--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Abstract_Rat.thy Thu Aug 09 15:52:49 2007 +0200
@@ -0,0 +1,502 @@
+(* Title: HOL/Library/Abstract_Rat.thy
+ ID: $Id$
+ Author: Amine Chaieb
+*)
+
+header {* Abstract rational numbers *}
+
+theory Abstract_Rat
+imports GCD
+begin
+
+types Num = "int \<times> int"
+syntax "_Num0" :: "Num" ("0\<^sub>N")
+translations "0\<^sub>N" \<rightleftharpoons> "(0, 0)"
+syntax "_Numi" :: "int \<Rightarrow> Num" ("_\<^sub>N")
+translations "i\<^sub>N" \<rightleftharpoons> "(i, 1) \<Colon> Num"
+
+definition
+ isnormNum :: "Num \<Rightarrow> bool"
+where
+ "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> igcd a b = 1))"
+
+definition
+ normNum :: "Num \<Rightarrow> Num"
+where
+ "normNum = (\<lambda>(a,b). (if a=0 \<or> b = 0 then (0,0) else
+ (let g = igcd a b
+ in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
+
+lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
+proof -
+ have " \<exists> a b. x = (a,b)" by auto
+ then obtain a b where x[simp]: "x = (a,b)" by blast
+ {assume "a=0 \<or> b = 0" hence ?thesis by (simp add: normNum_def isnormNum_def)}
+ moreover
+ {assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
+ let ?g = "igcd a b"
+ let ?a' = "a div ?g"
+ let ?b' = "b div ?g"
+ let ?g' = "igcd ?a' ?b'"
+ from anz bnz have "?g \<noteq> 0" by simp with igcd_pos[of a b]
+ have gpos: "?g > 0" by arith
+ have gdvd: "?g dvd a" "?g dvd b" by (simp_all add: igcd_dvd1 igcd_dvd2)
+ from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]
+ anz bnz
+ have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"
+ by - (rule notI,simp add:igcd_def)+
+ from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by blast
+ from div_igcd_relprime[OF stupid] have gp1: "?g' = 1" .
+ from bnz have "b < 0 \<or> b > 0" by arith
+ moreover
+ {assume b: "b > 0"
+ from pos_imp_zdiv_nonneg_iff[OF gpos] b
+ have "?b' \<ge> 0" by simp
+ with nz' have b': "?b' > 0" by simp
+ from b b' anz bnz nz' gp1 have ?thesis
+ by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
+ moreover {assume b: "b < 0"
+ {assume b': "?b' \<ge> 0"
+ from gpos have th: "?g \<ge> 0" by arith
+ from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]
+ have False using b by simp }
+ hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
+ from anz bnz nz' b b' gp1 have ?thesis
+ by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
+ ultimately have ?thesis by blast
+ }
+ ultimately show ?thesis by blast
+qed
+
+text {* Arithmetic over Num *}
+
+definition
+ Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60)
+where
+ "Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b')
+ else if a'=0 \<or> b' = 0 then normNum(a,b)
+ else normNum(a*b' + b*a', b*b'))"
+
+definition
+ Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)
+where
+ "Nmul = (\<lambda>(a,b) (a',b'). let g = igcd (a*a') (b*b')
+ in (a*a' div g, b*b' div g))"
+
+definition
+ Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
+where
+ "Nneg \<equiv> (\<lambda>(a,b). (-a,b))"
+
+definition
+ Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)
+where
+ "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
+
+definition
+ Ninv :: "Num \<Rightarrow> Num"
+where
+ "Ninv \<equiv> \<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a)"
+
+definition
+ Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)
+where
+ "Ndiv \<equiv> \<lambda>a b. a *\<^sub>N Ninv b"
+
+lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
+ by(simp add: isnormNum_def Nneg_def split_def)
+lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
+ by (simp add: Nadd_def split_def)
+lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)"
+ by (simp add: Nsub_def split_def)
+lemma Nmul_normN[simp]: assumes xn:"isnormNum x" and yn: "isnormNum y"
+ shows "isnormNum (x *\<^sub>N y)"
+proof-
+ have "\<exists>a b. x = (a,b)" and "\<exists> a' b'. y = (a',b')" by auto
+ then obtain a b a' b' where ab: "x = (a,b)" and ab': "y = (a',b')" by blast
+ {assume "a = 0"
+ hence ?thesis using xn ab ab'
+ by (simp add: igcd_def isnormNum_def Let_def Nmul_def split_def)}
+ moreover
+ {assume "a' = 0"
+ hence ?thesis using yn ab ab'
+ by (simp add: igcd_def isnormNum_def Let_def Nmul_def split_def)}
+ moreover
+ {assume a: "a \<noteq>0" and a': "a'\<noteq>0"
+ hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)
+ from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a*a', b*b')"
+ using ab ab' a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
+ hence ?thesis by simp}
+ ultimately show ?thesis by blast
+qed
+
+lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
+by (simp add: Ninv_def isnormNum_def split_def)
+(cases "fst x = 0",auto simp add: igcd_commute)
+
+lemma isnormNum_int[simp]:
+ "isnormNum 0\<^sub>N" "isnormNum (1::int)\<^sub>N" "i \<noteq> 0 \<Longrightarrow> isnormNum i\<^sub>N"
+ by (simp_all add: isnormNum_def igcd_def)
+
+
+text {* Relations over Num *}
+
+definition
+ Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
+where
+ "Nlt0 = (\<lambda>(a,b). a < 0)"
+
+definition
+ Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
+where
+ "Nle0 = (\<lambda>(a,b). a \<le> 0)"
+
+definition
+ Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
+where
+ "Ngt0 = (\<lambda>(a,b). a > 0)"
+
+definition
+ Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
+where
+ "Nge0 = (\<lambda>(a,b). a \<ge> 0)"
+
+definition
+ Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)
+where
+ "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
+
+definition
+ Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)
+where
+ "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
+
+definition
+ "INum = (\<lambda>(a,b). of_int a / of_int b)"
+
+lemma INum_int [simp]: "INum i\<^sub>N = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)"
+ by (simp_all add: INum_def)
+
+lemma isnormNum_unique[simp]:
+ assumes na: "isnormNum x" and nb: "isnormNum y"
+ shows "((INum x ::'a::{ring_char_0,field, division_by_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
+proof
+ have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto
+ then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast
+ assume H: ?lhs
+ {assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" hence ?rhs
+ using na nb H
+ apply (simp add: INum_def split_def isnormNum_def)
+ apply (cases "a = 0", simp_all)
+ apply (cases "b = 0", simp_all)
+ apply (cases "a' = 0", simp_all)
+ apply (cases "a' = 0", simp_all add: of_int_eq_0_iff)
+ done}
+ moreover
+ { assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"
+ from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)
+ from prems have eq:"a * b' = a'*b"
+ by (simp add: INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
+ from prems have gcd1: "igcd a b = 1" "igcd b a = 1" "igcd a' b' = 1" "igcd b' a' = 1"
+ by (simp_all add: isnormNum_def add: igcd_commute)
+ from eq have raw_dvd: "a dvd a'*b" "b dvd b'*a" "a' dvd a*b'" "b' dvd b*a'"
+ apply(unfold dvd_def)
