# Theory Abstract_Rat

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theory Abstract_Rat
imports Complex_Main
`(*  Title:      HOL/Library/Abstract_Rat.thy    Author:     Amine Chaieb*)header {* Abstract rational numbers *}theory Abstract_Ratimports Complex_Mainbegintype_synonym Num = "int × int"abbreviation Num0_syn :: Num  ("0⇩N")  where "0⇩N ≡ (0, 0)"abbreviation Numi_syn :: "int => Num"  ("'((_)')⇩N")  where "(i)⇩N ≡ (i, 1)"definition isnormNum :: "Num => bool" where  "isnormNum = (λ(a,b). (if a = 0 then b = 0 else b > 0 ∧ gcd a b = 1))"definition normNum :: "Num => Num" where  "normNum = (λ(a,b).    (if a=0 ∨ b = 0 then (0,0) else      (let g = gcd a b       in if b > 0 then (a div g, b div g) else (- (a div g), - (b div g)))))"declare gcd_dvd1_int[presburger] gcd_dvd2_int[presburger]lemma normNum_isnormNum [simp]: "isnormNum (normNum x)"proof -  obtain a b where x: "x = (a, b)" by (cases x)  { assume "a=0 ∨ b = 0" hence ?thesis by (simp add: x normNum_def isnormNum_def) }  moreover  { assume anz: "a ≠ 0" and bnz: "b ≠ 0"    let ?g = "gcd a b"    let ?a' = "a div ?g"    let ?b' = "b div ?g"    let ?g' = "gcd ?a' ?b'"    from anz bnz have "?g ≠ 0" by simp  with gcd_ge_0_int[of a b]    have gpos: "?g > 0" by arith    have gdvd: "?g dvd a" "?g dvd b" by arith+    from dvd_mult_div_cancel[OF gdvd(1)] dvd_mult_div_cancel[OF gdvd(2)] anz bnz    have nz': "?a' ≠ 0" "?b' ≠ 0" by - (rule notI, simp)+    from anz bnz have stupid: "a ≠ 0 ∨ b ≠ 0" by arith    from div_gcd_coprime_int[OF stupid] have gp1: "?g' = 1" .    from bnz have "b < 0 ∨ b > 0" by arith    moreover    { assume b: "b > 0"      from b have "?b' ≥ 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' ≥ 0"        from gpos have th: "?g ≥ 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 blastqedtext {* Arithmetic over Num *}definition Nadd :: "Num => Num => Num"  (infixl "+⇩N" 60) where  "Nadd = (λ(a,b) (a',b'). if a = 0 ∨ b = 0 then normNum(a',b')    else if a'=0 ∨ b' = 0 then normNum(a,b)    else normNum(a*b' + b*a', b*b'))"definition Nmul :: "Num => Num => Num"  (infixl "*⇩N" 60) where  "Nmul = (λ(a,b) (a',b'). let g = gcd (a*a') (b*b')    in (a*a' div g, b*b' div g))"definition Nneg :: "Num => Num" ("~⇩N")  where "Nneg ≡ (λ(a,b). (-a,b))"definition Nsub :: "Num => Num => Num"  (infixl "-⇩N" 60)  where "Nsub = (λa b. a +⇩N ~⇩N b)"definition Ninv :: "Num => Num"  where "Ninv = (λ(a,b). if a < 0 then (-b, ¦a¦) else (b,a))"definition Ndiv :: "Num => Num => Num"  (infixl "÷⇩N" 60)  where "Ndiv = (λa b. a *⇩N Ninv b)"lemma Nneg_normN[simp]: "isnormNum x ==> isnormNum (~⇩N x)"  by (simp add: isnormNum_def Nneg_def split_def)lemma Nadd_normN[simp]: "isnormNum (x +⇩N y)"  by (simp add: Nadd_def split_def)lemma Nsub_normN[simp]: "[| isnormNum y|] ==> isnormNum (x -⇩N y)"  by (simp add: Nsub_def split_def)lemma Nmul_normN[simp]:  assumes xn: "isnormNum x" and yn: "isnormNum y"  shows "isnormNum (x *⇩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 ≠0" and a': "a'≠0"    hence bp: "b > 0" "b' > 0" using xn yn x y by (simp_all add: isnormNum_def)    from mult_pos_pos[OF bp] have "x *⇩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 blastqedlemma Ninv_normN[simp]: "isnormNum x ==> isnormNum (Ninv x)"  by (simp add: Ninv_def isnormNum_def split_def)    (cases "fst x = 0", auto simp add: gcd_commute_int)lemma isnormNum_int[simp]:  "isnormNum 0⇩N" "isnormNum ((1::int)⇩N)" "i ≠ 0 ==> isnormNum (i)⇩N"  by (simp_all add: isnormNum_def)text {* Relations over Num *}definition Nlt0:: "Num => bool"  ("0>⇩N")  where "Nlt0 = (λ(a,b). a < 0)"definition Nle0:: "Num => bool"  ("0≥⇩N")  where "Nle0 = (λ(a,b). a ≤ 0)"definition Ngt0:: "Num => bool"  ("0<⇩N")  where "Ngt0 = (λ(a,b). a > 0)"definition Nge0:: "Num => bool"  ("0≤⇩N")  where "Nge0 = (λ(a,b). a ≥ 0)"definition Nlt :: "Num => Num => bool"  (infix "<⇩N" 55)  where "Nlt = (λa b. 0>⇩N (a -⇩N b))"definition Nle :: "Num => Num => bool"  (infix "≤⇩N" 55)  where "Nle = (λa b. 0≥⇩N (a -⇩N b))"definition "INum = (λ(a,b). of_int a / of_int b)"lemma INum_int [simp]: "INum (i)⇩N = ((of_int i) ::'a::field)" "INum 0⇩N = (0::'a::field)"  by (simp_all add: INum_def)lemma isnormNum_unique[simp]:  assumes na: "isnormNum x" and nb: "isnormNum y"  shows "((INum x ::'a::{field_char_0, field_inverse_zero}) = INum y) = (x = y)" (is "?lhs = ?rhs")proof  obtain a b where x: "x = (a, b)" by (cases x)  obtain a' b' where y: "y = (a', b')" by (cases y)  assume H: ?lhs  { assume "a = 0 ∨ b = 0 ∨ a' = 0 ∨ 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 ≠ 0" and bz: "b ≠ 0" and a'z: "a'≠0" and b'z: "b'≠0"    from az bz a'z b'z na nb have pos: "b > 0" "b' > 0" by (simp_all add: x y isnormNum_def)    from H bz b'z have eq: "a * b' = a'*b"      by (simp add: x y INum_def eq_divide_eq divide_eq_eq of_int_mult[symmetric] del: of_int_mult)    from az a'z na nb have gcd1: "gcd a b = 1" "gcd b a = 1" "gcd a' b' = 1" "gcd b' a' = 1"      by (simp_all add: x y isnormNum_def add: gcd_commute_int)    from eq have raw_dvd: "a dvd a' * b" "b dvd b' * a" "a' dvd a * b'" "b' dvd b * a'"      apply -      apply algebra      apply algebra      apply simp      apply algebra      done    from zdvd_antisym_abs[OF coprime_dvd_mult_int[OF gcd1(2) raw_dvd(2)]        coprime_dvd_mult_int[OF gcd1(4) raw_dvd(4)]]      have eq1: "b = b'" using pos by arith      with eq have "a = a'" using pos by simp      with eq1 have ?rhs by (simp add: x y) }  ultimately show ?rhs by blastnext  assume ?rhs thus ?lhs by simpqedlemma isnormNum0[simp]:    "isnormNum x ==> (INum x = (0::'a::{field_char_0, field_inverse_zero})) = (x = 0⇩N)"  unfolding INum_int(2)[symmetric]  by (rule isnormNum_unique) simp_alllemma of_int_div_aux: "d ~= 0 ==> ((of_int x)::'a::field_char_0) / (of_int d) =    of_int (x div d) + (of_int (x mod d)) / ((of_int d)::'a)"proof -  assume "d ~= 0"  let ?t = "of_int (x div d) * ((of_int d)::'a) + of_int(x mod d)"  let ?f = "λx. x / of_int d"  have "x = (x div d) * d + x mod d"    by auto  then have eq: "of_int x = ?t"    by (simp only: of_int_mult[symmetric] of_int_add [symmetric])  then have "of_int x / of_int d = ?t / of_int d"    using cong[OF refl[of ?f] eq] by simp  then show ?thesis by (simp add: add_divide_distrib algebra_simps `d ~= 0`)qedlemma of_int_div: "(d::int) ~= 0 ==> d dvd n ==>    (of_int(n div d)::'a::field_char_0) = of_int n / of_int d"  apply (frule of_int_div_aux [of d n, where ?'a = 'a])  apply simp  apply (simp add: dvd_eq_mod_eq_0)  donelemma normNum[simp]: "INum (normNum x) = (INum x :: 'a::{field_char_0, field_inverse_zero})"proof -  obtain a b where x: "x = (a, b)" by (cases x)  { assume "a = 0 ∨ b = 0"    hence ?thesis by (simp add: x INum_def normNum_def split_def Let_def) }  moreover  { assume a: "a ≠ 0" and b: "b ≠ 0"    let ?g = "gcd a b"    from a b have g: "?g ≠ 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 blastqedlemma INum_normNum_iff:  "(INum x ::'a::{field_char_0, field_inverse_zero}) = INum y <-> normNum x = normNum y"  (is "?lhs = ?rhs")proof -  have "normNum x = normNum y <-> (INum (normNum x) :: 'a) = INum (normNum y)"    by (simp del: normNum)  also have "… = ?lhs" by simp  finally show ?thesis by simpqedlemma Nadd[simp]: "INum (x +⇩N y) = INum x + (INum y :: 'a :: {field_char_0, field_inverse_zero})"proof -  let ?z = "0:: 'a"  obtain a b where x: "x = (a, b)" by (cases x)  obtain a' b' where y: "y = (a', b')" by (cases y)  { assume "a=0 ∨ a'= 0 ∨ b =0 ∨ 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 ≠ 0" "a'≠ 0" and bb': "b ≠ 0" "b' ≠ 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' ≠ 0"      let ?g = "gcd (a * b' + b * a') (b*b')"      have gz: "?g ≠ 0" using z by simp      have ?thesis using aa' bb' z gz        of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a * b' + b * a'" and y="b*b'"]]        of_int_div[where ?'a = 'a, OF gz gcd_dvd2_int[where x="a * b' + b * a'" and y="b*b'"]]        by (simp add: x y Nadd_def INum_def normNum_def Let_def add_divide_distrib) }    ultimately have ?thesis using aa' bb'      by (simp add: x y Nadd_def INum_def normNum_def Let_def) }  ultimately show ?thesis by blastqedlemma Nmul[simp]: "INum (x *⇩N y) = INum x * (INum y:: 'a :: {field_char_0, field_inverse_zero})"proof -  let ?z = "0::'a"  obtain a b where x: "x = (a, b)" by (cases x)  obtain a' b' where y: "y = (a', b')" by (cases y)  { assume "a=0 ∨ a'= 0 ∨ b = 0 ∨ 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 ≠ 0" "a' ≠ 0" "b ≠ 0" "b' ≠ 0"    let ?g="gcd (a*a') (b*b')"    have gz: "?g ≠ 0" using z by simp    from z of_int_div[where ?'a = 'a, OF gz gcd_dvd1_int[where x="a*a'" and y="b*b'"]]      of_int_div[where ?'a = 'a , OF gz gcd_dvd2_int[where x="a*a'" and y="b*b'"]]    have ?thesis by (simp add: Nmul_def x y Let_def INum_def) }  ultimately show ?thesis by blastqedlemma Nneg[simp]: "INum (~⇩N x) = - (INum x ::'a:: field)"  by (simp add: Nneg_def split_def INum_def)lemma Nsub[simp]: "INum (x -⇩N y) = INum x - (INum y:: 'a :: {field_char_0, field_inverse_zero})"  by (simp add: Nsub_def split_def)lemma Ninv[simp]: "INum (Ninv x) = (1::'a :: field_inverse_zero) / (INum x)"  by (simp add: Ninv_def INum_def split_def)lemma Ndiv[simp]: "INum (x ÷⇩N y) = INum x / (INum y ::'a :: {field_char_0, field_inverse_zero})"  by (simp add: Ndiv_def)lemma Nlt0_iff[simp]:  assumes nx: "isnormNum x"  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})< 0) = 