--- a/src/HOL/Decision_Procs/Rat_Pair.thy Sat Jun 20 20:17:29 2015 +0200
+++ b/src/HOL/Decision_Procs/Rat_Pair.thy Sat Jun 20 20:55:31 2015 +0200
@@ -16,12 +16,14 @@
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 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
+definition normNum :: "Num \<Rightarrow> Num"
+where
"normNum = (\<lambda>(a,b).
- (if a=0 \<or> b = 0 then (0,0) else
+ (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)))))"
@@ -30,61 +32,72 @@
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"
+ consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0" by blast
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ by (simp add: x normNum_def isnormNum_def)
+ next
+ case 2
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]
+ from 2 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
+ from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] 2
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 2 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"
+ from 2 consider "b < 0" | "b > 0" by arith
+ then show ?thesis
+ proof cases
+ case 1
+ have False if b': "?b' \<ge> 0"
+ proof -
+ from gpos have th: "?g \<ge> 0" by arith
+ from mult_nonneg_nonneg[OF th b'] dvd_mult_div_cancel[OF gdvd(2)]
+ show ?thesis using 1 by arith
+ qed
+ then have b': "?b' < 0" by (presburger add: linorder_not_le[symmetric])
+ from \<open>a \<noteq> 0\<close> nz' 1 b' gp1 show ?thesis
+ by (simp add: x isnormNum_def normNum_def Let_def split_def)
+ next
+ case 2
+ then 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
+ from 2 b' \<open>a \<noteq> 0\<close> nz' gp1 show ?thesis
+ by (simp add: x isnormNum_def normNum_def Let_def split_def)
+ qed
+ qed
qed
text \<open>Arithmetic over Num\<close>
-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 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 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))"
+ where "Nneg = (\<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))"
+ 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)"
@@ -95,53 +108,59 @@
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)"
+lemma Nsub_normN[simp]: "isnormNum y \<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"
+ 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)
+ consider "a = 0" | "a' = 0" | "a \<noteq> 0" "a' \<noteq> 0" by blast
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ using xn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
+ next
+ case 2
+ then show ?thesis
+ using yn x y by (simp add: isnormNum_def Let_def Nmul_def split_def)
+ next
+ case 3
+ then have bp: "b > 0" "b' > 0"
+ using xn yn x y by (simp_all add: isnormNum_def)
from 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
+ using x y 3 bp by (simp add: Nmul_def Let_def split_def normNum_def)
+ then show ?thesis by simp
+ qed
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)
+ apply (simp add: Ninv_def isnormNum_def split_def)
+ apply (cases "fst x = 0")
+ apply (auto simp add: gcd_commute_int)
+ done
-lemma isnormNum_int[simp]:
- "isnormNum 0\<^sub>N" "isnormNum ((1::int)\<^sub>N)" "i \<noteq> 0 \<Longrightarrow> isnormNum (i)\<^sub>N"
+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 \<open>Relations over Num\<close>
definition Nlt0:: "Num \<Rightarrow> bool" ("0>\<^sub>N")
- where "Nlt0 = (\<lambda>(a,b). a < 0)"
+ where "Nlt0 = (\<lambda>(a, b). a < 0)"
definition Nle0:: "Num \<Rightarrow> bool" ("0\<ge>\<^sub>N")
- where "Nle0 = (\<lambda>(a,b). a \<le> 0)"
+ where "Nle0 = (\<lambda>(a, b). a \<le> 0)"
definition Ngt0:: "Num \<Rightarrow> bool" ("0<\<^sub>N")
- where "Ngt0 = (\<lambda>(a,b). a > 0)"
+ where "Ngt0 = (\<lambda>(a, b). a > 0)"
definition Nge0:: "Num \<Rightarrow> bool" ("0\<le>\<^sub>N")
- where "Nge0 = (\<lambda>(a,b). a \<ge> 0)"
+ 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))"
@@ -149,27 +168,34 @@
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)"
+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)"