+ apply (rule_tac x="b'" in exI, simp add: mult_ac)
+ apply (rule_tac x="a'" in exI, simp add: mult_ac)
+ apply (rule_tac x="b" in exI, simp add: mult_ac)
+ apply (rule_tac x="a" in exI, simp add: mult_ac)
+ done
+ from zdvd_dvd_eq[OF bz zrelprime_dvd_mult[OF gcd1(2) raw_dvd(2)]
+ zrelprime_dvd_mult[OF gcd1(4) raw_dvd(4)]]
+ have eq1: "b = b'" using pos by simp_all
+ with eq have "a = a'" using pos by simp
+ with eq1 have ?rhs by simp}
+ ultimately show ?rhs by blast
+next
+ assume ?rhs thus ?lhs by simp
+qed
+
+
+lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{ring_char_0, field,division_by_zero})) = (x = 0\<^sub>N)"
+ unfolding INum_int(2)[symmetric]
+ by (rule isnormNum_unique, simp_all)
+
+lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::{field, ring_char_0}) / (of_int d) =
+ of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
+proof -
+ assume "d ~= 0"
+ hence dz: "of_int d \<noteq> (0::'a)" by (simp add: of_int_eq_0_iff)
+ let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"
+ let ?f = "\<lambda>x. x / of_int d"
+ have "x = (x div d) * d + x mod d"
+ by auto
+ then have eq: "of_int x = ?t"
+ by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
+ then have "of_int x / of_int d = ?t / of_int d"
+ using cong[OF refl[of ?f] eq] by simp
+ then show ?thesis by (simp add: add_divide_distrib ring_simps prems)
+qed
+
+lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
+ (of_int(n div d)::'a::{field, ring_char_0}) = of_int n / of_int d"
+ apply (frule of_int_div_aux [of d n, where ?'a = 'a])
+ apply simp
+ apply (simp add: zdvd_iff_zmod_eq_0)
+done
+
+
+lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{ring_char_0,field, division_by_zero})"
+proof-
+ have "\<exists> a b. x = (a,b)" by auto
+ then obtain a b where x[simp]: "x = (a,b)" by blast
+ {assume "a=0 \<or> b = 0" hence ?thesis
+ by (simp add: INum_def normNum_def split_def Let_def)}
+ moreover
+ {assume a: "a\<noteq>0" and b: "b\<noteq>0"
+ let ?g = "igcd a b"
+ from a b have g: "?g \<noteq> 0"by simp
+ from of_int_div[OF g, where ?'a = 'a]
+ have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}
+ ultimately show ?thesis by blast
+qed
+
+lemma INum_normNum_iff [code]: "(INum x ::'a::{field, division_by_zero, ring_char_0}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
+proof -
+ have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
+ by (simp del: normNum)
+ also have "\<dots> = ?lhs" by simp
+ finally show ?thesis by simp
+qed
+
+lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {ring_char_0,division_by_zero,field})"
+proof-
+let ?z = "0:: 'a"
+ have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
+ then obtain a b a' b' where x[simp]: "x = (a,b)"
+ and y[simp]: "y = (a',b')" by blast
+ {assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" hence ?thesis
+ apply (cases "a=0",simp_all add: Nadd_def)
+ apply (cases "b= 0",simp_all add: INum_def)
+ apply (cases "a'= 0",simp_all)
+ apply (cases "b'= 0",simp_all)
+ done }
+ moreover
+ {assume aa':"a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
+ {assume z: "a * b' + b * a' = 0"
+ hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp
+ hence "of_int b' * of_int a / (of_int b * of_int b') + of_int b * of_int a' / (of_int b * of_int b') = ?z" by (simp add:add_divide_distrib)
+ hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' by simp
+ from z aa' bb' have ?thesis
+ by (simp add: th Nadd_def normNum_def INum_def split_def)}
+ moreover {assume z: "a * b' + b * a' \<noteq> 0"
+ let ?g = "igcd (a * b' + b * a') (b*b')"
+ have gz: "?g \<noteq> 0" using z by simp
+ have ?thesis using aa' bb' z gz
+ of_int_div[where ?'a = 'a,
+ OF gz igcd_dvd1[where i="a * b' + b * a'" and j="b*b'"]]
+ of_int_div[where ?'a = 'a,
+ OF gz igcd_dvd2[where i="a * b' + b * a'" and j="b*b'"]]
+ by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
+ ultimately have ?thesis using aa' bb'
+ by (simp add: Nadd_def INum_def normNum_def x y Let_def) }
+ ultimately show ?thesis by blast
+qed
+
+lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {ring_char_0,division_by_zero,field}) "
+proof-
+ let ?z = "0::'a"
+ have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
+ then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast
+ {assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis
+ apply (cases "a=0",simp_all add: x y Nmul_def INum_def Let_def)
+ apply (cases "b=0",simp_all)
+ apply (cases "a'=0",simp_all)
+ done }
+ moreover
+ {assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
+ let ?g="igcd (a*a') (b*b')"
+ have gz: "?g \<noteq> 0" using z by simp
+ from z of_int_div[where ?'a = 'a, OF gz igcd_dvd1[where i="a*a'" and j="b*b'"]]
+ of_int_div[where ?'a = 'a , OF gz igcd_dvd2[where i="a*a'" and j="b*b'"]]
+ have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}
+ ultimately show ?thesis by blast
+qed
+
+lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
+ by (simp add: Nneg_def split_def INum_def)
+
+lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {ring_char_0,division_by_zero,field})"
+by (simp add: Nsub_def split_def)
+
+lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: {division_by_zero,field}) / (INum x)"
+ by (simp add: Ninv_def INum_def split_def)
+
+lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {ring_char_0, division_by_zero,field})" by (simp add: Ndiv_def)
+
+lemma Nlt0_iff[simp]: assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})< 0) = 0>\<^sub>N x "
+proof-
+ have " \<exists> a b. x = (a,b)" by simp
+ then obtain a b where x[simp]:"x = (a,b)" by blast
+ {assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }
+ moreover
+ {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
+ from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: Nlt0_def INum_def)}
+ ultimately show ?thesis by blast
+qed
+
+lemma Nle0_iff[simp]:assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
+proof-
+ have " \<exists> a b. x = (a,b)" by simp
+ then obtain a b where x[simp]:"x = (a,b)" by blast
+ {assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }
+ moreover
+ {assume a: "a\<noteq>0" hence b: "(of_int b :: 'a) > 0" using nx by (simp add: isnormNum_def)
+ from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: Nle0_def INum_def)}
+ ultimately show ?thesis by blast
+qed
+
+lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})> 0) = 0<\<^sub>N x"
+proof-
+ have " \<exists> a b. x = (a,b)" by simp
+ then obtain a b where x[simp]:"x = (a,b)" by blast
+ {assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }
+ moreover
+ {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
+ from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: Ngt0_def INum_def)}
+ ultimately show ?thesis by blast
+qed
+lemma Nge0_iff[simp]:assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
+proof-
+ have " \<exists> a b. x = (a,b)" by simp
+ then obtain a b where x[simp]:"x = (a,b)" by blast
+ {assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }
