author | wenzelm |
Sat, 12 Jan 2013 14:53:56 +0100 | |
changeset 50843 | 1465521b92a1 |
parent 50282 | fe4d4bb9f4c2 |
permissions | -rw-r--r-- |
24197 | 1 |
(* Title: HOL/Library/Abstract_Rat.thy |
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Author: Amine Chaieb |
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*) |
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header {* Abstract rational numbers *} |
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theory Abstract_Rat |
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imports Complex_Main |
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begin |
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type_synonym Num = "int \<times> int" |
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abbreviation Num0_syn :: Num ("0\<^sub>N") |
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where "0\<^sub>N \<equiv> (0, 0)" |
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abbreviation Numi_syn :: "int \<Rightarrow> Num" ("'((_)')\<^sub>N") |
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where "(i)\<^sub>N \<equiv> (i, 1)" |
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definition isnormNum :: "Num \<Rightarrow> bool" where |
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"isnormNum = (\<lambda>(a,b). (if a = 0 then b = 0 else b > 0 \<and> gcd a b = 1))" |
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definition normNum :: "Num \<Rightarrow> Num" where |
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"normNum = (\<lambda>(a,b). |
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(if a=0 \<or> b = 0 then (0,0) else |
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(let g = gcd a b |
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in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))" |
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|
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declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger] |
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||
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lemma normNum_isnormNum [simp]: "isnormNum (normNum x)" |
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proof - |
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obtain a b where x: "x = (a, b)" by (cases x) |
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{ assume "a=0 \<or> b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) } |
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moreover |
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{ assume anz: "a \<noteq> 0" and bnz: "b \<noteq> 0" |
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let ?g = "gcd a b" |
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let ?a' = "a div ?g" |
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let ?b' = "b div ?g" |
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let ?g' = "gcd ?a' ?b'" |
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from anz bnz have "?g \<noteq> 0" by simp with gcd_ge_0_int[of a b] |
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have gpos: "?g > 0" by arith |
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have gdvd: "?g dvd a" "?g dvd b" by arith+ |
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from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] anz bnz |
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have nz': "?a' \<noteq> 0" "?b' \<noteq> 0" by - (rule notI, simp)+ |
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from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith |
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from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" . |
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from bnz have "b < 0 \<or> b > 0" by arith |
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moreover |
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{ assume b: "b > 0" |
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from b have "?b' \<ge> 0" |
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by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos]) |
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with nz' have b': "?b' > 0" by arith |
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from b b' anz bnz nz' gp1 have ?thesis |
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by (simp add: x isnormNum_def normNum_def Let_def split_def) } |
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moreover { |
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assume b: "b < 0" |
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{ assume b': "?b' \<ge> 0" |
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from gpos have th: "?g \<ge> 0" by arith |
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from mult_nonneg_nonneg[OF th b'] dvd_mult_div_cancel[OF gdvd(2)] |
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have False using b by arith } |
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hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric]) |
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from anz bnz nz' b b' gp1 have ?thesis |
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by (simp add: x isnormNum_def normNum_def Let_def split_def) } |
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ultimately have ?thesis by blast |
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} |
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ultimately show ?thesis by blast |
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qed |
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text {* Arithmetic over Num *} |
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definition Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60) where |
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"Nadd = (\<lambda>(a,b) (a',b'). if a = 0 \<or> b = 0 then normNum(a',b') |
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else if a'=0 \<or> b' = 0 then normNum(a,b) |
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else normNum(a*b' + b*a', b*b'))" |
