume "q \<noteq> 0" |
46573 | 210 |
then show "inverse q * q = 1" |
211 |
by (cases q rule: Fract_cases_nonzero) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
54863
diff
changeset
|
212 |
(simp_all add: fract_expand eq_fract mult.commute) |
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
213 |
next |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
214 |
fix q r :: "'a fract" |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
215 |
show "q div r = q * inverse r" by (simp add: divide_fract_def) |
36409 | 216 |
next |
46573 | 217 |
show "inverse 0 = (0:: 'a fract)" |
218 |
by (simp add: fract_expand) (simp add: fract_collapse) |
|
31761
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
219 |
qed |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
220 |
|
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
221 |
end |
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
222 |
|
3585bebe49a8
Added Library/Fraction_Field.thy: The fraction field of any integral
chaieb
parents:
diff
changeset
|
223 |
|
60500 | 224 |
subsubsection \<open>The ordered field of fractions over an ordered idom\<close> |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
225 |
|
61106 | 226 |
instantiation fract :: (linordered_idom) linorder |
227 |
begin |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
228 |
|
61106 | 229 |
lemma less_eq_fract_respect: |
230 |
fixes a b a' b' c d c' d' :: 'a |
|
231 |
assumes neq: "b \<noteq> 0" "b' \<noteq> 0" "d \<noteq> 0" "d' \<noteq> 0" |
|
232 |
assumes eq1: "a * b' = a' * b" |
|
233 |
assumes eq2: "c * d' = c' * d" |
|
234 |
shows "((a * d) * (b * d) \<le> (c * b) * (b * d)) \<longleftrightarrow> ((a' * d') * (b' * d') \<le> (c' * b') * (b' * d'))" |
|
235 |
proof - |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
236 |
let ?le = "\<lambda>a b c d. ((a * d) * (b * d) \<le> (c * b) * (b * d))" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
237 |
{ |
54463 | 238 |
fix a b c d x :: 'a |
239 |
assume x: "x \<noteq> 0" |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
240 |
have "?le a b c d = ?le (a * x) (b * x) c d" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
241 |
proof - |
54463 | 242 |
from x have "0 < x * x" |
243 |
by (auto simp add: zero_less_mult_iff) |
|
46573 | 244 |
then have "?le a b c d = |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
245 |
((a * d) * (b * d) * (x * x) \<le> (c * b) * (b * d) * (x * x))" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
246 |
by (simp add: mult_le_cancel_right) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
247 |
also have "... = ?le (a * x) (b * x) c d" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
248 |
by (simp add: ac_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
249 |
finally show ?thesis . |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
250 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
251 |
} note le_factor = this |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
252 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
253 |
let ?D = "b * d" and ?D' = "b' * d'" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
254 |
from neq have D: "?D \<noteq> 0" by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
255 |
from neq have "?D' \<noteq> 0" by simp |
46573 | 256 |
then have "?le a b c d = ?le (a * ?D') (b * ?D') c d" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
257 |
by (rule le_factor) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
258 |
also have "... = ((a * b') * ?D * ?D' * d * d' \<le> (c * d') * ?D * ?D' * b * b')" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
259 |
by (simp add: ac_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
260 |
also have "... = ((a' * b) * ?D * ?D' * d * d' \<le> (c' * d) * ?D * ?D' * b * b')" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
261 |
by (simp only: eq1 eq2) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
262 |
also have "... = ?le (a' * ?D) (b' * ?D) c' d'" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
263 |
by (simp add: ac_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
264 |
also from D have "... = ?le a' b' c' d'" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
265 |
by (rule le_factor [symmetric]) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
266 |
finally show "?le a b c d = ?le a' b' c' d'" . |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
267 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
268 |
|
61106 | 269 |
lift_definition less_eq_fract :: "'a fract \<Rightarrow> 'a fract \<Rightarrow> bool" |
270 |
is "\<lambda>q r. (fst q * snd r) * (snd q * snd r) \<le> (fst r * snd q) * (snd q * snd r)" |
|
271 |
by (clarsimp simp add: less_eq_fract_respect) |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
272 |
|
46573 | 273 |
definition less_fract_def: "z < (w::'a fract) \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
274 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
275 |
lemma le_fract [simp]: |
61106 | 276 |