+ moreover
+ {assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
+ from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: Nge0_def INum_def)}
+ ultimately show ?thesis by blast
+qed
+
+lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field}) < INum y) = (x <\<^sub>N y)"
+proof-
+ let ?z = "0::'a"
+ have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp
+ also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
+ finally show ?thesis by (simp add: Nlt_def)
+qed
+
+lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
+ shows "((INum x :: 'a :: {ring_char_0,division_by_zero,ordered_field})\<le> INum y) = (x \<le>\<^sub>N y)"
+proof-
+ have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp
+ also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp
+ finally show ?thesis by (simp add: Nle_def)
+qed
+
+lemma Nadd_commute: "x +\<^sub>N y = y +\<^sub>N x"
+proof-
+ have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
+ have "(INum (x +\<^sub>N y)::'a :: {ring_char_0,division_by_zero,field}) = INum (y +\<^sub>N x)" by simp
+ with isnormNum_unique[OF n] show ?thesis by simp
+qed
+
+lemma[simp]: "(0, b) +\<^sub>N y = normNum y" "(a, 0) +\<^sub>N y = normNum y"
+ "x +\<^sub>N (0, b) = normNum x" "x +\<^sub>N (a, 0) = normNum x"
+ apply (simp add: Nadd_def split_def, simp add: Nadd_def split_def)
+ apply (subst Nadd_commute,simp add: Nadd_def split_def)
+ apply (subst Nadd_commute,simp add: Nadd_def split_def)
+ done
+
+lemma normNum_nilpotent_aux[simp]: assumes nx: "isnormNum x"
+ shows "normNum x = x"
+proof-
+ let ?a = "normNum x"
+ have n: "isnormNum ?a" by simp
+ have th:"INum ?a = (INum x ::'a :: {ring_char_0, division_by_zero,field})" by simp
+ with isnormNum_unique[OF n nx]
+ show ?thesis by simp
+qed
+
+lemma normNum_nilpotent[simp]: "normNum (normNum x) = normNum x"
+ by simp
+lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
+ by (simp_all add: normNum_def)
+lemma normNum_Nadd: "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
+lemma Nadd_normNum1[simp]: "normNum x +\<^sub>N y = x +\<^sub>N y"
+proof-
+ have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
+ have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a :: {ring_char_0, division_by_zero,field})" by simp
+ also have "\<dots> = INum (x +\<^sub>N y)" by simp
+ finally show ?thesis using isnormNum_unique[OF n] by simp
+qed
+lemma Nadd_normNum2[simp]: "x +\<^sub>N normNum y = x +\<^sub>N y"
+proof-
+ have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
+ have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a :: {ring_char_0, division_by_zero,field})" by simp
+ also have "\<dots> = INum (x +\<^sub>N y)" by simp
+ finally show ?thesis using isnormNum_unique[OF n] by simp
+qed
+
+lemma Nadd_assoc: "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
+proof-
+ have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
+ have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a :: {ring_char_0, division_by_zero,field})" by simp
+ with isnormNum_unique[OF n] show ?thesis by simp
+qed
+
+lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
+ by (simp add: Nmul_def split_def Let_def igcd_commute mult_commute)
+
+lemma Nmul_assoc: assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"
+ shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
+proof-
+ from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
+ by simp_all
+ have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a :: {ring_char_0, division_by_zero,field})" by simp
+ with isnormNum_unique[OF n] show ?thesis by simp
+qed
+
+lemma Nsub0: assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
+proof-
+ {fix h :: "'a :: {ring_char_0,division_by_zero,ordered_field}"
+ from isnormNum_unique[where ?'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
+ have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
+ also have "\<dots> = (INum x = (INum y:: 'a))" by simp
+ also have "\<dots> = (x = y)" using x y by simp
+ finally show ?thesis .}
+qed
+
+lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
+ by (simp_all add: Nmul_def Let_def split_def)
+
+lemma Nmul_eq0[simp]: assumes nx:"isnormNum x" and ny: "isnormNum y"
+ shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"
+proof-
+ {fix h :: "'a :: {ring_char_0,division_by_zero,ordered_field}"
+ have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto
+ then obtain a b a' b' where xy[simp]: "x = (a,b)" "y = (a',b')" by blast
+ have n0: "isnormNum 0\<^sub>N" by simp
+ show ?thesis using nx ny
+ apply (simp only: isnormNum_unique[where ?'a = 'a, OF Nmul_normN[OF nx ny] n0, symmetric] Nmul[where ?'a = 'a])
+ apply (simp add: INum_def split_def isnormNum_def fst_conv snd_conv)
+ apply (cases "a=0",simp_all)
+ apply (cases "a'=0",simp_all)
+ done }
+qed
+lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
+ by (simp add: Nneg_def split_def)
+
+lemma Nmul1[simp]:
+ "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c"
+ "isnormNum c \<Longrightarrow> c *\<^sub>N 1\<^sub>N = c"
+ apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
+ by (cases "fst c = 0", simp_all,cases c, simp_all)+
+
+end
\ No newline at end of file
--- a/src/HOL/Library/Executable_Rat.thy Thu Aug 09 15:52:47 2007 +0200
+++ b/src/HOL/Library/Executable_Rat.thy Thu Aug 09 15:52:49 2007 +0200
@@ -3,107 +3,95 @@
Author: Florian Haftmann, TU Muenchen
*)
-header {* Executable implementation of rational numbers in HOL *}
+header {* Implementation of rational numbers as pairs of integers *}
theory Executable_Rat
-imports "~~/src/HOL/Real/Rational" "~~/src/HOL/NumberTheory/IntPrimes"
+imports Abstract_Rat "~~/src/HOL/Real/Rational"
begin
-text {*
- Actually \emph{nothing} is proved about this implementation.
-*}
-
-subsection {* Representation and operations of executable rationals *}
-
-datatype erat = Rat bool nat nat
-
-axiomatization
- div_zero :: erat
-
-fun
- common :: "(nat * nat) \<Rightarrow> (nat * nat) \<Rightarrow> (nat * nat) * nat" where
- "common (p1, q1) (p2, q2) = (
- let
- q' = q1 * q2 div gcd (q1, q2)
- in ((p1 * (q' div q1), p2 * (q' div q2)), q'))"
-
-definition
- minus_sign :: "nat \<Rightarrow> nat \<Rightarrow> bool * nat" where
- "minus_sign n m = (if n < m then (False, m - n) else (True, n - m))"
-
-fun
- add_sign :: "bool * nat \<Rightarrow> bool * nat \<Rightarrow> bool * nat" where
- "add_sign (True, n) (True, m) = (True, n + m)"
-| "add_sign (False, n) (False, m) = (False, n + m)"
-| "add_sign (True, n) (False, m) = minus_sign n m"
-| "add_sign (False, n) (True, m) = minus_sign m n"
+hide (open) const Rat
definition
- erat_of_quotient :: "int \<Rightarrow> int \<Rightarrow> erat" where
- "erat_of_quotient k1 k2 = (
- let
- l1 = nat (abs k1);
- l2 = nat (abs k2);
- m = gcd (l1, l2)
- in Rat (k1 \<le> 0 \<longleftrightarrow> k2 \<le> 0) (l1 div m) (l2 div m))"
+ Rat :: "int \<times> int \<Rightarrow> rat"
+where
+ "Rat = INum"
+
+code_datatype Rat
-instance erat :: zero
- zero_rat_def: "0 \<equiv> Rat True 0 1" ..