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definition Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60) where |
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"Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b') |
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in (a*a' div g, b*b' div g))" |
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definition Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N") |
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where "Nneg \<equiv> (\<lambda>(a,b). (-a,b))" |
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definition Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60) |
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where "Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)" |
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definition Ninv :: "Num \<Rightarrow> Num" |
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where "Ninv = (\<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a))" |
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definition Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60) |
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where "Ndiv = (\<lambda>a b. a *\<^sub>N Ninv b)" |
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lemma Nneg_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (~\<^sub>N x)" |
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by (simp add: isnormNum_def Nneg_def split_def) |
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lemma Nadd_normN[simp]: "isnormNum (x +\<^sub>N y)" |
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by (simp add: Nadd_def split_def) |
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lemma Nsub_normN[simp]: "\<lbrakk> isnormNum y\<rbrakk> \<Longrightarrow> isnormNum (x -\<^sub>N y)" |
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by (simp add: Nsub_def split_def) |
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lemma Nmul_normN[simp]: |
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assumes xn: "isnormNum x" and yn: "isnormNum y" |
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shows "isnormNum (x *\<^sub>N y)" |
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proof - |
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obtain a b where x: "x = (a, b)" by (cases x) |
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obtain a' b' where y: "y = (a', b')" by (cases y) |
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{ assume "a = 0" |
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hence ?thesis using xn x y |
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by (simp add: isnormNum_def Let_def Nmul_def split_def) } |
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moreover |
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{ assume "a' = 0" |
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hence ?thesis using yn x y |
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by (simp add: isnormNum_def Let_def Nmul_def split_def) } |
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moreover |
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{ assume a: "a \<noteq>0" and a': "a'\<noteq>0" |
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hence bp: "b > 0" "b' > 0" using xn yn x y by (simp_all add: isnormNum_def) |
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from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a * a', b * b')" |
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using x y a a' bp by (simp add: Nmul_def Let_def split_def normNum_def) |
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hence ?thesis by simp } |
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ultimately show ?thesis by blast |
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qed |
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lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)" |
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by (simp add: Ninv_def isnormNum_def split_def) |
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(cases "fst x = 0", auto simp add: gcd_commute_int) |
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lemma isnormNum_int[simp]: |
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"isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i)\<^sub>N" |
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by (simp_all add: isnormNum_def) |
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text {* Relations over Num *} |
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definition Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N") |
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where "Nlt0 = (\<lambda>(a,b). a < 0)" |
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definition Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N") |
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where "Nle0 = (\<lambda>(a,b). a \<le> 0)" |
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definition Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N") |
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where "Ngt0 = (\<lambda>(a,b). a > 0)" |
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definition Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N") |
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where "Nge0 = (\<lambda>(a,b). a \<ge> 0)" |
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definition Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55) |
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where "Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))" |
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definition Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55) |
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where "Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))" |
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definition "INum = (\<lambda>(a,b). of_int a / of_int b)" |
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parents:
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lemma INum_int [simp]: "INum (i)\<^sub>N = ((of_int i) ::'a::field)" "INum 0\<^sub>N = (0::'a::field)" |
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by (simp_all add: INum_def) |
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lemma isnormNum_unique[simp]: |
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assumes na: "isnormNum x" and nb: "isnormNum y" |
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shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs") |
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proof |
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obtain a b where x: "x = (a, b)" by (cases x) |
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obtain a' b' where y: "y = (a', b')" by (cases y) |
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assume H: ?lhs |
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{ assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" |
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hence ?rhs using na nb H |
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by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) } |
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moreover |
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{ assume az: "a \<noteq> 0" and bz: "b \<noteq> 0" and a'z: "a'\<noteq>0" and b'z: "b'\<noteq>0" |
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from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: x y isnormNum_def) |
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from H bz b'z have eq: "a * b' = a'*b" |
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by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult) |
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from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1" |
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by (simp_all add: x y isnormNum_def add: gcd_commute_int) |
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from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'" |
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apply - |
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apply algebra |
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apply algebra |
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apply simp |
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apply algebra |
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done |
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from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)] |
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coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]] |
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have eq1: "b = b'" using pos by arith |
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with eq have "a = a'" using pos by simp |
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with eq1 have ?rhs by (simp add: x y) } |
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ultimately show ?rhs by blast |
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next |
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assume ?rhs thus ?lhs by simp |
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qed |
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lemma isnormNum0[simp]: |
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"isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)" |
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unfolding INum_int(2)[symmetric] |
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by (rule isnormNum_unique) simp_all |
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lemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) = |
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of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)" |
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proof - |
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assume "d ~= 0" |
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let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)" |
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let ?f = "\<lambda>x. x / of_int d" |
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have "x = (x div d) * d + x mod d" |
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by auto |
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then have eq: "of_int x = ?t" |
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by (simp only: of_int_mult[symmetric] of_int_add [symmetric]) |
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then have "of_int x / of_int d = ?t / of_int d" |
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using cong[OF refl[of ?f] eq] by simp |
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then show ?thesis by (simp add: add_divide_distrib algebra_simps `d ~= 0`) |
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qed |
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lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==> |
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(of_int(n div d)::'a::field_char_0) = of_int n / of_int d" |
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apply (frule of_int_div_aux [of d n, where ?'a = 'a]) |
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apply simp |
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apply (simp add: dvd_eq_mod_eq_0) |
44779 | 217 |
done |
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||
36409 | 220 |
lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})" |
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proof - |
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obtain a b where x: "x = (a, b)" by (cases x) |
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{ assume "a = 0 \<or> b = 0" |
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hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) } |
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moreover |
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{ assume a: "a \<noteq> 0" and b: "b \<noteq> 0" |
