haftmann@24197
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(* Title: HOL/Library/Abstract_Rat.thy
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haftmann@24197
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Author: Amine Chaieb
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haftmann@24197
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*)
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header {* Abstract rational numbers *}
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theory Abstract_Rat
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haftmann@36411
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imports Complex_Main
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haftmann@24197
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begin
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haftmann@24197
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wenzelm@43334
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type_synonym Num = "int \<times> int"
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wenzelm@25005
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wenzelm@25005
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abbreviation
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wenzelm@25005
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Num0_syn :: Num ("0\<^sub>N")
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wenzelm@25005
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where "0\<^sub>N \<equiv> (0, 0)"
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wenzelm@25005
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wenzelm@25005
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abbreviation
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wenzelm@25005
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Numi_syn :: "int \<Rightarrow> Num" ("_\<^sub>N")
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wenzelm@25005
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where "i\<^sub>N \<equiv> (i, 1)"
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definition
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isnormNum :: "Num \<Rightarrow> bool"
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where
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huffman@31704
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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
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normNum :: "Num \<Rightarrow> Num"
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where
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"normNum = (\<lambda>(a,b). (if a=0 \<or> b = 0 then (0,0) else
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huffman@31704
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(let g = gcd a b
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haftmann@24197
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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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haftmann@24197
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nipkow@31952
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declare gcd_dvd1_int[presburger]
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nipkow@31952
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declare gcd_dvd2_int[presburger]
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lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"
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proof -
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have " \<exists> a b. x = (a,b)" by auto
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haftmann@24197
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then obtain a b where x[simp]: "x = (a,b)" by blast
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haftmann@24197
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{assume "a=0 \<or> b = 0" hence ?thesis by (simp add: 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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huffman@31704
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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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huffman@31704
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let ?g' = "gcd ?a' ?b'"
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nipkow@31952
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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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wenzelm@41776
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have gpos: "?g > 0" by arith
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chaieb@27668
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have gdvd: "?g dvd a" "?g dvd b" by arith+
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from zdvd_mult_div_cancel[OF gdvd(1)] zdvd_mult_div_cancel[OF gdvd(2)]
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anz bnz
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huffman@31704
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have nz':"?a' \<noteq> 0" "?b' \<noteq> 0"
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huffman@31704
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by - (rule notI, simp)+
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chaieb@27668
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from anz bnz have stupid: "a \<noteq> 0 \<or> b \<noteq> 0" by arith
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nipkow@31952
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from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .
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haftmann@24197