+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}) = INum y) = (x = y)" (is "?lhs = ?rhs")
+ assumes na: "isnormNum x"
+ and nb: "isnormNum y"
+ shows "(INum x ::'a::{field_char_0,field}) = INum y \<longleftrightarrow> 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"
+ consider "a = 0 \<or> b = 0 \<or> a' = 0 \<or> b' = 0" | "a \<noteq> 0" "b \<noteq> 0" "a' \<noteq> 0" "b' \<noteq> 0"
+ by blast
+ then show ?rhs if H: ?lhs
+ proof cases
+ case 1
+ then show ?thesis
+ using na nb H by (simp add: x y INum_def split_def isnormNum_def split: split_if_asm)
+ next
+ case 2
+ with na nb have pos: "b > 0" "b' > 0"
+ by (simp_all add: x y isnormNum_def)
+ from H \<open>b \<noteq> 0\<close> \<open>b' \<noteq> 0\<close> 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"
+ from \<open>a \<noteq> 0\<close> \<open>a' \<noteq> 0\<close> 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 -
@@ -182,23 +208,22 @@
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
+ with eq1 show ?thesis by (simp add: x y)
+ qed
+ show ?lhs if ?rhs
+ using that by simp
qed
-
-lemma isnormNum0[simp]:
- "isnormNum x \<Longrightarrow> (INum x = (0::'a::{field_char_0, field})) = (x = 0\<^sub>N)"
+lemma isnormNum0[simp]: "isnormNum x \<Longrightarrow> INum x = (0::'a::{field_char_0,field}) \<longleftrightarrow> 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)"
+lemma of_int_div_aux:
+ assumes "d \<noteq> 0"
+ shows "(of_int x ::'a::field_char_0) / of_int d =
+ of_int (x div d) + (of_int (x mod d)) / of_int d"
proof -
- assume "d ~= 0"
- let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"
+ 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
@@ -206,30 +231,36 @@
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 \<open>d ~= 0\<close>)
+ then show ?thesis
+ by (simp add: add_divide_distrib algebra_simps \<open>d \<noteq> 0\<close>)
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"
- using of_int_div_aux [of d n, where ?'a = 'a] by simp
+lemma of_int_div:
+ fixes d :: int
+ assumes "d \<noteq> 0" "d dvd n"
+ shows "(of_int (n div d) ::'a::field_char_0) = of_int n / of_int d"
+ using assms of_int_div_aux [of d n, where ?'a = 'a] by simp
-lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field})"
+lemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0,field})"
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"
+ consider "a = 0 \<or> b = 0" | "a \<noteq> 0" "b \<noteq> 0" by blast
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ by (simp add: x INum_def normNum_def split_def Let_def)
+ next
+ case 2
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
+ from 2 have g: "?g \<noteq> 0"by simp
+ from of_int_div[OF g, where ?'a = 'a] show ?thesis
+ by (auto simp add: x INum_def normNum_def split_def Let_def)
+ qed
qed
-lemma INum_normNum_iff:
- "(INum x ::'a::{field_char_0, field}) = INum y \<longleftrightarrow> normNum x = normNum y"
- (is "?lhs = ?rhs")
+lemma INum_normNum_iff: "(INum x ::'a::{field_char_0,field}) = INum y \<longleftrightarrow> normNum x = normNum y"
+ (is "?lhs \<longleftrightarrow> _")
proof -
have "normNum x = normNum y \<longleftrightarrow> (INum (normNum x) :: 'a) = INum (normNum y)"
by (simp del: normNum)
@@ -237,178 +268,231 @@
finally show ?thesis by simp
qed
-lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0, field})"
-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 [of "a * b' + b * a'" "b * b'"]]
- of_int_div [where ?'a = 'a, OF gz gcd_dvd2 [of "a * b' + b * a'" "b * b'"]]
- by (simp add: x y Nadd_def INum_def normNum_def Let_def) (simp add: field_simps)
- }
- 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})"
+lemma Nadd[simp]: "INum (x +\<^sub>N y) = INum x + (INum y :: 'a :: {field_char_0,field})"
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 [of "a * a'" "b * b'"]]
- of_int_div [where ?'a = 'a , OF gz gcd_dvd2 [of "a * a'" "b * b'"]]
- have ?thesis by (simp add: Nmul_def x y Let_def INum_def) }
- ultimately show ?thesis by blast