"\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b \<le> Fract c d \<longleftrightarrow> (a * d) * (b * d) \<le> (c * b) * (b * d)" |
277 |
by transfer simp |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
278 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
279 |
lemma less_fract [simp]: |
61106 | 280 |
"\<lbrakk> b \<noteq> 0; d \<noteq> 0 \<rbrakk> \<Longrightarrow> Fract a b < Fract c d \<longleftrightarrow> (a * d) * (b * d) < (c * b) * (b * d)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
281 |
by (simp add: less_fract_def less_le_not_le ac_simps assms) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
282 |
|
47252 | 283 |
instance |
284 |
proof |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
285 |
fix q r s :: "'a fract" |
54463 | 286 |
assume "q \<le> r" and "r \<le> s" |
287 |
then show "q \<le> s" |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
288 |
proof (induct q, induct r, induct s) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
289 |
fix a b c d e f :: 'a |
54463 | 290 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" |
291 |
assume 1: "Fract a b \<le> Fract c d" |
|
292 |
assume 2: "Fract c d \<le> Fract e f" |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
293 |
show "Fract a b \<le> Fract e f" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
294 |
proof - |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
295 |
from neq obtain bb: "0 < b * b" and dd: "0 < d * d" and ff: "0 < f * f" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
296 |
by (auto simp add: zero_less_mult_iff linorder_neq_iff) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
297 |
have "(a * d) * (b * d) * (f * f) \<le> (c * b) * (b * d) * (f * f)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
298 |
proof - |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
299 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
300 |
by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
301 |
with ff show ?thesis by (simp add: mult_le_cancel_right) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
302 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
303 |
also have "... = (c * f) * (d * f) * (b * b)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
304 |
by (simp only: ac_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
305 |
also have "... \<le> (e * d) * (d * f) * (b * b)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
306 |
proof - |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
307 |
from neq 2 have "(c * f) * (d * f) \<le> (e * d) * (d * f)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
308 |
by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
309 |
with bb show ?thesis by (simp add: mult_le_cancel_right) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
310 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
311 |
finally have "(a * f) * (b * f) * (d * d) \<le> e * b * (b * f) * (d * d)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
312 |
by (simp only: ac_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
313 |
with dd have "(a * f) * (b * f) \<le> (e * b) * (b * f)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
314 |
by (simp add: mult_le_cancel_right) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
315 |
with neq show ?thesis by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
316 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
317 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
318 |
next |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
319 |
fix q r :: "'a fract" |
54463 | 320 |
assume "q \<le> r" and "r \<le> q" |
321 |
then show "q = r" |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
322 |
proof (induct q, induct r) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
323 |
fix a b c d :: 'a |
54463 | 324 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" |
325 |
assume 1: "Fract a b \<le> Fract c d" |
|
326 |
assume 2: "Fract c d \<le> Fract a b" |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
327 |
show "Fract a b = Fract c d" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
328 |
proof - |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
329 |
from neq 1 have "(a * d) * (b * d) \<le> (c * b) * (b * d)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
330 |
by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
331 |
also have "... \<le> (a * d) * (b * d)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
332 |
proof - |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
333 |
from neq 2 have "(c * b) * (d * b) \<le> (a * d) * (d * b)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
334 |
by simp |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
335 |
then show ?thesis by (simp only: ac_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
336 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
337 |
finally have "(a * d) * (b * d) = (c * b) * (b * d)" . |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
338 |
moreover from neq have "b * d \<noteq> 0" by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
339 |
ultimately have "a * d = c * b" by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