-
-instance erat :: one
- one_rat_def: "1 \<equiv> Rat True 1 1" ..
+lemma Rat_simp:
+ "Rat (k, l) = rat_of_int k / rat_of_int l"
+ unfolding Rat_def INum_def by simp
-instance erat :: plus
- add_rat_def: "r + s \<equiv> case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
- let
- ((r1, r2), den) = common (p1, q1) (p2, q2);
- (sign, num) = add_sign (a1, r1) (a2, r2)
- in Rat sign num den" ..
+lemma Rat_zero [simp]: "Rat 0\<^sub>N = 0"
+ by (simp add: Rat_simp)
+
+lemma Rat_lit [simp]: "Rat i\<^sub>N = rat_of_int i"
+ by (simp add: Rat_simp)
+
+lemma zero_rat_code [code]:
+ "0 = Rat 0\<^sub>N" by simp
-instance erat :: minus
- uminus_rat_def: "- r \<equiv> case r of Rat a p q \<Rightarrow>
- if p = 0 then Rat True 0 1
- else Rat (\<not> a) p q" ..
-
-instance erat :: times
- times_rat_def: "r * s \<equiv> case r of Rat a1 p1 q1 \<Rightarrow> case s of Rat a2 p2 q2 \<Rightarrow>
- let
- p = p1 * p2;
- q = q1 * q2;
- m = gcd (p, q)
- in Rat (a1 = a2) (p div m) (q div m)" ..
+lemma zero_rat_code [code]:
+ "1 = Rat 1\<^sub>N" by simp
-instance erat :: inverse
- inverse_rat_def: "inverse r \<equiv> case r of Rat a p q \<Rightarrow>
- if p = 0 then div_zero
- else Rat a q p" ..
+lemma [code, code unfold]:
+ "number_of k = rat_of_int (number_of k)"
+ by (simp add: number_of_is_id rat_number_of_def)
+
+definition
+ [code func del]: "Fract' (b\<Colon>bool) k l = Fract k l"
-instance erat :: ord
- le_rat_def: "r1 \<le> r2 \<equiv> case r1 of Rat a1 p1 q1 \<Rightarrow> case r2 of Rat a2 p2 q2 \<Rightarrow>
- (\<not> a1 \<and> a2) \<or>
- (\<not> (a1 \<and> \<not> a2) \<and>
- (let
- ((r1, r2), dummy) = common (p1, q1) (p2, q2)
- in if a1 then r1 \<le> r2 else r2 \<le> r1))" ..
-
-
-subsection {* Code generator setup *}
+lemma [code]:
+ "Fract k l = Fract' (l \<noteq> 0) k l"
+ unfolding Fract'_def ..
-subsubsection {* code lemmas *}
-
-lemma number_of_rat [code unfold]:
- "(number_of k \<Colon> rat) = Fract k 1"
- unfolding Fract_of_int_eq rat_number_of_def by simp
+lemma [code]:
+ "Fract' True k l = (if l \<noteq> 0 then Rat (k, l) else Fract 1 0)"
+ by (simp add: Fract'_def Rat_simp Fract_of_int_quotient [of k l])
-lemma rat_minus [code func]:
- "(a\<Colon>rat) - b = a + (- b)" unfolding diff_minus ..
-
-lemma rat_divide [code func]:
- "(a\<Colon>rat) / b = a * inverse b" unfolding divide_inverse ..
+lemma [code]:
+ "of_rat (Rat (k, l)) = (if l \<noteq> 0 then of_int k / of_int l else 0)"
+ by (cases "l = 0")
+ (auto simp add: Rat_simp of_rat_rat [simplified Fract_of_int_quotient [of k l], symmetric])
instance rat :: eq ..
-subsubsection {* names *}
+lemma rat_eq_code [code]: "Rat x = Rat y \<longleftrightarrow> normNum x = normNum y"
+ unfolding Rat_def INum_normNum_iff ..
+
+lemma rat_less_eq_code [code]: "Rat x \<le> Rat y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y"
+proof -
+ have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> Rat (normNum x) \<le> Rat (normNum y)"
+ by (simp add: Rat_def del: normNum)
+ also have "\<dots> = (Rat x \<le> Rat y)" by (simp add: Rat_def)
+ finally show ?thesis by simp
+qed
+
+lemma rat_less_code [code]: "Rat x < Rat y \<longleftrightarrow> normNum x <\<^sub>N normNum y"
+proof -
+ have "normNum x <\<^sub>N normNum y \<longleftrightarrow> Rat (normNum x) < Rat (normNum y)"
+ by (simp add: Rat_def del: normNum)
+ also have "\<dots> = (Rat x < Rat y)" by (simp add: Rat_def)
+ finally show ?thesis by simp
+qed
+
+lemma rat_add_code [code]: "Rat x + Rat y = Rat (x +\<^sub>N y)"
+ unfolding Rat_def by simp
+
+lemma rat_mul_code [code]: "Rat x * Rat y = Rat (x *\<^sub>N y)"
+ unfolding Rat_def by simp
+
+lemma rat_neg_code [code]: "- Rat x = Rat (~\<^sub>N x)"
+ unfolding Rat_def by simp
+
+lemma rat_sub_code [code]: "Rat x - Rat y = Rat (x -\<^sub>N y)"
+ unfolding Rat_def by simp
+
+lemma rat_inv_code [code]: "inverse (Rat x) = Rat (Ninv x)"
+ unfolding Rat_def Ninv divide_rat_def by simp
+
+lemma rat_div_code [code]: "Rat x / Rat y = Rat (x \<div>\<^sub>N y)"
+ unfolding Rat_def by simp
code_modulename SML
Executable_Rat Rational
@@ -114,37 +102,4 @@
code_modulename Haskell
Executable_Rat Rational
-subsubsection {* rat as abstype *}
-
-code_const div_zero
- (SML "raise/ Fail/ \"Division by zero\"")
- (OCaml "failwith \"Division by zero\"")
- (Haskell "error/ \"Division by zero\"")
-
-code_abstype rat erat where
- Fract \<equiv> erat_of_quotient
- "0 \<Colon> rat" \<equiv> "0 \<Colon> erat"
- "1 \<Colon> rat" \<equiv> "1 \<Colon> erat"
- "op + \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat" \<equiv> "op + \<Colon> erat \<Rightarrow> erat \<Rightarrow> erat"
- "uminus \<Colon> rat \<Rightarrow> rat" \<equiv> "uminus \<Colon> erat \<Rightarrow> erat"
- "op * \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat" \<equiv> "op * \<Colon> erat \<Rightarrow> erat \<Rightarrow> erat"
- "inverse \<Colon> rat \<Rightarrow> rat" \<equiv> "inverse \<Colon> erat \<Rightarrow> erat"
- "op \<le> \<Colon> rat \<Rightarrow> rat \<Rightarrow> bool" \<equiv> "op \<le> \<Colon> erat \<Rightarrow> erat \<Rightarrow> bool"
- "op = \<Colon> rat \<Rightarrow> rat \<Rightarrow> bool" \<equiv> "op = \<Colon> erat \<Rightarrow> erat \<Rightarrow> bool"
-
-types_code
- rat ("{*erat*}")
-
-consts_code
- div_zero ("(raise/ (Fail/ \"Division by zero\"))")
- Fract ("({*erat_of_quotient*} (_) (_))")
- "0 \<Colon> rat" ("({*Rat True 0 1*})")
- "1 \<Colon> rat" ("({*Rat True 1 1*})")
- "plus \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat" ("({*op + \<Colon> erat \<Rightarrow> erat \<Rightarrow> erat*} (_) (_))")
- "uminus \<Colon> rat \<Rightarrow> rat" ("({*uminus \<Colon> erat \<Rightarrow> erat*} (_))")
- "op * \<Colon> rat \<Rightarrow> rat \<Rightarrow> rat" ("({*op * \<Colon> erat \<Rightarrow> erat \<Rightarrow> erat*} (_) (_))")