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from bnz have "b < 0 \<or> b > 0" by arith
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moreover
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haftmann@24197
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{assume b: "b > 0"
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chaieb@27668
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from b have "?b' \<ge> 0"
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wenzelm@32962
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by (presburger add: pos_imp_zdiv_nonneg_iff[OF gpos])
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chaieb@27668
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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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wenzelm@32962
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by (simp add: isnormNum_def normNum_def Let_def split_def fst_conv snd_conv)}
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haftmann@24197
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moreover {assume b: "b < 0"
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{assume b': "?b' \<ge> 0"
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wenzelm@32962
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from gpos have th: "?g \<ge> 0" by arith
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wenzelm@32962
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from mult_nonneg_nonneg[OF th b'] zdvd_mult_div_cancel[OF gdvd(2)]
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wenzelm@32962
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have False using b by arith }
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haftmann@24197
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hence b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
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haftmann@24197
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from anz bnz nz' b b' gp1 have ?thesis
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wenzelm@32962
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by (simp add: isnormNum_def normNum_def Let_def split_def)}
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haftmann@24197
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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
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Nadd :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "+\<^sub>N" 60)
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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
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Nmul :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "*\<^sub>N" 60)
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where
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huffman@31704
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"Nmul = (\<lambda>(a,b) (a',b'). let g = gcd (a*a') (b*b')
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haftmann@24197
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in (a*a' div g, b*b' div g))"
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definition
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Nneg :: "Num \<Rightarrow> Num" ("~\<^sub>N")
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where
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"Nneg \<equiv> (\<lambda>(a,b). (-a,b))"
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definition
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Nsub :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "-\<^sub>N" 60)
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where
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"Nsub = (\<lambda>a b. a +\<^sub>N ~\<^sub>N b)"
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definition
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Ninv :: "Num \<Rightarrow> Num"
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where
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"Ninv \<equiv> \<lambda>(a,b). if a < 0 then (-b, \<bar>a\<bar>) else (b,a)"
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definition
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Ndiv :: "Num \<Rightarrow> Num \<Rightarrow> Num" (infixl "\<div>\<^sub>N" 60)
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where
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"Ndiv \<equiv> \<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]: 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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haftmann@24197
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have "\<exists>a b. x = (a,b)" and "\<exists> a' b'. y = (a',b')" by auto
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haftmann@24197
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then obtain a b a' b' where ab: "x = (a,b)" and ab': "y = (a',b')" by blast
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haftmann@24197
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{assume "a = 0"
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haftmann@24197
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hence ?thesis using xn ab ab'
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huffman@31704
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by (simp add: isnormNum_def Let_def Nmul_def split_def)}
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haftmann@24197
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moreover
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haftmann@24197
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{assume "a' = 0"