340 |
with neq show ?thesis by (simp add: eq_fract) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
341 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
342 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
343 |
next |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
344 |
fix q r :: "'a fract" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
345 |
show "q \<le> q" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
346 |
by (induct q) simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
347 |
show "(q < r) = (q \<le> r \<and> \<not> r \<le> q)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
348 |
by (simp only: less_fract_def) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
349 |
show "q \<le> r \<or> r \<le> q" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
350 |
by (induct q, induct r) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
54863
diff
changeset
|
351 |
(simp add: mult.commute, rule linorder_linear) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
352 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
353 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
354 |
end |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
355 |
|
54463 | 356 |
instantiation fract :: (linordered_idom) "{distrib_lattice,abs_if,sgn_if}" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
357 |
begin |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
358 |
|
61106 | 359 |
definition abs_fract_def2: "\<bar>q\<bar> = (if q < 0 then -q else (q::'a fract))" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
360 |
|
46573 | 361 |
definition sgn_fract_def: |
54463 | 362 |
"sgn (q::'a fract) = (if q = 0 then 0 else if 0 < q then 1 else - 1)" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
363 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
364 |
theorem abs_fract [simp]: "\<bar>Fract a b\<bar> = Fract \<bar>a\<bar> \<bar>b\<bar>" |
61106 | 365 |
unfolding abs_fract_def2 not_le[symmetric] |
366 |
by transfer(auto simp add: zero_less_mult_iff le_less) |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
367 |
|
46573 | 368 |
definition inf_fract_def: |
61076 | 369 |
"(inf :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = min" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
370 |
|
46573 | 371 |
definition sup_fract_def: |
61076 | 372 |
"(sup :: 'a fract \<Rightarrow> 'a fract \<Rightarrow> 'a fract) = max" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
373 |
|
46573 | 374 |
instance |
61106 | 375 |
by intro_classes (simp_all add: abs_fract_def2 sgn_fract_def inf_fract_def sup_fract_def max_min_distrib2) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
376 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
377 |
end |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
378 |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
58881
diff
changeset
|
379 |
instance fract :: (linordered_idom) linordered_field |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
380 |
proof |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
381 |
fix q r s :: "'a fract" |
53196 | 382 |
assume "q \<le> r" |
383 |
then show "s + q \<le> s + r" |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
384 |
proof (induct q, induct r, induct s) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
385 |
fix a b c d e f :: 'a |
53196 | 386 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
387 |
assume le: "Fract a b \<le> Fract c d" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
388 |
show "Fract e f + Fract a b \<le> Fract e f + Fract c d" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
389 |
proof - |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
390 |
let ?F = "f * f" from neq have F: "0 < ?F" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
391 |
by (auto simp add: zero_less_mult_iff) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
392 |
from neq le have "(a * d) * (b * d) \<le> (c * b) * (b * d)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
393 |
by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
394 |
with F have "(a * d) * (b * d) * ?F * ?F \<le> (c * b) * (b * d) * ?F * ?F" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
395 |
by (simp add: mult_le_cancel_right) |
36348
89c54f51f55a
dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents:
36331
diff
changeset
|
396 |
with neq show ?thesis by (simp add: field_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
397 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
398 |
qed |
53196 | 399 |
next |
400 |
fix q r s :: "'a fract" |
|
401 |
assume "q < r" and "0 < s" |
|
402 |
then show "s * q < s * r" |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
403 |
proof (induct q, induct r, induct s) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
404 |
fix a b c d e f :: 'a |
54463 | 405 |
assume neq: "b \<noteq> 0" "d \<noteq> 0" "f \<noteq> 0" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
406 |
assume le: "Fract a b < Fract c d" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