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let ?g = "gcd a b" |
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from a b have g: "?g \<noteq> 0"by simp |
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from of_int_div[OF g, where ?'a = 'a] |
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44779 | 230 |
have ?thesis by (auto simp add: x INum_def normNum_def split_def Let_def) } |
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ultimately show ?thesis by blast |
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qed |
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||
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lemma INum_normNum_iff: |
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"(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y" |
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(is "?lhs = ?rhs") |
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24197 | 237 |
proof - |
238 |
have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)" |
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by (simp del: normNum) |
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also have "\<dots> = ?lhs" by simp |
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finally show ?thesis by simp |
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qed |
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||
36409 | 244 |
lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})" |
44779 | 245 |
proof - |
246 |
let ?z = "0:: 'a" |
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44780 | 247 |
obtain a b where x: "x = (a, b)" by (cases x) |
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obtain a' b' where y: "y = (a', b')" by (cases y) |
|
44779 | 249 |
{ assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" |
44780 | 250 |
hence ?thesis |
251 |
apply (cases "a=0", simp_all add: x y Nadd_def) |
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44779 | 252 |
apply (cases "b= 0", simp_all add: INum_def) |
253 |
apply (cases "a'= 0", simp_all) |
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254 |
apply (cases "b'= 0", simp_all) |
|
24197 | 255 |
done } |
44780 | 256 |
moreover |
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{ assume aa': "a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0" |
|
44779 | 258 |
{ assume z: "a * b' + b * a' = 0" |
24197 | 259 |
hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp |
44780 | 260 |
hence "of_int b' * of_int a / (of_int b * of_int b') + |
261 |
of_int b * of_int a' / (of_int b * of_int b') = ?z" |
|
262 |
by (simp add:add_divide_distrib) |
|
44779 | 263 |
hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' |
44780 | 264 |
by simp |
265 |
from z aa' bb' have ?thesis |
|
266 |
by (simp add: x y th Nadd_def normNum_def INum_def split_def) } |
|
44779 | 267 |
moreover { |
268 |
assume z: "a * b' + b * a' \<noteq> 0" |
|
31706 | 269 |
let ?g = "gcd (a * b' + b * a') (b*b')" |
24197 | 270 |
have gz: "?g \<noteq> 0" using z by simp |
271 |
have ?thesis using aa' bb' z gz |
|
44779 | 272 |
of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]] |
273 |
of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]] |
|
44780 | 274 |
by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib) } |
44779 | 275 |
ultimately have ?thesis using aa' bb' |
44780 | 276 |
by (simp add: x y Nadd_def INum_def normNum_def Let_def) } |
24197 | 277 |
ultimately show ?thesis by blast |
278 |
qed |
|
279 |
||
44779 | 280 |
lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})" |
281 |
proof - |
|
24197 | 282 |
let ?z = "0::'a" |
44780 | 283 |
obtain a b where x: "x = (a, b)" by (cases x) |
284 |
obtain a' b' where y: "y = (a', b')" by (cases y) |
|
44779 | 285 |
{ assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" |
44780 | 286 |
hence ?thesis |
44779 | 287 |
apply (cases "a=0", simp_all add: x y Nmul_def INum_def Let_def) |
288 |
apply (cases "b=0", simp_all) |
|
44780 | 289 |
apply (cases "a'=0", simp_all) |
24197 | 290 |
done } |
291 |
moreover |
|
44779 | 292 |
{ assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0" |
31706 | 293 |
let ?g="gcd (a*a') (b*b')" |
24197 | 294 |
have gz: "?g \<noteq> 0" using z by simp |
44779 | 295 |
from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]] |
44780 | 296 |
of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]] |
44779 | 297 |
have ?thesis by (simp add: Nmul_def x y Let_def INum_def) } |
24197 | 298 |
ultimately show ?thesis by blast |
299 |
qed |
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||
301 |
lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)" |
|
302 |
by (simp add: Nneg_def split_def INum_def) |
|
303 |
||
44779 | 304 |
lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})" |
305 |
by (simp add: Nsub_def split_def) |
|
24197 | 306 |
|
36409 | 307 |
lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)" |
24197 | 308 |
by (simp add: Ninv_def INum_def split_def) |
309 |
||
44779 | 310 |
lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})" |
311 |
by (simp add: Ndiv_def) |
|
24197 | 312 |
|
44779 | 313 |
lemma Nlt0_iff[simp]: |
44780 | 314 |
assumes nx: "isnormNum x" |
44779 | 315 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x" |
316 |
proof - |
|
44780 | 317 |
obtain a b where x: "x = (a, b)" by (cases x) |
318 |
{ assume "a = 0" hence ?thesis by (simp add: x Nlt0_def INum_def) } |
|
24197 | 319 |
moreover |
44780 | 320 |
{ assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" |
321 |
using nx by (simp add: x isnormNum_def) |
|
24197 | 322 |
from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"] |
44780 | 323 |
have ?thesis by (simp add: x Nlt0_def INum_def) } |
24197 | 324 |
ultimately show ?thesis by blast |
325 |
qed |
|
326 |
||
44779 | 327 |
lemma Nle0_iff[simp]: |
328 |
assumes nx: "isnormNum x" |
|
36409 | 329 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x" |
44779 | 330 |
proof - |
44780 | 331 |
obtain a b where x: "x = (a, b)" by (cases x) |
332 |
{ assume "a = 0" hence ?thesis by (simp add: x Nle0_def INum_def) } |
|
24197 | 333 |
moreover |