- "inverse \<Colon> rat \<Rightarrow> rat" ("({*inverse \<Colon> erat \<Rightarrow> erat*} (_))")
- "op \<le> \<Colon> rat \<Rightarrow> rat \<Rightarrow> bool" ("({*op \<le> \<Colon> erat \<Rightarrow> erat \<Rightarrow> bool*} (_) (_))")
- "op = \<Colon> rat \<Rightarrow> rat \<Rightarrow> bool" ("({*op = \<Colon> erat \<Rightarrow> erat \<Rightarrow> bool*} (_) (_))")
-
end
--- a/src/HOL/Library/Executable_Real.thy Thu Aug 09 15:52:47 2007 +0200
+++ b/src/HOL/Library/Executable_Real.thy Thu Aug 09 15:52:49 2007 +0200
@@ -1,472 +1,81 @@
(* Title: HOL/Library/Executable_Real.thy
ID: $Id$
- Author: Amine Chaieb, TU Muenchen
+ Author: Florian Haftmann, TU Muenchen
*)
header {* Implementation of rational real numbers as pairs of integers *}
theory Executable_Real
-imports GCD "~~/src/HOL/Real/Real"
+imports Abstract_Rat "~~/src/HOL/Real/Real"
begin
-subsection {* Implementation of operations on pair of integers *}
-
-types Num = "int * int"
-syntax "_Num0" :: "Num" ("0\<^sub>N")
-translations "0\<^sub>N" \<rightleftharpoons> "(0,0)"
-syntax "_Numi" :: "int \<Rightarrow> Num" ("_\<^sub>N")
-translations "i\<^sub>N" \<rightleftharpoons> "(i,1)::Num"
-
-constdefs isnormNum :: "Num \<Rightarrow> bool"
- "isnormNum \<equiv> \<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> igcd a b = 1)"
-
-constdefs normNum :: "Num \<Rightarrow> Num"
- "normNum \<equiv> \<lambda>(a,b). (if a=0 \<or> b = 0 then (0,0) else
- (let g = igcd a b
- in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g))))"
-
-lemma normNum_isnormNum[simp]: "isnormNum (normNum x)"
-proof-
- have " \<exists> a b. x = (a,b)" by auto
- then obtain a b where x[simp]: "x = (a,b)" by blast
- {assume "a=0 \<or> b = 0" hence ?thesis by (simp add: normNum_def isnormNum_def)}
- moreover
- {assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
- let ?g = "igcd a b"
- let ?a' = "a div ?g"
- let ?b' = "b div ?g"
- let ?g' = "igcd ?a' ?b'"
- from anz bnz have "?g \<noteq> 0" by simp with igcd_pos[of a b]
- have gpos: "?g > 0" by arith
- have gdvd: "?g dvd a" "?g dvd b" by (simp_all add: igcd_dvd1 igcd_dvd2)
- from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]
- anz bnz
- have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"
- by - (rule notI,simp add:igcd_def)+
- from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by blast
- from div_igcd_relprime[OF stupid] have gp1: "?g' = 1" .
- from bnz have "b < 0 \<or> b > 0" by arith
- moreover
- {assume b: "b > 0"
- from pos_imp_zdiv_nonneg_iff[OF gpos] b
- have "?b' \<ge> 0" by simp
- with nz' have b': "?b' > 0" by simp
- from b b' anz bnz nz' gp1 have ?thesis
- by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
- moreover {assume b: "b < 0"
- {assume b': "?b' \<ge> 0"
- from gpos have th: "?g \<ge> 0" by arith
- from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]
- have False using b by simp }
- hence b': "?b' < 0" by (auto simp add: linorder_not_le[symmetric])
- from anz bnz nz' b b' gp1 have ?thesis
- by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
- ultimately have ?thesis by blast
- }
- ultimately show ?thesis by blast
-qed
- (* Arithmetic over Num *)
-constdefs Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60)
- "Nadd \<equiv> \<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b')
- else if a'=0 \<or> b' = 0 then normNum(a,b)
- else normNum(a*b' + b*a', b*b')"
-constdefs Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)
- "Nmul \<equiv> \<lambda>(a,b) (a',b'). let g = igcd (a*a') (b*b')
- in (a*a' div g, b*b' div g)"
-constdefs Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
- "Nneg \<equiv> \<lambda>(a,b). (-a,b)"
-constdefs Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)
- "Nsub \<equiv> \<lambda>a b. a +\<^sub>N ~\<^sub>N b"
-constdefs Ninv :: "Num \<Rightarrow> Num"
-"Ninv \<equiv> \<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a)"
-constdefs Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)
- "Ndiv \<equiv> \<lambda>a b. a *\<^sub>N Ninv b"
-
-lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)"
- by(simp add: isnormNum_def Nneg_def split_def)
-lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)"
- by (simp add: Nadd_def split_def)
-lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)"
- by (simp add: Nsub_def split_def)
-lemma Nmul_normN[simp]: assumes xn:"isnormNum x" and yn: "isnormNum y"
- shows "isnormNum (x *\<^sub>N y)"
-proof-
- have "\<exists>a b. x = (a,b)" and "\<exists> a' b'. y = (a',b')" by auto
- then obtain a b a' b' where ab: "x = (a,b)" and ab': "y = (a',b')" by blast
- {assume "a = 0"
- hence ?thesis using xn ab ab'
- by (simp add: igcd_def isnormNum_def Let_def Nmul_def split_def)}
- moreover
- {assume "a' = 0"
- hence ?thesis using yn ab ab'
- by (simp add: igcd_def isnormNum_def Let_def Nmul_def split_def)}
- moreover
- {assume a: "a \<noteq>0" and a': "a'\<noteq>0"
- hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)
- from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a*a', b*b')"
- using ab ab' a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
- hence ?thesis by simp}
- ultimately show ?thesis by blast
-qed
-
-lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
-by (simp add: Ninv_def isnormNum_def split_def)
-(cases "fst x = 0",auto simp add: igcd_commute)
-
-lemma isnormNum_int[simp]:
- "isnormNum 0\<^sub>N" "isnormNum (1::int)\<^sub>N" "i \<noteq> 0 \<Longrightarrow> isnormNum i\<^sub>N"
- by (simp_all add: isnormNum_def igcd_def)
-
- (* Relations over Num *)
-constdefs Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
- "Nlt0 \<equiv> \<lambda>(a,b). a < 0"
-constdefs Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
- "Nle0 \<equiv> \<lambda>(a,b). a \<le> 0"
-constdefs Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
- "Ngt0 \<equiv> \<lambda>(a,b). a > 0"
