--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Rat_Pair.thy Thu Oct 31 11:44:20 2013 +0100
@@ -0,0 +1,521 @@
+(* Title: HOL/Decision_Procs/Rat_Pair.thy
+ Author: Amine Chaieb
+*)
+
+header {* Rational numbers as pairs *}
+
+theory Rat_Pair
+imports Complex_Main
+begin
+
+type_synonym Num = "int \<times> int"
+
+abbreviation Num0_syn :: Num ("0\<^sub>N")
+ where "0\<^sub>N \<equiv> (0, 0)"
+
+abbreviation Numi_syn :: "int \<Rightarrow> Num" ("'((_)')\<^sub>N")
+ where "(i)\<^sub>N \<equiv> (i, 1)"
+
+definition isnormNum :: "Num \<Rightarrow> bool" where
+ "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd 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 = gcd a b
+ in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
+
+declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger]
+
+lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) }
+ moreover
+ { assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
+ let ?g = "gcd a b"
+ let ?a' = "a div ?g"
+ let ?b' = "b div ?g"
+ let ?g' = "gcd ?a' ?b'"
+ from anz bnz have "?g \<noteq> 0" by simp with gcd_ge_0_int[of a b]
+ have gpos: "?g > 0" by arith
+ have gdvd: "?g dvd a" "?g dvd b" by arith+
+ from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] anz bnz
+ have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
+ from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
+ from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .
+ from bnz have "b < 0 \<or> b > 0" by arith
+ moreover
+ { assume b: "b > 0"
+ from b have "?b' \<ge> 0"
+ by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
+ with nz' have b': "?b' > 0" by arith
+ from b b' anz bnz nz' gp1 have ?thesis
+ by (simp add: x isnormNum_def normNum_def Let_def split_def) }
+ 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'] dvd_mult_div_cancel[OF gdvd(2)]
+ have False using b by arith }
+ hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
+ from anz bnz nz' b b' gp1 have ?thesis
+ by (simp add: x isnormNum_def normNum_def Let_def split_def) }
+ 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 = gcd (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 = (\<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 = (\<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 -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ { assume "a = 0"
+ hence ?thesis using xn x y
+ by (simp add: isnormNum_def Let_def Nmul_def split_def) }
+ moreover
+ { assume "a' = 0"
+ hence ?thesis using yn x y
+ by (simp add: 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 x y by (simp_all add: isnormNum_def)
+ from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')"
+ using x y 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: gcd_commute_int)
+
+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)
+
+
+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::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
+proof
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ assume H: ?lhs
+ { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
+ hence ?rhs using na nb H
+ by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) }
+ 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: x y isnormNum_def)
+ from H bz b'z have eq: "a * b' = a'*b"
+ by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
+ from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
+ by (simp_all add: x y isnormNum_def add: gcd_commute_int)
+ from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
+ apply -
+ apply algebra
+ apply algebra
+ apply simp
+ apply algebra
+ done
+ from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]
+ coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]
+ have eq1: "b = b'" using pos by arith
+ with eq have "a = a'" using pos by simp
+ with eq1 have ?rhs by (simp add: x y) }
+ ultimately show ?rhs by blast
+next
+ assume ?rhs thus ?lhs by simp
+qed
+
+
+lemma isnormNum0[simp]:
+ "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_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_char_0) / (of_int d) =
+ of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
+proof -
+ assume "d ~= 0"
+ 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 algebra_simps `d ~= 0`)
+qed
+
+lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
+ (of_int(n div d)::'a::field_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: dvd_eq_mod_eq_0)
+ done
+
+
+lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0 \<or> b = 0"
+ hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) }
+ moreover
+ { assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
+ let ?g = "gcd 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: x INum_def normNum_def split_def Let_def) }
+ ultimately show ?thesis by blast
+qed
+
+lemma INum_normNum_iff:
+ "(INum x ::'a::{field_char_0, field_inverse_zero}) = 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 :: {field_char_0, field_inverse_zero})"
+proof -
+ let ?z = "0:: 'a"
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"
+ hence ?thesis
+ apply (cases "a=0", simp_all add: x y 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: x y th Nadd_def normNum_def INum_def split_def) }
+ moreover {
+ assume z: "a * b' + b * a' \<noteq> 0"
+ let ?g = "gcd (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 gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]]
+ of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="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: x y Nadd_def INum_def normNum_def Let_def) }
+ ultimately show ?thesis by blast
+qed
+
+lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})"
+proof -
+ let ?z = "0::'a"
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ { 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="gcd (a*a') (b*b')"
+ have gz: "?g \<noteq> 0" using z by simp
+ from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
+ of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="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]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
+ by (simp add: Nsub_def split_def)
+
+lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (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 :: {field_char_0, field_inverse_zero})"