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haftmann@24197
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hence ?thesis using yn ab ab'
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huffman@31704
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by (simp add: isnormNum_def Let_def Nmul_def split_def)}
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haftmann@24197
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moreover
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haftmann@24197
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{assume a: "a \<noteq>0" and a': "a'\<noteq>0"
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haftmann@24197
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hence bp: "b > 0" "b' > 0" using xn yn ab ab' by (simp_all add: isnormNum_def)
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haftmann@24197
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from mult_pos_pos[OF bp] have "x *\<^sub>N y = normNum (a*a', b*b')"
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haftmann@24197
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using ab ab' a a' bp by (simp add: Nmul_def Let_def split_def normNum_def)
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haftmann@24197
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hence ?thesis by simp}
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haftmann@24197
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ultimately show ?thesis by blast
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haftmann@24197
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qed
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haftmann@24197
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haftmann@24197
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lemma Ninv_normN[simp]: "isnormNum x \<Longrightarrow> isnormNum (Ninv x)"
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wenzelm@25005
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by (simp add: Ninv_def isnormNum_def split_def)
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nipkow@31952
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(cases "fst x = 0", auto simp add: gcd_commute_int)
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haftmann@24197
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haftmann@24197
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lemma isnormNum_int[simp]:
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wenzelm@41776
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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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huffman@31704
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by (simp_all add: isnormNum_def)
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haftmann@24197
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haftmann@24197
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haftmann@24197
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text {* Relations over Num *}
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haftmann@24197
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definition
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haftmann@24197
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Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
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haftmann@24197
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where
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haftmann@24197
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"Nlt0 = (\<lambda>(a,b). a < 0)"
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haftmann@24197
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haftmann@24197
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definition
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haftmann@24197
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Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
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haftmann@24197
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where
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haftmann@24197
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"Nle0 = (\<lambda>(a,b). a \<le> 0)"
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haftmann@24197
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haftmann@24197
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definition
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haftmann@24197
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Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
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haftmann@24197
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where
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haftmann@24197
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"Ngt0 = (\<lambda>(a,b). a > 0)"
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haftmann@24197
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haftmann@24197
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definition
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haftmann@24197
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Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
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haftmann@24197
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where
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haftmann@24197
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"Nge0 = (\<lambda>(a,b). a \<ge> 0)"
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haftmann@24197
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haftmann@24197