+ consider "a = 0 \<or> a'= 0 \<or> b =0 \<or> b' = 0" | "a \<noteq> 0" "a'\<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
+ by blast
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ apply (cases "a = 0")
+ apply (simp_all add: x y Nadd_def)
+ apply (cases "b = 0")
+ apply (simp_all add: INum_def)
+ apply (cases "a'= 0")
+ apply simp_all
+ apply (cases "b'= 0")
+ apply simp_all
+ done
+ next
+ case 2
+ show ?thesis
+ proof (cases "a * b' + b * a' = 0")
+ case True
+ then have "of_int (a * b' + b * a') / (of_int b * of_int b') = ?z"
+ by simp
+ then have "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)
+ then have th: "of_int a / of_int b + of_int a' / of_int b' = ?z"
+ using 2 by simp
+ from True 2 show ?thesis
+ by (simp add: x y th Nadd_def normNum_def INum_def split_def)
+ next
+ case False
+ let ?g = "gcd (a * b' + b * a') (b * b')"
+ have gz: "?g \<noteq> 0"
+ using False by simp
+ show ?thesis
+ using 2 False gz
+ of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * b' + b * a'" "b * b'"]]
+ of_int_div [where ?'a = 'a, OF gz gcd_dvd2 [of "a * b' + b * a'" "b * b'"]]
+ by (simp add: x y Nadd_def INum_def normNum_def Let_def) (simp add: field_simps)
+ qed
+ qed
qed
-lemma Nneg[simp]: "INum (~\<^sub>N x) = - (INum x ::'a:: field)"
+lemma Nmul[simp]: "INum (x *\<^sub>N y) = INum x * (INum y:: 'a::{field_char_0,field})"
+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)
+ consider "a = 0 \<or> a' = 0 \<or> b = 0 \<or> b' = 0" | "a \<noteq> 0" "a' \<noteq> 0" "b \<noteq> 0" "b' \<noteq> 0"
+ by blast
+ then show ?thesis
+ proof cases
+ case 1
+ then show ?thesis
+ apply (cases "a = 0")
+ apply (simp_all add: x y Nmul_def INum_def Let_def)
+ apply (cases "b = 0")
+ apply simp_all
+ apply (cases "a' = 0")
+ apply simp_all
+ done
+ next
+ case 2
+ let ?g = "gcd (a * a') (b * b')"
+ have gz: "?g \<noteq> 0"
+ using 2 by simp
+ from 2 of_int_div [where ?'a = 'a, OF gz gcd_dvd1 [of "a * a'" "b * b'"]]
+ of_int_div [where ?'a = 'a , OF gz gcd_dvd2 [of "a * a'" "b * b'"]]
+ show ?thesis
+ by (simp add: Nmul_def x y Let_def INum_def)
+ qed
+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})"
+lemma Nsub[simp]: "INum (x -\<^sub>N y) = INum x - (INum y:: 'a :: {field_char_0,field})"
by (simp add: Nsub_def split_def)
lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field) / (INum x)"
by (simp add: Ninv_def INum_def split_def)
-lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0, field})"
+lemma Ndiv[simp]: "INum (x \<div>\<^sub>N y) = INum x / (INum y ::'a :: {field_char_0,field})"
by (simp add: Ndiv_def)
lemma Nlt0_iff[simp]:
assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field})< 0) = 0>\<^sub>N x"
+ shows "((INum x :: 'a::{field_char_0,linordered_field})< 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"
+ show ?thesis
+ proof (cases "a = 0")
+ case True
+ then show ?thesis
+ by (simp add: x Nlt0_def INum_def)
+ next
+ case False
+ then have 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
+ show ?thesis
+ by (simp add: x Nlt0_def INum_def)
+ qed
qed
lemma Nle0_iff[simp]:
assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field}) \<le> 0) = 0\<ge>\<^sub>N x"
+ shows "((INum x :: 'a::{field_char_0,linordered_field}) \<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"
+ show ?thesis
+ proof (cases "a = 0")
+ case True
+ then show ?thesis
+ by (simp add: x Nle0_def INum_def)
+ next
+ case False
+ then have 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
+ show ?thesis
+ by (simp add: x Nle0_def INum_def)
+ qed
qed
lemma Ngt0_iff[simp]:
assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field})> 0) = 0<\<^sub>N x"
+ shows "((INum x :: 'a::{field_char_0,linordered_field})> 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)
+ show ?thesis
+ proof (cases "a = 0")
+ case True
+ then show ?thesis
+ by (simp add: x Ngt0_def INum_def)
+ next
+ case False
+ then have 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
+ show ?thesis