+ by (simp add: Ndiv_def)
+
+lemma Nlt0_iff[simp]:
+ assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) }
+ moreover
+ { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0"
+ using nx by (simp add: x isnormNum_def)
+ from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: x Nlt0_def INum_def) }
+ ultimately show ?thesis by blast
+qed
+
+lemma Nle0_iff[simp]:
+ assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) }
+ moreover
+ { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0"
+ using nx by (simp add: x isnormNum_def)
+ from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: x Nle0_def INum_def) }
+ ultimately show ?thesis by blast
+qed
+
+lemma Ngt0_iff[simp]:
+ assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
+ moreover
+ { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
+ by (simp add: x isnormNum_def)
+ from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: x Ngt0_def INum_def) }
+ ultimately show ?thesis by blast
+qed
+
+lemma Nge0_iff[simp]:
+ assumes nx: "isnormNum x"
+ shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
+proof -
+ obtain a b where x: "x = (a, b)" by (cases x)
+ { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
+ moreover
+ { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
+ by (simp add: x isnormNum_def)
+ from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
+ have ?thesis by (simp add: x 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 :: {field_char_0, linordered_field_inverse_zero}) < 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 :: {field_char_0, linordered_field_inverse_zero})\<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:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ shows "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) = INum (y +\<^sub>N x)" by simp
+ with isnormNum_unique[OF n] show ?thesis by simp
+qed
+
+lemma [simp]:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ shows "(0, b) +\<^sub>N y = normNum y"
+ and "(a, 0) +\<^sub>N y = normNum y"
+ and "x +\<^sub>N (0, b) = normNum x"
+ and "x +\<^sub>N (a, 0) = normNum x"
+ apply (simp add: Nadd_def split_def)
+ apply (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 "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ 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)" by simp
+ with isnormNum_unique[OF n nx] show ?thesis by simp
+qed
+
+lemma normNum_nilpotent[simp]:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ shows "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:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
+
+lemma Nadd_normNum1[simp]:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ shows "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)" 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]:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ shows "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)" 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:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ shows "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)" 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 gcd_commute_int mult_commute)
+
+lemma Nmul_assoc:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ 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)" by simp
+ with isnormNum_unique[OF n] show ?thesis by simp
+qed
+
+lemma Nsub0:
+ assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ assumes x: "isnormNum x" and y: "isnormNum y"
+ shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
+proof -
+ fix h :: 'a
+ 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 "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+ assumes nx: "isnormNum x" and ny: "isnormNum y"
+ shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
+proof -
+ fix h :: 'a
+ obtain a b where x: "x = (a, b)" by (cases x)
+ obtain a' b' where y: "y = (a', b')" by (cases y)
+ 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: x y INum_def split_def isnormNum_def split: split_if_asm)
+ 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)
+ apply (cases "fst c = 0", simp_all, cases c, simp_all)+
+ done
+
+end
--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Thu Oct 31 11:44:20 2013 +0100
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy Thu Oct 31 11:44:20 2013 +0100
@@ -5,7 +5,7 @@
header {* Implementation and verification of multivariate polynomials *}
theory Reflected_Multivariate_Polynomial
-imports Complex_Main "~~/src/HOL/Library/Abstract_Rat" Polynomial_List
+imports Complex_Main Rat_Pair Polynomial_List
begin
subsection{* Datatype of polynomial expressions *}
--- a/src/HOL/Library/Abstract_Rat.thy Thu Oct 31 11:44:20 2013 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,521 +0,0 @@
-(* Title: HOL/Library/Abstract_Rat.thy
- Author: Amine Chaieb
-*)
-
-header {* Abstract rational numbers *}
-
-theory Abstract_Rat
-imports Complex_Main
-begin
-
-type_synonym Num = "int \<times> int"
-
-abbreviation Num0_syn :: Num ("0\<^sub>N")
- where "0\<^sub>N \<equiv> (0, 0)"
-
-abbreviation Numi_syn :: "int \<Rightarrow> Num" ("'((_)')\<^sub>N")
- where "(i)\<^sub>N \<equiv> (i, 1)"
-
-definition isnormNum :: "Num \<Rightarrow> bool" where
- "isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd 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 = gcd a b
- in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"
-
-declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger]
-
-lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
-proof -
- obtain a b where x: "x = (a, b)" by (cases x)
- { assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) }
- moreover
- { assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0"
- let ?g = "gcd a b"
- let ?a' = "a div ?g"
- let ?b' = "b div ?g"
- let ?g' = "gcd ?a' ?b'"
- from anz bnz have "?g \<noteq> 0" by simp with gcd_ge_0_int[of a b]
- have gpos: "?g > 0" by arith
- have gdvd: "?g dvd a" "?g dvd b" by arith+
- from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] anz bnz
- have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+
- from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
- from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .
- from bnz have "b < 0 \<or> b > 0" by arith
- moreover
- { assume b: "b > 0"
- from b have "?b' \<ge> 0"
- by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
- with nz' have b': "?b' > 0" by arith
- from b b' anz bnz nz' gp1 have ?thesis
- by (simp add: x isnormNum_def normNum_def Let_def split_def) }
- 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'] dvd_mult_div_cancel[OF gdvd(2)]
- have False using b by arith }
- hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
- from anz bnz nz' b b' gp1 have ?thesis
- by (simp add: x isnormNum_def normNum_def Let_def split_def) }
- 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 = gcd (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 = (\<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 = (\<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 -
- obtain a b where x: "x = (a, b)" by (cases x)
- obtain a' b' where y: "y = (a', b')" by (cases y)
- { assume "a = 0"
- hence ?thesis using xn x y
- by (simp add: isnormNum_def Let_def Nmul_def split_def) }
- moreover
- { assume "a' = 0"
- hence ?thesis using yn x y
- by (simp add: 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 x y by (simp_all add: isnormNum_def)
- from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')"
- using x y 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: gcd_commute_int)
-
-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)
-
-
-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::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")
-proof
- obtain a b where x: "x = (a, b)" by (cases x)
- obtain a' b' where y: "y = (a', b')" by (cases y)
- assume H: ?lhs
- { assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
- hence ?rhs using na nb H
- by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) }
- 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: x y isnormNum_def)
- from H bz b'z have eq: "a * b' = a'*b"
- by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
- from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"
- by (simp_all add: x y isnormNum_def add: gcd_commute_int)
- from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"
- apply -
- apply algebra
- apply algebra
- apply simp
- apply algebra
- done
- from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]
- coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]
- have eq1: "b = b'" using pos by arith
- with eq have "a = a'" using pos by simp
- with eq1 have ?rhs by (simp add: x y) }
- ultimately show ?rhs by blast
-next
- assume ?rhs thus ?lhs by simp
-qed
-
-
-lemma isnormNum0[simp]:
- "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_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_char_0) / (of_int d) =
- of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
-proof -
- assume "d ~= 0"
- 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 algebra_simps `d ~= 0`)
-qed
-
-lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
- (of_int(n div d)::'a::field_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: dvd_eq_mod_eq_0)
- done
-
-
-lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
-proof -
- obtain a b where x: "x = (a, b)" by (cases x)
- { assume "a = 0 \<or> b = 0"
- hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) }
- moreover
- { assume a: "a \<noteq> 0" and b: "b \<noteq> 0"
- let ?g = "gcd 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: x INum_def normNum_def split_def Let_def) }
- ultimately show ?thesis by blast
-qed
-
-lemma INum_normNum_iff:
- "(INum x ::'a::{field_char_0, field_inverse_zero}) = 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 :: {field_char_0, field_inverse_zero})"
-proof -
- let ?z = "0:: 'a"
- obtain a b where x: "x = (a, b)" by (cases x)
- obtain a' b' where y: "y = (a', b')" by (cases y)
- { assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0"
- hence ?thesis
- apply (cases "a=0", simp_all add: x y 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: x y th Nadd_def normNum_def INum_def split_def) }
- moreover {
- assume z: "a * b' + b * a' \<noteq> 0"
- let ?g = "gcd (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 gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]]
- of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="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: x y Nadd_def INum_def normNum_def Let_def) }
- ultimately show ?thesis by blast
-qed
-
-lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})"
-proof -
- let ?z = "0::'a"
- obtain a b where x: "x = (a, b)" by (cases x)
- obtain a' b' where y: "y = (a', b')" by (cases y)
- { 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="gcd (a*a') (b*b')"
- have gz: "?g \<noteq> 0" using z by simp
- from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
- of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="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]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