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definition
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haftmann@24197
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Nlt :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "<\<^sub>N" 55)
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haftmann@24197
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where
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haftmann@24197
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"Nlt = (\<lambda>a b. 0>\<^sub>N (a -\<^sub>N b))"
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haftmann@24197
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haftmann@24197
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definition
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haftmann@24197
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Nle :: "Num \<Rightarrow> Num \<Rightarrow> bool" (infix "\<le>\<^sub>N" 55)
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haftmann@24197
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where
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haftmann@24197
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"Nle = (\<lambda>a b. 0\<ge>\<^sub>N (a -\<^sub>N b))"
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haftmann@24197
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haftmann@24197
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definition
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haftmann@24197
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"INum = (\<lambda>(a,b). of_int a / of_int b)"
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haftmann@24197
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wenzelm@41776
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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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haftmann@24197
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by (simp_all add: INum_def)
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haftmann@24197
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haftmann@24197
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lemma isnormNum_unique[simp]:
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haftmann@24197
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assumes na: "isnormNum x" and nb: "isnormNum y"
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haftmann@36409
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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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haftmann@24197
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proof
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haftmann@24197
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have "\<exists> a b a' b'. x = (a,b) \<and> y = (a',b')" by auto
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haftmann@24197
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then obtain a b a' b' where xy[simp]: "x = (a,b)" "y=(a',b')" by blast
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haftmann@24197
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assume H: ?lhs
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nipkow@32456
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{assume "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0"
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nipkow@32456
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hence ?rhs using na nb H
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nipkow@32456
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by (simp add: INum_def split_def isnormNum_def split: split_if_asm)}
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haftmann@24197
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moreover
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haftmann@24197
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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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haftmann@24197
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from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: isnormNum_def)
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wenzelm@41776
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from H bz b'z have eq:"a * b' = a'*b"
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haftmann@24197
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by (simp add: INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)
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wenzelm@41776
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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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nipkow@31952
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by (simp_all add: isnormNum_def add: gcd_commute_int)
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chaieb@27668
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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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chaieb@27668
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apply -
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chaieb@27668
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apply algebra
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chaieb@27668
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apply algebra
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chaieb@27668
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apply simp
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chaieb@27668
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apply algebra