407 |
assume gt: "0 < Fract e f" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
408 |
show "Fract e f * Fract a b < Fract e f * Fract c d" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
409 |
proof - |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
410 |
let ?E = "e * f" and ?F = "f * f" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
411 |
from neq gt have "0 < ?E" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
412 |
by (auto simp add: Zero_fract_def order_less_le eq_fract) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
413 |
moreover from neq have "0 < ?F" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
414 |
by (auto simp add: zero_less_mult_iff) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
415 |
moreover from neq le have "(a * d) * (b * d) < (c * b) * (b * d)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
416 |
by simp |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
417 |
ultimately have "(a * d) * (b * d) * ?E * ?F < (c * b) * (b * d) * ?E * ?F" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
418 |
by (simp add: mult_less_cancel_right) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
419 |
with neq show ?thesis |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
420 |
by (simp add: ac_simps) |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
421 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
422 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
423 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
424 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
425 |
lemma fract_induct_pos [case_names Fract]: |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
426 |
fixes P :: "'a::linordered_idom fract \<Rightarrow> bool" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
427 |
assumes step: "\<And>a b. 0 < b \<Longrightarrow> P (Fract a b)" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
428 |
shows "P q" |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
429 |
proof (cases q) |
54463 | 430 |
case (Fract a b) |
431 |
{ |
|
432 |
fix a b :: 'a |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
433 |
assume b: "b < 0" |
54463 | 434 |
have "P (Fract a b)" |
435 |
proof - |
|
436 |
from b have "0 < - b" by simp |
|
437 |
then have "P (Fract (- a) (- b))" |
|
438 |
by (rule step) |
|
439 |
then show "P (Fract a b)" |
|
440 |
by (simp add: order_less_imp_not_eq [OF b]) |
|
441 |
qed |
|
442 |
} |
|
443 |
with Fract show "P q" |
|
444 |
by (auto simp add: linorder_neq_iff step) |
|
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
445 |
qed |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
446 |
|
53196 | 447 |
lemma zero_less_Fract_iff: "0 < b \<Longrightarrow> 0 < Fract a b \<longleftrightarrow> 0 < a" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
448 |
by (auto simp add: Zero_fract_def zero_less_mult_iff) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
449 |
|
53196 | 450 |
lemma Fract_less_zero_iff: "0 < b \<Longrightarrow> Fract a b < 0 \<longleftrightarrow> a < 0" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
451 |
by (auto simp add: Zero_fract_def mult_less_0_iff) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
452 |
|
53196 | 453 |
lemma zero_le_Fract_iff: "0 < b \<Longrightarrow> 0 \<le> Fract a b \<longleftrightarrow> 0 \<le> a" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
454 |
by (auto simp add: Zero_fract_def zero_le_mult_iff) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
455 |
|
53196 | 456 |
lemma Fract_le_zero_iff: "0 < b \<Longrightarrow> Fract a b \<le> 0 \<longleftrightarrow> a \<le> 0" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
457 |
by (auto simp add: Zero_fract_def mult_le_0_iff) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
458 |
|
53196 | 459 |
lemma one_less_Fract_iff: "0 < b \<Longrightarrow> 1 < Fract a b \<longleftrightarrow> b < a" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
460 |
by (auto simp add: One_fract_def mult_less_cancel_right_disj) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
461 |
|
53196 | 462 |
lemma Fract_less_one_iff: "0 < b \<Longrightarrow> Fract a b < 1 \<longleftrightarrow> a < b" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
463 |
by (auto simp add: One_fract_def mult_less_cancel_right_disj) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
464 |
|
53196 | 465 |
lemma one_le_Fract_iff: "0 < b \<Longrightarrow> 1 \<le> Fract a b \<longleftrightarrow> b \<le> a" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
466 |
by (auto simp add: One_fract_def mult_le_cancel_right) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
467 |
|
53196 | 468 |
lemma Fract_le_one_iff: "0 < b \<Longrightarrow> Fract a b \<le> 1 \<longleftrightarrow> a \<le> b" |
36331
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
469 |
by (auto simp add: One_fract_def mult_le_cancel_right) |
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
470 |
|
6b9e487450ba
Library/Fraction_Field.thy: ordering relations for fractions
huffman
parents:
36312
diff
changeset
|
471 |
end |