-constdefs Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
- "Nge0 \<equiv> \<lambda>(a,b). a \<ge> 0"
-constdefs Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)
- "Nlt \<equiv> \<lambda>a b. 0>\<^sub>N (a -\<^sub>N b)"
-constdefs Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)
- "Nle \<equiv> \<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b)"
-
-
-subsection {* Interpretation of the normalized rats in reals *}
+hide (open) const Real
definition
- INum:: "Num \<Rightarrow> real"
+ Real :: "int \<times> int \<Rightarrow> real"
where
- INum_def: "INum \<equiv> \<lambda>(a,b). real a / real b"
+ "Real = INum"
+
+code_datatype Real
+
+lemma Real_simp:
+ "Real (k, l) = real_of_int k / real_of_int l"
+ unfolding Real_def INum_def by simp
+
+lemma Real_zero [simp]: "Real 0\<^sub>N = 0"
+ by (simp add: Real_simp)
-code_datatype INum
+lemma Real_lit [simp]: "Real i\<^sub>N = real_of_int i"
+ by (simp add: Real_simp)
+
+lemma zero_real_code [code]:
+ "0 = Real 0\<^sub>N" by simp
+
+lemma zero_real_code [code]:
+ "1 = Real 1\<^sub>N" by simp
+
+lemma [code, code unfold]:
+ "number_of k = real_of_int (number_of k)"
+ by (simp add: number_of_is_id real_number_of_def)
+
instance real :: eq ..
-definition
- real_int :: "int \<Rightarrow> real"
-where
- "real_int = real"
-lemmas [code unfold] = real_int_def [symmetric]
-
-lemma [code unfold]:
- "real = real_int o int"
- by (auto simp add: real_int_def expand_fun_eq)
-
-lemma INum_int [simp]: "INum i\<^sub>N = real i" "INum 0\<^sub>N = 0"
- by (simp_all add: INum_def)
-lemmas [code, code unfold] = INum_int [unfolded real_int_def [symmetric], symmetric]
-
-lemma [code, code unfold]: "1 = INum 1\<^sub>N" by simp
+lemma real_eq_code [code]: "Real x = Real y \<longleftrightarrow> normNum x = normNum y"
+ unfolding Real_def INum_normNum_iff ..
-lemma isnormNum_unique[simp]:
- assumes na: "isnormNum x" and nb: "isnormNum y"
- shows "(INum x = INum y) = (x = y)" (is "?lhs = ?rhs")
-proof
- have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto
- then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast
- assume H: ?lhs
- {assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" hence ?rhs
- using na nb H
- by (simp add: INum_def split_def isnormNum_def)
- (cases "a = 0", simp_all,cases "b = 0", simp_all,
- cases "a' = 0", simp_all,cases "a' = 0", simp_all)}
- moreover
- { assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0"
- from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)
- from prems have eq:"a * b' = a'*b"
- by (simp add: INum_def eq_divide_eq divide_eq_eq real_of_int_mult[symmetric] del: real_of_int_mult)
- from prems have gcd1: "igcd a b = 1" "igcd b a = 1" "igcd a' b' = 1" "igcd b' a' = 1"
- by (simp_all add: isnormNum_def add: igcd_commute)
- from eq have raw_dvd: "a dvd a'*b" "b dvd b'*a" "a' dvd a*b'" "b' dvd b*a'"
- apply(unfold dvd_def)
- apply (rule_tac x="b'" in exI, simp add: mult_ac)
- apply (rule_tac x="a'" in exI, simp add: mult_ac)
- apply (rule_tac x="b" in exI, simp add: mult_ac)
- apply (rule_tac x="a" in exI, simp add: mult_ac)
- done
- from zdvd_dvd_eq[OF bz zrelprime_dvd_mult[OF gcd1(2) raw_dvd(2)]
- zrelprime_dvd_mult[OF gcd1(4) raw_dvd(4)]]
- have eq1: "b = b'" using pos by simp_all
- with eq have "a = a'" using pos by simp
- with eq1 have ?rhs by simp}
- ultimately show ?rhs by blast
-next
- assume ?rhs thus ?lhs by simp
-qed
-
-
-lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = 0) = (x = 0\<^sub>N)"
- unfolding INum_int(2)[symmetric]
- by (rule isnormNum_unique, simp_all)
-
-lemma normNum[simp]: "INum (normNum x) = INum x"
-proof-
- have "\<exists> a b. x = (a,b)" by auto
- then obtain a b where x[simp]: "x = (a,b)" by blast
- {assume "a=0 \<or> b = 0" hence ?thesis
- by (simp add: INum_def normNum_def split_def Let_def)}
- moreover
- {assume a: "a\<noteq>0" and b: "b\<noteq>0"
- let ?g = "igcd a b"
- from a b have g: "?g \<noteq> 0"by simp
- from real_of_int_div[OF g]
- have ?thesis by (simp add: INum_def normNum_def split_def Let_def)}
- ultimately show ?thesis by blast
-qed
-
-lemma INum_normNum_iff [code]: "INum x = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
+lemma real_less_eq_code [code]: "Real x \<le> Real y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y"
proof -
- have "normNum x = normNum y \<longleftrightarrow> INum (normNum x) = INum (normNum y)"
- by (simp del: normNum)
- also have "\<dots> = ?lhs" by simp
+ have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> Real (normNum x) \<le> Real (normNum y)"
+ by (simp add: Real_def del: normNum)
+ also have "\<dots> = (Real x \<le> Real y)" by (simp add: Real_def)
finally show ?thesis by simp
qed
-lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + INum y"
-proof-
- have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
- then obtain a b a' b' where x[simp]: "x = (a,b)"
- and y[simp]: "y = (a',b')" by blast
- {assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" hence ?thesis
- apply (cases "a=0",simp_all add: Nadd_def)
- apply (cases "b= 0",simp_all add: INum_def)
- apply (cases "a'= 0",simp_all)
- apply (cases "b'= 0",simp_all)
- done }
- moreover
- {assume aa':"a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
- {assume z: "a * b' + b * a' = 0"
- hence "real (a*b' + b*a') / (real b* real b') = 0" by simp
- hence "real b' * real a / (real b * real b') + real b * real a' / (real b * real b') = 0" by (simp add:add_divide_distrib)
- hence th: "real a / real b + real a' / real b' = 0" using bb' aa' by simp
- from z aa' bb' have ?thesis
- by (simp add: th Nadd_def normNum_def INum_def split_def)}
- moreover {assume z: "a * b' + b * a' \<noteq> 0"
- let ?g = "igcd (a * b' + b * a') (b*b')"
- have gz: "?g \<noteq> 0" using z by simp
- have ?thesis using aa' bb' z gz