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haftmann@24197
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done
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nipkow@33657
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from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]
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nipkow@31952
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coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]
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wenzelm@41776
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have eq1: "b = b'" using pos by arith
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haftmann@24197
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with eq have "a = a'" using pos by simp
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haftmann@24197
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with eq1 have ?rhs by simp}
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haftmann@24197
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ultimately show ?rhs by blast
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haftmann@24197
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next
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haftmann@24197
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assume ?rhs thus ?lhs by simp
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haftmann@24197
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qed
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haftmann@24197
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haftmann@24197
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haftmann@36409
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lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0\<^sub>N)"
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haftmann@24197
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unfolding INum_int(2)[symmetric]
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haftmann@24197
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by (rule isnormNum_unique, simp_all)
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haftmann@24197
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223 |
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haftmann@36409
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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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haftmann@24197
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of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"
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haftmann@24197
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proof -
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haftmann@24197
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227 |
assume "d ~= 0"
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haftmann@24197
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let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"
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haftmann@24197
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let ?f = "\<lambda>x. x / of_int d"
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haftmann@24197
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have "x = (x div d) * d + x mod d"
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haftmann@24197
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by auto
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haftmann@24197
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then have eq: "of_int x = ?t"
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haftmann@24197
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by (simp only: of_int_mult[symmetric] of_int_add [symmetric])
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haftmann@24197
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then have "of_int x / of_int d = ?t / of_int d"
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haftmann@24197
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using cong[OF refl[of ?f] eq] by simp
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wenzelm@41776
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then show ?thesis by (simp add: add_divide_distrib algebra_simps `d ~= 0`)
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haftmann@24197
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qed
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haftmann@24197
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haftmann@24197
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lemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
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haftmann@36409
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(of_int(n div d)::'a::field_char_0) = of_int n / of_int d"
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haftmann@24197
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241 |
apply (frule of_int_div_aux [of d n, where ?'a = 'a])
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haftmann@24197
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apply simp
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nipkow@29979
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apply (simp add: dvd_eq_mod_eq_0)
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haftmann@24197
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244 |
done
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haftmann@24197
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haftmann@24197
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246 |
|
haftmann@36409
|
247 |
lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"