0>⇩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 ≠ 0" hence b: "(of_int b::'a) > 0"      using nx by (simp add: x isnormNum_def)    from pos_divide_less_eq[OF b, where b="of_int a" and a="0::'a"]    have ?thesis by (simp add: x Nlt0_def INum_def) }  ultimately show ?thesis by blastqedlemma Nle0_iff[simp]:  assumes nx: "isnormNum x"  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) ≤ 0) = 0≥⇩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 ≠ 0" hence b: "(of_int b :: 'a) > 0"      using nx by (simp add: x isnormNum_def)    from pos_divide_le_eq[OF b, where b="of_int a" and a="0::'a"]    have ?thesis by (simp add: x Nle0_def INum_def) }  ultimately show ?thesis by blastqedlemma Ngt0_iff[simp]:  assumes nx: "isnormNum x"  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})> 0) = 0<⇩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 ≠ 0" hence b: "(of_int b::'a) > 0" using nx      by (simp add: x isnormNum_def)    from pos_less_divide_eq[OF b, where b="of_int a" and a="0::'a"]    have ?thesis by (simp add: x Ngt0_def INum_def) }  ultimately show ?thesis by blastqedlemma Nge0_iff[simp]:  assumes nx: "isnormNum x"  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) ≥ 0) = 0≤⇩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 ≠ 0" hence b: "(of_int b::'a) > 0" using nx      by (simp add: x isnormNum_def)    from pos_le_divide_eq[OF b, where b="of_int a" and a="0::'a"]    have ?thesis by (simp add: x Nge0_def INum_def) }  ultimately show ?thesis by blastqedlemma Nlt_iff[simp]:  assumes nx: "isnormNum x" and ny: "isnormNum y"  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero}) < INum y) = (x <⇩N y)"proof -  let ?z = "0::'a"  have "((INum x ::'a) < INum y) = (INum (x -⇩N y) < ?z)"    using nx ny by simp  also have "… = (0>⇩N (x -⇩N y))"    using Nlt0_iff[OF Nsub_normN[OF ny]] by simp  finally show ?thesis by (simp add: Nlt_def)qedlemma Nle_iff[simp]:  assumes nx: "isnormNum x" and ny: "isnormNum y"  shows "((INum x :: 'a :: {field_char_0, linordered_field_inverse_zero})≤ INum y) = (x ≤⇩N y)"proof -  have "((INum x ::'a) ≤ INum y) = (INum (x -⇩N y) ≤ (0::'a))"    using nx ny by simp  also have "… = (0≥⇩N (x -⇩N y))"    using Nle0_iff[OF Nsub_normN[OF ny]] by simp  finally show ?thesis by (simp add: Nle_def)qedlemma Nadd_commute:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  shows "x +⇩N y = y +⇩N x"proof -  have n: "isnormNum (x +⇩N y)" "isnormNum (y +⇩N x)" by simp_all  have "(INum (x +⇩N y)::'a) = INum (y +⇩N x)" by simp  with isnormNum_unique[OF n] show ?thesis by simpqedlemma [simp]:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  shows "(0, b) +⇩N y = normNum y"    and "(a, 0) +⇩N y = normNum y"    and "x +⇩N (0, b) = normNum x"    and "x +⇩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)  donelemma normNum_nilpotent_aux[simp]:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  assumes nx: "isnormNum x"  shows "normNum x = x"proof -  let ?a = "normNum x"  have n: "isnormNum ?a" by simp  have th: "INum ?a = (INum x ::'a)" by simp  with isnormNum_unique[OF n nx] show ?thesis by simpqedlemma