- real_of_int_div[OF gz igcd_dvd1[where i="a * b' + b * a'" and j="b*b'"]]
- real_of_int_div[OF gz igcd_dvd2[where i="a * b' + b * a'" and j="b*b'"]]
- by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
- ultimately have ?thesis using aa' bb'
- by (simp add: Nadd_def INum_def normNum_def x y Let_def) }
- ultimately show ?thesis by blast
-qed
-lemmas [code] = Nadd [symmetric]
-
-lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * INum y"
-proof-
- have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
- then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast
- {assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis
- apply (cases "a=0",simp_all add: x y Nmul_def INum_def Let_def)
- apply (cases "b=0",simp_all)
- apply (cases "a'=0",simp_all)
- done }
- moreover
- {assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
- let ?g="igcd (a*a') (b*b')"
- have gz: "?g \<noteq> 0" using z by simp
- from z real_of_int_div[OF gz igcd_dvd1[where i="a*a'" and j="b*b'"]]
- real_of_int_div[OF gz igcd_dvd2[where i="a*a'" and j="b*b'"]]
- have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}
- ultimately show ?thesis by blast
-qed
-lemmas [code] = Nmul [symmetric]
-
-lemma Nneg[simp]: "INum (~\<^sub>N x) = - INum x"
- by (simp add: Nneg_def split_def INum_def)
-lemmas [code] = Nneg [symmetric]
-
-lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - INum y"
- by (simp add: Nsub_def split_def)
-lemmas [code] = Nsub [symmetric]
-
-lemma Ninv[simp]: "INum (Ninv x) = 1 / (INum x)"
- by (simp add: Ninv_def INum_def split_def)
-lemmas [code] = Ninv [symmetric]
-
-lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / INum y" by (simp add: Ndiv_def)
-lemmas [code] = Ndiv [symmetric]
-
-lemma Nlt0_iff[simp]: assumes nx: "isnormNum x" shows "(INum x < 0) = 0>\<^sub>N x "
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }
- moreover
- {assume a: "a\<noteq>0" hence b: "real b > 0" using nx by (simp add: isnormNum_def)
- from pos_divide_less_eq[OF b, where b="real a" and a="0"]
- have ?thesis by (simp add: Nlt0_def INum_def)}
- ultimately show ?thesis by blast
-qed
-
-lemma Nle0_iff[simp]:assumes nx: "isnormNum x" shows "(INum x \<le> 0) = 0\<ge>\<^sub>N x"
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }
- moreover
- {assume a: "a\<noteq>0" hence b: "real b > 0" using nx by (simp add: isnormNum_def)
- from pos_divide_le_eq[OF b, where b="real a" and a="0"]
- have ?thesis by (simp add: Nle0_def INum_def)}
- ultimately show ?thesis by blast
-qed
-
-lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "(INum x > 0) = 0<\<^sub>N x"
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }
- moreover
- {assume a: "a\<noteq>0" hence b: "real b > 0" using nx by (simp add: isnormNum_def)
- from pos_less_divide_eq[OF b, where b="real a" and a="0"]
- have ?thesis by (simp add: Ngt0_def INum_def)}
- ultimately show ?thesis by blast
-qed
-lemma Nge0_iff[simp]:assumes nx: "isnormNum x" shows "(INum x \<ge> 0) = 0\<le>\<^sub>N x"
-proof-
- have " \<exists> a b. x = (a,b)" by simp
- then obtain a b where x[simp]:"x = (a,b)" by blast
- {assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }
- moreover
- {assume a: "a\<noteq>0" hence b: "real b > 0" using nx by (simp add: isnormNum_def)
- from pos_le_divide_eq[OF b, where b="real a" and a="0"]
- have ?thesis by (simp add: Nge0_def INum_def)}
- ultimately show ?thesis by blast
-qed
-
-lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
- shows "(INum x < INum y) = (x <\<^sub>N y)"
-proof-
- have "(INum x < INum y) = (INum (x -\<^sub>N y) < 0)" using nx ny by simp
- also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
- finally show ?thesis by (simp add: Nlt_def)
-qed
-
-lemma [code]: "INum x < INum y \<longleftrightarrow> normNum x <\<^sub>N normNum y"
+lemma real_less_code [code]: "Real x < Real y \<longleftrightarrow> normNum x <\<^sub>N normNum y"
proof -
- have "normNum x <\<^sub>N normNum y \<longleftrightarrow> INum (normNum x) < INum (normNum y)"
- by (simp del: normNum)
- also have "\<dots> = (INum x < INum y)" by simp
+ have "normNum x <\<^sub>N normNum y \<longleftrightarrow> Real (normNum x) < Real (normNum y)"
+ by (simp add: Real_def del: normNum)
+ also have "\<dots> = (Real x < Real y)" by (simp add: Real_def)
finally show ?thesis by simp
qed
-lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
- shows "(INum x \<le> INum y) = (x \<le>\<^sub>N y)"
-proof-
- have "(INum x \<le> INum y) = (INum (x -\<^sub>N y) \<le> 0)" using nx ny by simp
- also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp
- finally show ?thesis by (simp add: Nle_def)
-qed
-
-lemma [code]: "INum x \<le> INum y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y"
-proof -
- have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> INum (normNum x) \<le> INum (normNum y)"
- by (simp del: normNum)
- also have "\<dots> = (INum x \<le> INum y)" by simp
- finally show ?thesis by simp
-qed
+lemma real_add_code [code]: "Real x + Real y = Real (x +\<^sub>N y)"
+ unfolding Real_def by simp
-lemma Nadd_commute: "x +\<^sub>N y = y +\<^sub>N x"
-proof-
- have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
- have "INum (x +\<^sub>N y) = INum (y +\<^sub>N x)" by simp
- with isnormNum_unique[OF n] show ?thesis by simp
-qed
+lemma real_mul_code [code]: "Real x * Real y = Real (x *\<^sub>N y)"
+ unfolding Real_def by simp
-lemma[simp]: "(0, b) +\<^sub>N y = normNum y" "(a, 0) +\<^sub>N y = normNum y"
- "x +\<^sub>N (0, b) = normNum x" "x +\<^sub>N (a, 0) = normNum x"
- apply (simp add: Nadd_def split_def, simp add: Nadd_def split_def)
- apply (subst Nadd_commute,simp add: Nadd_def split_def)
- apply (subst Nadd_commute,simp add: Nadd_def split_def)
- done
-
-lemma normNum_nilpotent_aux[simp]: assumes nx: "isnormNum x"
- shows "normNum x = x"