|
haftmann@24197
|
248 |
proof-
|
haftmann@24197
|
249 |
have "\<exists> a b. x = (a,b)" by auto
|
haftmann@24197
|
250 |
then obtain a b where x[simp]: "x = (a,b)" by blast
|
haftmann@24197
|
251 |
{assume "a=0 \<or> b = 0" hence ?thesis
|
haftmann@24197
|
252 |
by (simp add: INum_def normNum_def split_def Let_def)}
|
haftmann@24197
|
253 |
moreover
|
haftmann@24197
|
254 |
{assume a: "a\<noteq>0" and b: "b\<noteq>0"
|
huffman@31704
|
255 |
let ?g = "gcd a b"
|
haftmann@24197
|
256 |
from a b have g: "?g \<noteq> 0"by simp
|
haftmann@24197
|
257 |
from of_int_div[OF g, where ?'a = 'a]
|
haftmann@24197
|
258 |
have ?thesis by (auto simp add: INum_def normNum_def split_def Let_def)}
|
haftmann@24197
|
259 |
ultimately show ?thesis by blast
|
haftmann@24197
|
260 |
qed
|
haftmann@24197
|
261 |
|
haftmann@36409
|
262 |
lemma INum_normNum_iff: "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y \<longleftrightarrow> normNum x = normNum y" (is "?lhs = ?rhs")
|
haftmann@24197
|
263 |
proof -
|
haftmann@24197
|
264 |
have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
|
haftmann@24197
|
265 |
by (simp del: normNum)
|
haftmann@24197
|
266 |
also have "\<dots> = ?lhs" by simp
|
haftmann@24197
|
267 |
finally show ?thesis by simp
|
haftmann@24197
|
268 |
qed
|
haftmann@24197
|
269 |
|
haftmann@36409
|
270 |
lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})"
|
haftmann@24197
|
271 |
proof-
|
haftmann@24197
|
272 |
let ?z = "0:: 'a"
|
haftmann@24197
|
273 |
have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
|
haftmann@24197
|
274 |
then obtain a b a' b' where x[simp]: "x = (a,b)"
|
haftmann@24197
|
275 |
and y[simp]: "y = (a',b')" by blast
|
haftmann@24197
|
276 |
{assume "a=0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" hence ?thesis
|
haftmann@24197
|
277 |
apply (cases "a=0",simp_all add: Nadd_def)
|
haftmann@24197
|
278 |
apply (cases "b= 0",simp_all add: INum_def)
|
haftmann@24197
|
279 |
apply (cases "a'= 0",simp_all)
|
haftmann@24197
|
280 |
apply (cases "b'= 0",simp_all)
|
haftmann@24197
|
281 |
done }
|
haftmann@24197
|
282 |
moreover
|
haftmann@24197
|
283 |
{assume aa':"a \<noteq> 0" "a'\<noteq> 0" and bb': "b \<noteq> 0" "b' \<noteq> 0"
|
haftmann@24197
|
284 |
{assume z: "a * b' + b * a' = 0"
|
haftmann@24197
|
285 |
hence "of_int (a*b' + b*a') / (of_int b* of_int b') = ?z" by simp
|
haftmann@24197
|
286 |
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)
|
haftmann@24197
|
287 |
hence th: "of_int a / of_int b + of_int a' / of_int b' = ?z" using bb' aa' by simp
|
haftmann@24197
|
288 |
from z aa' bb' have ?thesis
|
wenzelm@32962
|
289 |
by (simp add: th Nadd_def normNum_def INum_def split_def)}
|
haftmann@24197
|
290 |
moreover {assume z: "a * b' + b * a' \<noteq> 0"
|
huffman@31704
|
291 |
let ?g = "gcd (a * b' + b * a') (b*b')"
|
haftmann@24197
|
292 |
have gz: "?g \<noteq> 0" using z by simp
|
haftmann@24197
|
293 |
have ?thesis using aa' bb' z gz
|
wenzelm@32962
|
294 |
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,
|
wenzelm@32962
|
295 |
OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]
|
wenzelm@41776
|
296 |
by (simp add: Nadd_def INum_def normNum_def Let_def add_divide_distrib)}
|
haftmann@24197
|
297 |
ultimately have ?thesis using aa' bb'
|
wenzelm@41776
|
298 |
by (simp add: Nadd_def INum_def normNum_def Let_def) }
|
haftmann@24197
|
299 |
ultimately show ?thesis by blast
|
haftmann@24197
|
300 |
qed
|
haftmann@24197
|
301 |
|
haftmann@36409
|
302 |
lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero}) "
|
haftmann@24197
|
303 |
proof-
|
haftmann@24197
|
304 |
let ?z = "0::'a"
|
haftmann@24197
|
305 |
have " \<exists> a b. x = (a,b)" " \<exists> a' b'. y = (a',b')" by auto
|
haftmann@24197
|
306 |
then obtain a b a' b' where x: "x = (a,b)" and y: "y = (a',b')" by blast
|
haftmann@24197
|
307 |
{assume "a=0 \<or> a'= 0 \<or> b = 0 \<or> b' = 0" hence ?thesis
|
haftmann@24197
|
308 |
apply (cases "a=0",simp_all add: x y Nmul_def INum_def Let_def)
|
haftmann@24197
|
309 |
apply (cases "b=0",simp_all)
|
haftmann@24197
|
310 |
apply (cases "a'=0",simp_all)
|
haftmann@24197
|
311 |
done }
|
haftmann@24197
|
312 |
moreover
|
haftmann@24197
|
313 |
{assume z: "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
|
huffman@31704
|
314 |
let ?g="gcd (a*a') (b*b')"
|
haftmann@24197
|
315 |
have gz: "?g \<noteq> 0" using z by simp
|
nipkow@31952
|
316 |
from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]
|
nipkow@31952
|
317 |
of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]]
|
haftmann@24197
|
318 |
have ?thesis by (simp add: Nmul_def x y Let_def INum_def)}
|
haftmann@24197
|
319 |
ultimately show ?thesis by blast
|
haftmann@24197
|
320 |
qed
|
haftmann@24197
|
321 |
|
haftmann@24197
|
322 |
lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
|
haftmann@24197
|
323 |
by (simp add: Nneg_def split_def INum_def)
|
haftmann@24197
|
324 |
|
haftmann@36409
|
325 |
lemma Nsub[simp]: shows "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"
|
haftmann@24197
|
326 |
by (simp add: Nsub_def split_def)
|
haftmann@24197
|
327 |
|
haftmann@36409
|
328 |
lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)"
|
haftmann@24197
|
329 |
by (simp add: Ninv_def INum_def split_def)
|
haftmann@24197
|
330 |
|
haftmann@36409
|
331 |
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)
|
haftmann@24197
|
332 |
|
haftmann@24197
|
333 |
lemma Nlt0_iff[simp]: assumes nx: "isnormNum x"
|
haftmann@36409
|
334 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>\<^sub>N x "
|
haftmann@24197
|
335 |
proof-
|
haftmann@24197
|
336 |
have " \<exists> a b. x = (a,b)" by simp
|
haftmann@24197
|
337 |
then obtain a b where x[simp]:"x = (a,b)" by blast
|
haftmann@24197
|
338 |