- by (simp add: Nsub_def split_def)
-
-lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (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 :: {field_char_0, field_inverse_zero})"
- by (simp add: Ndiv_def)
-
-lemma Nlt0_iff[simp]:
- assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x"
-proof -
- obtain a b where x: "x = (a, b)" by (cases x)
- { assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) }
- moreover
- { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0"
- using nx by (simp add: x isnormNum_def)
- from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: x Nlt0_def INum_def) }
- ultimately show ?thesis by blast
-qed
-
-lemma Nle0_iff[simp]:
- assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
-proof -
- obtain a b where x: "x = (a, b)" by (cases x)
- { assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) }
- moreover
- { assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0"
- using nx by (simp add: x isnormNum_def)
- from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: x Nle0_def INum_def) }
- ultimately show ?thesis by blast
-qed
-
-lemma Ngt0_iff[simp]:
- assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
-proof -
- obtain a b where x: "x = (a, b)" by (cases x)
- { assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) }
- moreover
- { assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
- by (simp add: x isnormNum_def)
- from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: x Ngt0_def INum_def) }
- ultimately show ?thesis by blast
-qed
-
-lemma Nge0_iff[simp]:
- assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
-proof -
- obtain a b where x: "x = (a, b)" by (cases x)
- { assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) }
- moreover
- { assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx
- by (simp add: x isnormNum_def)
- from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
- have ?thesis by (simp add: x 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 :: {field_char_0, linordered_field_inverse_zero}) < 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 :: {field_char_0, linordered_field_inverse_zero})\<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:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- shows "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) = INum (y +\<^sub>N x)" by simp
- with isnormNum_unique[OF n] show ?thesis by simp
-qed
-
-lemma [simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- shows "(0, b) +\<^sub>N y = normNum y"
- and "(a, 0) +\<^sub>N y = normNum y"
- and "x +\<^sub>N (0, b) = normNum x"
- and "x +\<^sub>N (a, 0) = normNum x"
- apply (simp add: Nadd_def split_def)
- apply (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 "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- 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)" by simp
- with isnormNum_unique[OF n nx] show ?thesis by simp
-qed
-
-lemma normNum_nilpotent[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- shows "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:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
-
-lemma Nadd_normNum1[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- shows "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)" 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]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- shows "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)" 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:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- shows "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)" 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 gcd_commute_int mult_commute)
-
-lemma Nmul_assoc:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- 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)" by simp
- with isnormNum_unique[OF n] show ?thesis by simp
-qed
-
-lemma Nsub0:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- assumes x: "isnormNum x" and y: "isnormNum y"
- shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y"
-proof -
- fix h :: 'a
- 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 "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
- assumes nx: "isnormNum x" and ny: "isnormNum y"
- shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N"
-proof -
- fix h :: 'a
- obtain a b where x: "x = (a, b)" by (cases x)
- obtain a' b' where y: "y = (a', b')" by (cases y)
- 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: x y INum_def split_def isnormNum_def split: split_if_asm)
- 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)
- apply (cases "fst c = 0", simp_all, cases c, simp_all)+
- done
-
-end
--- a/src/HOL/Library/Library.thy Thu Oct 31 11:44:20 2013 +0100
+++ b/src/HOL/Library/Library.thy Thu Oct 31 11:44:20 2013 +0100
@@ -1,7 +1,6 @@
(*<*)
theory Library
imports
- Abstract_Rat
AList
BigO
Binomial