+ by (simp add: x Ngt0_def INum_def)
+ qed
qed
lemma Nge0_iff[simp]:
assumes nx: "isnormNum x"
- shows "((INum x :: 'a :: {field_char_0, linordered_field}) \<ge> 0) = 0\<le>\<^sub>N x"
+ shows "((INum x :: 'a::{field_char_0,linordered_field}) \<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)
+ show ?thesis
+ proof (cases "a = 0")
+ case True
+ then show ?thesis
+ by (simp add: x Nge0_def INum_def)
+ next
+ case False
+ then have 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
+ show ?thesis
+ by (simp add: x Nge0_def INum_def)
+ qed
qed
lemma Nlt_iff[simp]:
- assumes nx: "isnormNum x" and ny: "isnormNum y"
- shows "((INum x :: 'a :: {field_char_0, linordered_field}) < INum y) = (x <\<^sub>N y)"
+ assumes nx: "isnormNum x"
+ and ny: "isnormNum y"
+ shows "((INum x :: 'a::{field_char_0,linordered_field}) < INum y) = (x <\<^sub>N y)"
proof -
let ?z = "0::'a"
have "((INum x ::'a) < INum y) = (INum (x -\<^sub>N y) < ?z)"
using nx ny by simp
also have "\<dots> = (0>\<^sub>N (x -\<^sub>N y))"
using Nlt0_iff[OF Nsub_normN[OF ny]] by simp
- finally show ?thesis by (simp add: Nlt_def)
+ 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})\<le> INum y) = (x \<le>\<^sub>N y)"
+ assumes nx: "isnormNum x"
+ and ny: "isnormNum y"
+ shows "((INum x :: 'a::{field_char_0,linordered_field})\<le> INum y) = (x \<le>\<^sub>N y)"
proof -
have "((INum x ::'a) \<le> INum y) = (INum (x -\<^sub>N y) \<le> (0::'a))"
using nx ny by simp
also have "\<dots> = (0\<ge>\<^sub>N (x -\<^sub>N y))"
using Nle0_iff[OF Nsub_normN[OF ny]] by simp
- finally show ?thesis by (simp add: Nle_def)
+ finally show ?thesis
+ by (simp add: Nle_def)
qed
lemma Nadd_commute:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
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
+ 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})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
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)
+ apply (subst Nadd_commute)
+ apply (simp add: Nadd_def split_def)
+ apply (subst Nadd_commute)
+ apply (simp add: Nadd_def split_def)
done
lemma normNum_nilpotent_aux[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
assumes nx: "isnormNum x"
shows "normNum x = x"
proof -
@@ -419,7 +503,7 @@
qed
lemma normNum_nilpotent[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
shows "normNum (normNum x) = normNum x"
by simp
@@ -427,11 +511,12 @@
by (simp_all add: normNum_def)
lemma normNum_Nadd:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
- shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
+ shows "normNum (x +\<^sub>N y) = x +\<^sub>N y"
+ by simp
lemma Nadd_normNum1[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
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
@@ -441,7 +526,7 @@
qed
lemma Nadd_normNum2[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
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
@@ -451,7 +536,7 @@
qed
lemma Nadd_assoc:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
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
@@ -463,8 +548,10 @@
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})"
- assumes nx: "isnormNum x" and ny: "isnormNum y" and nz: "isnormNum z"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
+ 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))"
@@ -474,11 +561,11 @@
qed
lemma Nsub0:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
- assumes x: "isnormNum x" and y: "isnormNum y"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
+ 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
@@ -490,11 +577,11 @@
by (simp_all add: Nmul_def Let_def split_def)
lemma Nmul_eq0[simp]:
- assumes "SORT_CONSTRAINT('a::{field_char_0, field})"
- assumes nx: "isnormNum x" and ny: "isnormNum y"
+ assumes "SORT_CONSTRAINT('a::{field_char_0,field})"
+ 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
@@ -508,9 +595,7 @@
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"
+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