44780 | 334 |
{ assume a: "a \<noteq> 0" hence b: "(of_int b :: 'a) > 0" |
335 |
using nx by (simp add: x isnormNum_def) |
|
24197 | 336 |
from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"] |
44780 | 337 |
have ?thesis by (simp add: x Nle0_def INum_def) } |
24197 | 338 |
ultimately show ?thesis by blast |
339 |
qed |
|
340 |
||
44779 | 341 |
lemma Ngt0_iff[simp]: |
342 |
assumes nx: "isnormNum x" |
|
343 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x" |
|
344 |
proof - |
|
44780 | 345 |
obtain a b where x: "x = (a, b)" by (cases x) |
346 |
{ assume "a = 0" hence ?thesis by (simp add: x Ngt0_def INum_def) } |
|
24197 | 347 |
moreover |
44780 | 348 |
{ assume a: "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx |
349 |
by (simp add: x isnormNum_def) |
|
24197 | 350 |
from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"] |
44780 | 351 |
have ?thesis by (simp add: x Ngt0_def INum_def) } |
24197 | 352 |
ultimately show ?thesis by blast |
353 |
qed |
|
354 |
||
44779 | 355 |
lemma Nge0_iff[simp]: |
356 |
assumes nx: "isnormNum x" |
|
357 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x" |
|
358 |
proof - |
|
44780 | 359 |
obtain a b where x: "x = (a, b)" by (cases x) |
360 |
{ assume "a = 0" hence ?thesis by (simp add: x Nge0_def INum_def) } |
|
44779 | 361 |
moreover |
362 |
{ assume "a \<noteq> 0" hence b: "(of_int b::'a) > 0" using nx |
|
44780 | 363 |
by (simp add: x isnormNum_def) |
44779 | 364 |
from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"] |
44780 | 365 |
have ?thesis by (simp add: x Nge0_def INum_def) } |
44779 | 366 |
ultimately show ?thesis by blast |
367 |
qed |
|
368 |
||
369 |
lemma Nlt_iff[simp]: |
|
370 |
assumes nx: "isnormNum x" and ny: "isnormNum y" |
|
36409 | 371 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)" |
44779 | 372 |
proof - |
24197 | 373 |
let ?z = "0::'a" |
44779 | 374 |
have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" |
375 |
using nx ny by simp |
|
376 |
also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" |
|
377 |
using Nlt0_iff[OF Nsub_normN[OF ny]] by simp |
|
24197 | 378 |
finally show ?thesis by (simp add: Nlt_def) |
379 |
qed |
|
380 |
||
44779 | 381 |
lemma Nle_iff[simp]: |
382 |
assumes nx: "isnormNum x" and ny: "isnormNum y" |
|
36409 | 383 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)" |
44779 | 384 |
proof - |
385 |
have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" |
|
386 |
using nx ny by simp |
|
387 |
also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" |
|
388 |
using Nle0_iff[OF Nsub_normN[OF ny]] by simp |
|
24197 | 389 |
finally show ?thesis by (simp add: Nle_def) |
390 |
qed |
|
391 |
||
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|
392 |
lemma Nadd_commute: |
36409 | 393 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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|
394 |
shows "x +\<^sub>N y = y +\<^sub>N x" |
44779 | 395 |
proof - |
24197 | 396 |
have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all |
31964 | 397 |
have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp |
24197 | 398 |
with isnormNum_unique[OF n] show ?thesis by simp |
399 |
qed |
|
400 |
||
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|
401 |
lemma [simp]: |
36409 | 402 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
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|
403 |
shows "(0, b) +\<^sub>N y = normNum y" |
44780 | 404 |
and "(a, 0) +\<^sub>N y = normNum y" |
28615
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|
405 |
and "x +\<^sub>N (0, b) = normNum x" |
4c8fa015ec7f
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changeset
|
406 |
and "x +\<^sub>N (a, 0) = normNum x" |
4c8fa015ec7f
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parents:
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changeset
|
407 |
apply (simp add: Nadd_def split_def) |
4c8fa015ec7f
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parents:
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diff
changeset
|
408 |
apply (simp add: Nadd_def split_def) |
4c8fa015ec7f
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wenzelm
parents:
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diff
changeset
|
409 |
apply (subst Nadd_commute, simp add: Nadd_def split_def) |
4c8fa015ec7f
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diff
changeset
|
410 |
apply (subst Nadd_commute, simp add: Nadd_def split_def) |
24197 | 411 |
done |
412 |
||
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|
413 |
lemma normNum_nilpotent_aux[simp]: |
36409 | 414 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
44780 | 415 |
assumes nx: "isnormNum x" |
24197 | 416 |
shows "normNum x = x" |
44779 | 417 |
proof - |
24197 | 418 |
let ?a = "normNum x" |
419 |
have n: "isnormNum ?a" by simp |
|
44779 | 420 |
have th: "INum ?a = (INum x ::'a)" by simp |
421 |
with isnormNum_unique[OF n nx] show ?thesis by simp |
|
24197 | 422 |
qed |
423 |
||
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|
424 |
lemma normNum_nilpotent[simp]: |
36409 | 425 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
28615
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changeset
|
426 |
shows "normNum (normNum x) = normNum x" |
24197 | 427 |
by simp |
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changeset
|
428 |
|
24197 | 429 |
lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N" |
430 |
by (simp_all add: normNum_def) |
|
28615
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changeset
|
431 |
|
4c8fa015ec7f
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|
432 |
lemma normNum_Nadd: |
36409 | 433 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
28615
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changeset
|
434 |
shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp |
4c8fa015ec7f
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parents:
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diff
changeset
|
435 |
|
4c8fa015ec7f
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parents:
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diff