{assume "a = 0" hence ?thesis by (simp add: Nlt0_def INum_def) }
|
haftmann@24197
|
339 |
moreover
|
haftmann@24197
|
340 |
{assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
|
haftmann@24197
|
341 |
from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]
|
haftmann@24197
|
342 |
have ?thesis by (simp add: Nlt0_def INum_def)}
|
haftmann@24197
|
343 |
ultimately show ?thesis by blast
|
haftmann@24197
|
344 |
qed
|
haftmann@24197
|
345 |
|
haftmann@24197
|
346 |
lemma Nle0_iff[simp]:assumes nx: "isnormNum x"
|
haftmann@36409
|
347 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<le> 0) = 0\<ge>\<^sub>N x"
|
haftmann@24197
|
348 |
proof-
|
haftmann@24197
|
349 |
have " \<exists> a b. x = (a,b)" by simp
|
haftmann@24197
|
350 |
then obtain a b where x[simp]:"x = (a,b)" by blast
|
haftmann@24197
|
351 |
{assume "a = 0" hence ?thesis by (simp add: Nle0_def INum_def) }
|
haftmann@24197
|
352 |
moreover
|
haftmann@24197
|
353 |
{assume a: "a\<noteq>0" hence b: "(of_int b :: 'a) > 0" using nx by (simp add: isnormNum_def)
|
haftmann@24197
|
354 |
from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]
|
haftmann@24197
|
355 |
have ?thesis by (simp add: Nle0_def INum_def)}
|
haftmann@24197
|
356 |
ultimately show ?thesis by blast
|
haftmann@24197
|
357 |
qed
|
haftmann@24197
|
358 |
|
haftmann@36409
|
359 |
lemma Ngt0_iff[simp]:assumes nx: "isnormNum x" shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<\<^sub>N x"
|
haftmann@24197
|
360 |
proof-
|
haftmann@24197
|
361 |
have " \<exists> a b. x = (a,b)" by simp
|
haftmann@24197
|
362 |
then obtain a b where x[simp]:"x = (a,b)" by blast
|
haftmann@24197
|
363 |
{assume "a = 0" hence ?thesis by (simp add: Ngt0_def INum_def) }
|
haftmann@24197
|
364 |
moreover
|
haftmann@24197
|
365 |
{assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
|
haftmann@24197
|
366 |
from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]
|
haftmann@24197
|
367 |
have ?thesis by (simp add: Ngt0_def INum_def)}
|
haftmann@24197
|
368 |
ultimately show ?thesis by blast
|
haftmann@24197
|
369 |
qed
|
haftmann@24197
|
370 |
lemma Nge0_iff[simp]:assumes nx: "isnormNum x"
|
haftmann@36409
|
371 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) \<ge> 0) = 0\<le>\<^sub>N x"
|
haftmann@24197
|
372 |
proof-
|
haftmann@24197
|
373 |
have " \<exists> a b. x = (a,b)" by simp
|
haftmann@24197
|
374 |
then obtain a b where x[simp]:"x = (a,b)" by blast
|
haftmann@24197
|
375 |
{assume "a = 0" hence ?thesis by (simp add: Nge0_def INum_def) }
|
haftmann@24197
|
376 |
moreover
|
haftmann@24197
|
377 |
{assume a: "a\<noteq>0" hence b: "(of_int b::'a) > 0" using nx by (simp add: isnormNum_def)
|
haftmann@24197
|
378 |
from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]
|
haftmann@24197
|
379 |
have ?thesis by (simp add: Nge0_def INum_def)}
|
haftmann@24197
|
380 |
ultimately show ?thesis by blast
|
haftmann@24197
|
381 |
qed
|
haftmann@24197
|
382 |
|
haftmann@24197
|
383 |
lemma Nlt_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
|
haftmann@36409
|
384 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <\<^sub>N y)"
|
haftmann@24197
|
385 |
proof-
|
haftmann@24197
|
386 |
let ?z = "0::'a"
|
haftmann@24197
|
387 |
have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)" using nx ny by simp
|
haftmann@24197
|
388 |
also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))" using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
|
haftmann@24197
|
389 |
finally show ?thesis by (simp add: Nlt_def)
|
haftmann@24197
|
390 |
qed
|
haftmann@24197
|
391 |
|
haftmann@24197
|
392 |
lemma Nle_iff[simp]: assumes nx: "isnormNum x" and ny: "isnormNum y"
|
haftmann@36409
|
393 |
shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})\<le> INum y) = (x \<le>\<^sub>N y)"
|
haftmann@24197
|
394 |
proof-
|
haftmann@24197
|
395 |
have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))" using nx ny by simp
|
haftmann@24197
|
396 |
also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))" using Nle0_iff[OF Nsub_normN[OF ny]] by simp
|
haftmann@24197
|
397 |
finally show ?thesis by (simp add: Nle_def)
|
haftmann@24197
|
398 |
qed
|
haftmann@24197
|
399 |
|
wenzelm@28615
|
400 |
lemma Nadd_commute:
|
haftmann@36409
|
401 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
402 |
shows "x +\<^sub>N y = y +\<^sub>N x"
|
haftmann@24197
|
403 |
proof-
|
haftmann@24197
|
404 |
have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
|
chaieb@31964
|
405 |
have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp
|
haftmann@24197
|
406 |
with isnormNum_unique[OF n] show ?thesis by simp
|
haftmann@24197
|
407 |
qed
|
haftmann@24197
|
408 |
|
wenzelm@28615
|
409 |
lemma [simp]:
|
haftmann@36409
|
410 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
411 |
shows "(0, b) +\<^sub>N y = normNum y"
|
wenzelm@28615
|
412 |
and "(a, 0) +\<^sub>N y = normNum y"
|
wenzelm@28615
|
413 |
and "x +\<^sub>N (0, b) = normNum x"
|
wenzelm@28615
|
414 |
and "x +\<^sub>N (a, 0) = normNum x"
|
wenzelm@28615
|
415 |
apply (simp add: Nadd_def split_def)
|
wenzelm@28615
|
416 |
apply (simp add: Nadd_def split_def)
|
wenzelm@28615
|
417 |
apply (subst Nadd_commute, simp add: Nadd_def split_def)
|
wenzelm@28615
|
418 |
apply (subst Nadd_commute, simp add: Nadd_def split_def)
|
haftmann@24197
|
419 |
done
|
haftmann@24197
|
420 |
|
wenzelm@28615
|
421 |
lemma normNum_nilpotent_aux[simp]:
|
haftmann@36409
|
422 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
423 |
assumes nx: "isnormNum x"
|
haftmann@24197
|
424 |
shows "normNum x = x"
|
haftmann@24197
|
425 |
proof-
|
haftmann@24197
|
426 |
let ?a = "normNum x"
|
haftmann@24197
|
427 |
have n: "isnormNum ?a" by simp
|
chaieb@31964
|
428 |
have th:"INum ?a = (INum x ::'a)" by simp
|
haftmann@24197