normNum_nilpotent[simp]:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  shows "normNum (normNum x) = normNum x"  by simplemma normNum0[simp]: "normNum (0,b) = 0⇩N" "normNum (a,0) = 0⇩N"  by (simp_all add: normNum_def)lemma normNum_Nadd:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  shows "normNum (x +⇩N y) = x +⇩N y" by simplemma Nadd_normNum1[simp]:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  shows "normNum x +⇩N y = x +⇩N y"proof -  have n: "isnormNum (normNum x +⇩N y)" "isnormNum (x +⇩N y)" by simp_all  have "INum (normNum x +⇩N y) = INum x + (INum y :: 'a)" by simp  also have "… = INum (x +⇩N y)" by simp  finally show ?thesis using isnormNum_unique[OF n] by simpqedlemma Nadd_normNum2[simp]:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  shows "x +⇩N normNum y = x +⇩N y"proof -  have n: "isnormNum (x +⇩N normNum y)" "isnormNum (x +⇩N y)" by simp_all  have "INum (x +⇩N normNum y) = INum x + (INum y :: 'a)" by simp  also have "… = INum (x +⇩N y)" by simp  finally show ?thesis using isnormNum_unique[OF n] by simpqedlemma Nadd_assoc:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  shows "x +⇩N y +⇩N z = x +⇩N (y +⇩N z)"proof -  have n: "isnormNum (x +⇩N y +⇩N z)" "isnormNum (x +⇩N (y +⇩N z))" by simp_all  have "INum (x +⇩N y +⇩N z) = (INum (x +⇩N (y +⇩N z)) :: 'a)" by simp  with isnormNum_unique[OF n] show ?thesis by simpqedlemma Nmul_commute: "isnormNum x ==> isnormNum y ==> x *⇩N y = y *⇩N x"  by (simp add: Nmul_def split_def Let_def gcd_commute_int mult_commute)lemma Nmul_assoc:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  assumes nx: "isnormNum x" and ny: "isnormNum y" and nz: "isnormNum z"  shows "x *⇩N y *⇩N z = x *⇩N (y *⇩N z)"proof -  from nx ny nz have n: "isnormNum (x *⇩N y *⇩N z)" "isnormNum (x *⇩N (y *⇩N z))"    by simp_all  have "INum (x +⇩N y +⇩N z) = (INum (x +⇩N (y +⇩N z)) :: 'a)" by simp  with isnormNum_unique[OF n] show ?thesis by simpqedlemma Nsub0:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  assumes x: "isnormNum x" and y: "isnormNum y"  shows "x -⇩N y = 0⇩N <-> x = y"proof -  fix h :: 'a  from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0⇩N"]  have "(x -⇩N y = 0⇩N) = (INum (x -⇩N y) = (INum 0⇩N :: 'a)) " by simp  also have "… = (INum x = (INum y :: 'a))" by simp  also have "… = (x = y)" using x y by simp  finally show ?thesis .qedlemma Nmul0[simp]: "c *⇩N 0⇩N = 0⇩N" " 0⇩N *⇩N c = 0⇩N"  by (simp_all add: Nmul_def Let_def split_def)lemma Nmul_eq0[simp]:  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"  assumes nx: "isnormNum x" and ny: "isnormNum y"  shows "x*⇩N y = 0⇩N <-> x = 0⇩N ∨ y = 0⇩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⇩N" by simp  show ?thesis using nx ny    apply (simp only: isnormNum_unique[where ?'a = 'a, OF  Nmul_normN[OF nx ny] n0, symmetric]      Nmul[where ?'a = 'a])    apply (simp add: x y INum_def split_def isnormNum_def split: split_if_asm)    doneqedlemma Nneg_Nneg[simp]: "~⇩N (~⇩N c) = c"  by (simp add: Nneg_def split_def)lemma Nmul1[simp]:    "isnormNum c ==> (1)⇩N *⇩N c = c"    "isnormNum c ==> c *⇩N (1)⇩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)+  doneend`