changeset
|
436 |
lemma Nadd_normNum1[simp]: |
36409 | 437 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
28615
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changeset
|
438 |
shows "normNum x +\<^sub>N y = x +\<^sub>N y" |
44779 | 439 |
proof - |
24197 | 440 |
have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all |
31964 | 441 |
have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp |
24197 | 442 |
also have "\<dots> = INum (x +\<^sub>N y)" by simp |
443 |
finally show ?thesis using isnormNum_unique[OF n] by simp |
|
444 |
qed |
|
445 |
||
28615
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changeset
|
446 |
lemma Nadd_normNum2[simp]: |
36409 | 447 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
28615
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wenzelm
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diff
changeset
|
448 |
shows "x +\<^sub>N normNum y = x +\<^sub>N y" |
44779 | 449 |
proof - |
28615
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parents:
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diff
changeset
|
450 |
have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all |
31964 | 451 |
have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp |
28615
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changeset
|
452 |
also have "\<dots> = INum (x +\<^sub>N y)" by simp |
4c8fa015ec7f
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wenzelm
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changeset
|
453 |
finally show ?thesis using isnormNum_unique[OF n] by simp |
4c8fa015ec7f
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diff
changeset
|
454 |
qed |
4c8fa015ec7f
explicit SORT_CONSTRAINT for proofs depending implicitly on certain sorts;
wenzelm
parents:
27668
diff
changeset
|
455 |
|
4c8fa015ec7f
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parents:
27668
diff
changeset
|
456 |
lemma Nadd_assoc: |
36409 | 457 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
28615
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wenzelm
parents:
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diff
changeset
|
458 |
shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)" |
44779 | 459 |
proof - |
24197 | 460 |
have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all |
31964 | 461 |
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp |
24197 | 462 |
with isnormNum_unique[OF n] show ?thesis by simp |
463 |
qed |
|
464 |
||
465 |
lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x" |
|
31952
40501bb2d57c
renamed lemmas: nat_xyz/int_xyz -> xyz_nat/xyz_int
nipkow
parents:
31706
diff
changeset
|
466 |
by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute) |
24197 | 467 |
|
28615
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changeset
|
468 |
lemma Nmul_assoc: |
36409 | 469 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
44780 | 470 |
assumes nx: "isnormNum x" and ny: "isnormNum y" and nz: "isnormNum z" |
24197 | 471 |
shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)" |
44779 | 472 |
proof - |
44780 | 473 |
from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))" |
24197 | 474 |
by simp_all |
31964 | 475 |
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp |
24197 | 476 |
with isnormNum_unique[OF n] show ?thesis by simp |
477 |
qed |
|
478 |
||
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changeset
|
479 |
lemma Nsub0: |
36409 | 480 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
44780 | 481 |
assumes x: "isnormNum x" and y: "isnormNum y" |
482 |
shows "x -\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = y" |
|
44779 | 483 |
proof - |
484 |
fix h :: 'a |
|
44780 | 485 |
from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"] |
44779 | 486 |
have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp |
487 |
also have "\<dots> = (INum x = (INum y :: 'a))" by simp |
|
488 |
also have "\<dots> = (x = y)" using x y by simp |
|
489 |
finally show ?thesis . |
|
24197 | 490 |
qed |
491 |
||
492 |
lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N" |
|
493 |
by (simp_all add: Nmul_def Let_def split_def) |
|
494 |
||
28615
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changeset
|
495 |
lemma Nmul_eq0[simp]: |
36409 | 496 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" |
44780 | 497 |
assumes nx: "isnormNum x" and ny: "isnormNum y" |
498 |
shows "x*\<^sub>N y = 0\<^sub>N \<longleftrightarrow> x = 0\<^sub>N \<or> y = 0\<^sub>N" |
|
44779 | 499 |
proof - |
500 |
fix h :: 'a |
|
44780 | 501 |
obtain a b where x: "x = (a, b)" by (cases x) |
502 |
obtain a' b' where y: "y = (a', b')" by (cases y) |
|
44779 | 503 |
have n0: "isnormNum 0\<^sub>N" by simp |
44780 | 504 |
show ?thesis using nx ny |
44779 | 505 |
apply (simp only: isnormNum_unique[where ?'a = 'a, OF Nmul_normN[OF nx ny] n0, symmetric] |
506 |
Nmul[where ?'a = 'a]) |
|
44780 | 507 |
apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm) |
44779 | 508 |
done |
24197 | 509 |
qed |
44779 | 510 |
|
24197 | 511 |
lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c" |
512 |
by (simp add: Nneg_def split_def) |
|
513 |
||
44780 | 514 |
lemma Nmul1[simp]: |
50282
fe4d4bb9f4c2
more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
parents:
47162
diff
changeset
|
515 |
"isnormNum c \<Longrightarrow> (1)\<^sub>N *\<^sub>N c = c" |
fe4d4bb9f4c2
more robust syntax that survives collapse of \<^isub> and \<^sub>;
wenzelm
parents:
47162
diff
changeset
|
516 |
"isnormNum c \<Longrightarrow> c *\<^sub>N (1)\<^sub>N = c" |
24197 | 517 |
apply (simp_all add: Nmul_def Let_def split_def isnormNum_def) |
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wenzelm
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changeset
|
518 |
apply (cases "fst c = 0", simp_all, cases c, simp_all)+ |
4c8fa015ec7f
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wenzelm
parents:
27668
diff
changeset
|
519 |
done |
24197 | 520 |
|
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wenzelm
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diff
changeset
|
521 |
end |