-proof-
- let ?a = "normNum x"
- have n: "isnormNum ?a" by simp
- have th:"INum ?a = INum x" by simp
- with isnormNum_unique[OF n nx]
- show ?thesis by simp
-qed
+lemma real_neg_code [code]: "- Real x = Real (~\<^sub>N x)"
+ unfolding Real_def by simp
-lemma normNum_nilpotent[simp]: "normNum (normNum x) = normNum x"
- by simp
-lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
- by (simp_all add: normNum_def)
-lemma normNum_Nadd: "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
-lemma Nadd_normNum1[simp]: "normNum x +\<^sub>N y = x +\<^sub>N y"
-proof-
- have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
- have "INum (normNum x +\<^sub>N y) = INum x + INum y" by simp
- also have "\<dots> = INum (x +\<^sub>N y)" by simp
- finally show ?thesis using isnormNum_unique[OF n] by simp
-qed
-lemma Nadd_normNum2[simp]: "x +\<^sub>N normNum y = x +\<^sub>N y"
-proof-
- have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
- have "INum (x +\<^sub>N normNum y) = INum x + INum y" by simp
- also have "\<dots> = INum (x +\<^sub>N y)" by simp
- finally show ?thesis using isnormNum_unique[OF n] by simp
-qed
-
-lemma Nadd_assoc: "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
-proof-
- have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
- have "INum (x +\<^sub>N y +\<^sub>N z) = INum (x +\<^sub>N (y +\<^sub>N z))" by simp
- with isnormNum_unique[OF n] show ?thesis by simp
-qed
-
-lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
- by (simp add: Nmul_def split_def Let_def igcd_commute mult_commute)
+lemma real_sub_code [code]: "Real x - Real y = Real (x -\<^sub>N y)"
+ unfolding Real_def by simp
-lemma Nmul_assoc: assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"
- shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
-proof-
- from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
- by simp_all
- have "INum (x +\<^sub>N y +\<^sub>N z) = INum (x +\<^sub>N (y +\<^sub>N z))" by simp
- with isnormNum_unique[OF n] show ?thesis by simp
-qed
-
-lemma Nsub0: assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
-proof-
- from isnormNum_unique[OF Nsub_normN[OF y], where y="0\<^sub>N"]
- have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = INum 0\<^sub>N)" by simp
- also have "\<dots> = (INum x = INum y)" by simp
- also have "\<dots> = (x = y)" using x y by simp
- finally show ?thesis .
-qed
-lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
- by (simp_all add: Nmul_def Let_def split_def)
+lemma real_inv_code [code]: "inverse (Real x) = Real (Ninv x)"
+ unfolding Real_def Ninv real_divide_def by simp
-lemma Nmul_eq0[simp]: assumes nx:"isnormNum x" and ny: "isnormNum y"
- shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"
-proof-
- have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto
- then obtain a b a' b' where xy[simp]: "x = (a,b)" "y = (a',b')" by blast
- have n0: "isnormNum 0\<^sub>N" by simp
- show ?thesis using nx ny
- apply (simp only: isnormNum_unique[OF Nmul_normN[OF nx ny] n0, symmetric] Nmul)
- apply (simp add: INum_def split_def isnormNum_def fst_conv snd_conv)
- apply (cases "a=0",simp_all)
- apply (cases "a'=0",simp_all)
- done
-qed
-lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
- by (simp add: Nneg_def split_def)
-
-lemma Nmul1[simp]:
- "isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c"
- "isnormNum c \<Longrightarrow> c *\<^sub>N 1\<^sub>N = c"
- apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
- by (cases "fst c = 0", simp_all,cases c, simp_all)+
-
-lemma [code, code unfold]:
- "number_of k = real_int (number_of k)"
- by (simp add: real_int_def)
+lemma real_div_code [code]: "Real x / Real y = Real (x \<div>\<^sub>N y)"
+ unfolding Real_def by simp
code_modulename SML
RealDef Real
@@ -480,19 +89,4 @@
RealDef Real
Executable_Real Real
-(* There is already an implementation in RealDef
-types_code real ("{* int * int *}")
-attach (term_of) {*
-fun term_of_real (p, q) =
- let
- val rT = HOLogic.realT;
-in if q = 1
- then HOLogic.mk_number rT p
- else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $
- HOLogic.mk_number rT p $ HOLogic.mk_number rT q
-end;
-*}
-
-consts_code INum ("")
-*)
end
--- a/src/HOL/Library/Library.thy Thu Aug 09 15:52:47 2007 +0200
+++ b/src/HOL/Library/Library.thy Thu Aug 09 15:52:49 2007 +0200
@@ -2,6 +2,7 @@
(*<*)
theory Library
imports
+ Abstract_Rat
AssocList
BigO
Binomial
--- a/src/HOL/ex/ExecutableContent.thy Thu Aug 09 15:52:47 2007 +0200
+++ b/src/HOL/ex/ExecutableContent.thy Thu Aug 09 15:52:49 2007 +0200
@@ -1,4 +1,3 @@
-
(* ID: $Id$
Author: Florian Haftmann, TU Muenchen
*)
@@ -14,6 +13,7 @@
Binomial
Commutative_Ring
"~~/src/HOL/ex/Commutative_Ring_Complete"
+ Executable_Rat
Executable_Real
GCD
List_Prefix
@@ -79,4 +79,44 @@
definition
"shadow keywords = keywords @ [ExecutableContent.keywords 0 0 0 0 0 0]"
+definition
+ foo :: "rat \<Rightarrow> rat \<Rightarrow> rat \<Rightarrow> rat" where
+ "foo r s t = (t + s) / t"
+
+definition
+ bar :: "rat \<Rightarrow> rat \<Rightarrow> rat \<Rightarrow> bool" where
+ "bar r s t \<longleftrightarrow> (r - s) \<le> t \<or> (s - t) \<le> r"
+
+definition
+ "R1 = Fract 3 7"
+
+definition
+ "R2 = Fract (-7) 5"
+
+definition
+ "R3 = Fract 11 (-9)"
+
+definition
+ "foobar = (foo R1 1 R3, bar R2 0 R3, foo R1 R3 R2)"
+
+definition
+ foo' :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real" where
+ "foo' r s t = (t + s) / t"
+
+definition
+ bar' :: "real \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool" where
+ "bar' r s t \<longleftrightarrow> (r - s) \<le> t \<or> (s - t) \<le> r"
+
+definition
+ "R1' = real_of_rat (Fract 3 7)"
+
+definition
+ "R2' = real_of_rat (Fract (-7) 5)"
+
+definition
+ "R3' = real_of_rat (Fract 11 (-9))"
+
+definition
+ "foobar' = (foo' R1' 1 R3', bar' R2' 0 R3', foo' R1' R3' R2')"
+
end