|
429 |
with isnormNum_unique[OF n nx]
|
haftmann@24197
|
430 |
show ?thesis by simp
|
haftmann@24197
|
431 |
qed
|
haftmann@24197
|
432 |
|
wenzelm@28615
|
433 |
lemma normNum_nilpotent[simp]:
|
haftmann@36409
|
434 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
435 |
shows "normNum (normNum x) = normNum x"
|
haftmann@24197
|
436 |
by simp
|
wenzelm@28615
|
437 |
|
haftmann@24197
|
438 |
lemma normNum0[simp]: "normNum (0,b) = 0\<^sub>N" "normNum (a,0) = 0\<^sub>N"
|
haftmann@24197
|
439 |
by (simp_all add: normNum_def)
|
wenzelm@28615
|
440 |
|
wenzelm@28615
|
441 |
lemma normNum_Nadd:
|
haftmann@36409
|
442 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
443 |
shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
|
wenzelm@28615
|
444 |
|
wenzelm@28615
|
445 |
lemma Nadd_normNum1[simp]:
|
haftmann@36409
|
446 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
447 |
shows "normNum x +\<^sub>N y = x +\<^sub>N y"
|
haftmann@24197
|
448 |
proof-
|
haftmann@24197
|
449 |
have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
|
chaieb@31964
|
450 |
have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
|
haftmann@24197
|
451 |
also have "\<dots> = INum (x +\<^sub>N y)" by simp
|
haftmann@24197
|
452 |
finally show ?thesis using isnormNum_unique[OF n] by simp
|
haftmann@24197
|
453 |
qed
|
haftmann@24197
|
454 |
|
wenzelm@28615
|
455 |
lemma Nadd_normNum2[simp]:
|
haftmann@36409
|
456 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
457 |
shows "x +\<^sub>N normNum y = x +\<^sub>N y"
|
wenzelm@28615
|
458 |
proof-
|
wenzelm@28615
|
459 |
have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
|
chaieb@31964
|
460 |
have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
|
wenzelm@28615
|
461 |
also have "\<dots> = INum (x +\<^sub>N y)" by simp
|
wenzelm@28615
|
462 |
finally show ?thesis using isnormNum_unique[OF n] by simp
|
wenzelm@28615
|
463 |
qed
|
wenzelm@28615
|
464 |
|
wenzelm@28615
|
465 |
lemma Nadd_assoc:
|
haftmann@36409
|
466 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
467 |
shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
|
haftmann@24197
|
468 |
proof-
|
haftmann@24197
|
469 |
have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
|
chaieb@31964
|
470 |
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
|
haftmann@24197
|
471 |
with isnormNum_unique[OF n] show ?thesis by simp
|
haftmann@24197
|
472 |
qed
|
haftmann@24197
|
473 |
|
haftmann@24197
|
474 |
lemma Nmul_commute: "isnormNum x \<Longrightarrow> isnormNum y \<Longrightarrow> x *\<^sub>N y = y *\<^sub>N x"
|
nipkow@31952
|
475 |
by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute)
|
haftmann@24197
|
476 |
|
wenzelm@28615
|
477 |
lemma Nmul_assoc:
|
haftmann@36409
|
478 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
479 |
assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"
|
haftmann@24197
|
480 |
shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
|
haftmann@24197
|
481 |
proof-
|
haftmann@24197
|
482 |
from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
|
haftmann@24197
|
483 |
by simp_all
|
chaieb@31964
|
484 |
have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
|
haftmann@24197
|
485 |
with isnormNum_unique[OF n] show ?thesis by simp
|
haftmann@24197
|
486 |
qed
|
haftmann@24197
|
487 |
|
wenzelm@28615
|
488 |
lemma Nsub0:
|
haftmann@36409
|
489 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
490 |
assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
|
haftmann@24197
|
491 |
proof-
|
wenzelm@28615
|
492 |
{ fix h :: 'a
|
chaieb@31964
|
493 |
from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
|
chaieb@31964
|
494 |
have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
|
chaieb@31964
|
495 |
also have "\<dots> = (INum x = (INum y :: 'a))" by simp
|
haftmann@24197
|
496 |
also have "\<dots> = (x = y)" using x y by simp
|
wenzelm@28615
|
497 |
finally show ?thesis . }
|
haftmann@24197
|
498 |
qed
|
haftmann@24197
|
499 |
|
haftmann@24197
|
500 |
lemma Nmul0[simp]: "c *\<^sub>N 0\<^sub>N = 0\<^sub>N" " 0\<^sub>N *\<^sub>N c = 0\<^sub>N"
|
haftmann@24197
|
501 |
by (simp_all add: Nmul_def Let_def split_def)
|
haftmann@24197
|
502 |
|
wenzelm@28615
|
503 |
lemma Nmul_eq0[simp]:
|
haftmann@36409
|
504 |
assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
|
wenzelm@28615
|
505 |
assumes nx:"isnormNum x" and ny: "isnormNum y"
|
haftmann@24197
|
506 |
shows "(x*\<^sub>N y = 0\<^sub>N) = (x = 0\<^sub>N \<or> y = 0\<^sub>N)"
|
haftmann@24197
|
507 |
proof-
|
wenzelm@28615
|
508 |
{ fix h :: 'a
|
wenzelm@28615
|
509 |
have " \<exists> a b a' b'. x = (a,b) \<and> y= (a',b')" by auto
|
wenzelm@28615
|
510 |
then obtain a b a' b' where xy[simp]: "x = (a,b)" "y = (a',b')" by blast
|
wenzelm@28615
|
511 |
have n0: "isnormNum 0\<^sub>N" by simp
|
wenzelm@28615
|
512 |
show ?thesis using nx ny
|
wenzelm@28615
|
513 |
apply (simp only: isnormNum_unique[where ?'a = 'a, OF Nmul_normN[OF nx ny] n0, symmetric] Nmul[where ?'a = 'a])
|
wenzelm@41776
|
514 |
by (simp add: INum_def split_def isnormNum_def split: split_if_asm)
|
wenzelm@28615
|
515 |
}
|
haftmann@24197
|
516 |
qed
|
haftmann@24197
|
517 |
lemma Nneg_Nneg[simp]: "~\<^sub>N (~\<^sub>N c) = c"
|
haftmann@24197
|
518 |
by (simp add: Nneg_def split_def)
|
haftmann@24197
|
519 |
|
haftmann@24197
|
520 |
lemma Nmul1[simp]:
|
haftmann@24197
|
521 |
"isnormNum c \<Longrightarrow> 1\<^sub>N *\<^sub>N c = c"
|
wenzelm@41776
|
522 |
"isnormNum c \<Longrightarrow> c *\<^sub>N (1\<^sub>N) = c"
|
haftmann@24197
|
523 |
apply (simp_all add: Nmul_def Let_def split_def isnormNum_def)
|
wenzelm@28615
|
524 |
apply (cases "fst c = 0", simp_all, cases c, simp_all)+
|
wenzelm@28615
|
525 |
done
|
haftmann@24197
|
526 |
|
wenzelm@28615
|
527 |
end
|