src/HOL/Integ/IntDef.thy
author berghofe
Tue, 27 Sep 2005 12:14:39 +0200
changeset 17667 2beb71c0f92e
parent 17551 2a747fc49a8c
child 18114 a36a9b2921e9
permissions -rw-r--r--
Inserted clause for nat in number_of_codegen again ("code unfold" turned out to be too inefficient).
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(*  Title:      IntDef.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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*)
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header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}
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theory IntDef
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imports Equiv_Relations NatArith
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begin
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constdefs
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  intrel :: "((nat * nat) * (nat * nat)) set"
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    --{*the equivalence relation underlying the integers*}
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    "intrel == {((x,y),(u,v)) | x y u v. x+v = u+y}"
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typedef (Integ)
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  int = "UNIV//intrel"
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    by (auto simp add: quotient_def)
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instance int :: "{ord, zero, one, plus, times, minus}" ..
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constdefs
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  int :: "nat => int"
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  "int m == Abs_Integ(intrel `` {(m,0)})"
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defs (overloaded)
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  Zero_int_def:  "0 == int 0"
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  One_int_def:   "1 == int 1"
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  minus_int_def:
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    "- z == Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. intrel``{(y,x)})"
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  add_int_def:
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   "z + w ==
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       Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. \<Union>(u,v) \<in> Rep_Integ w.
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		 intrel``{(x+u, y+v)})"
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  diff_int_def:  "z - (w::int) == z + (-w)"
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  mult_int_def:
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   "z * w ==
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       Abs_Integ (\<Union>(x,y) \<in> Rep_Integ z. \<Union>(u,v) \<in> Rep_Integ w.
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		  intrel``{(x*u + y*v, x*v + y*u)})"
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  le_int_def:
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   "z \<le> (w::int) == 
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    \<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Integ z & (u,v) \<in> Rep_Integ w"
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  less_int_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
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subsection{*Construction of the Integers*}
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subsubsection{*Preliminary Lemmas about the Equivalence Relation*}
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lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
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by (simp add: intrel_def)
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lemma equiv_intrel: "equiv UNIV intrel"
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by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
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text{*Reduces equality of equivalence classes to the @{term intrel} relation:
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  @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
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lemmas equiv_intrel_iff = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
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declare equiv_intrel_iff [simp]
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text{*All equivalence classes belong to set of representatives*}
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lemma [simp]: "intrel``{(x,y)} \<in> Integ"
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by (auto simp add: Integ_def intrel_def quotient_def)
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text{*Reduces equality on abstractions to equality on representatives:
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  @{term "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
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declare Abs_Integ_inject [simp]  Abs_Integ_inverse [simp]
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text{*Case analysis on the representation of an integer as an equivalence
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      class of pairs of naturals.*}
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lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
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     "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
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apply (rule Abs_Integ_cases [of z]) 
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apply (auto simp add: Integ_def quotient_def) 
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done
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subsubsection{*@{term int}: Embedding the Naturals into the Integers*}
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lemma inj_int: "inj int"
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by (simp add: inj_on_def int_def)
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lemma int_int_eq [iff]: "(int m = int n) = (m = n)"
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by (fast elim!: inj_int [THEN injD])
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subsubsection{*Integer Unary Negation*}
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lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
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proof -
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  have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
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    by (simp add: congruent_def) 
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  thus ?thesis
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    by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
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qed
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lemma zminus_zminus: "- (- z) = (z::int)"
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by (cases z, simp add: minus)
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lemma zminus_0: "- 0 = (0::int)"
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by (simp add: int_def Zero_int_def minus)
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subsection{*Integer Addition*}
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lemma add:
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     "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
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      Abs_Integ (intrel``{(x+u, y+v)})"
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proof -
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  have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z) 
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        respects2 intrel"
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    by (simp add: congruent2_def)
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  thus ?thesis
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    by (simp add: add_int_def UN_UN_split_split_eq
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                  UN_equiv_class2 [OF equiv_intrel equiv_intrel])
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qed
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lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)"
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by (cases z, cases w, simp add: minus add)
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lemma zadd_commute: "(z::int) + w = w + z"
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by (cases z, cases w, simp add: add_ac add)
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lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
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by (cases z1, cases z2, cases z3, simp add: add add_assoc)
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(*For AC rewriting*)
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lemma zadd_left_commute: "x + (y + z) = y + ((x + z)  ::int)"
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  apply (rule mk_left_commute [of "op +"])
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  apply (rule zadd_assoc)
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  apply (rule zadd_commute)
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  done
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lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
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lemmas zmult_ac = OrderedGroup.mult_ac
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lemma zadd_int: "(int m) + (int n) = int (m + n)"
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by (simp add: int_def add)
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lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"
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by (simp add: zadd_int zadd_assoc [symmetric])
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lemma int_Suc: "int (Suc m) = 1 + (int m)"
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by (simp add: One_int_def zadd_int)
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(*also for the instance declaration int :: comm_monoid_add*)
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lemma zadd_0: "(0::int) + z = z"
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apply (simp add: Zero_int_def int_def)
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apply (cases z, simp add: add)
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done
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lemma zadd_0_right: "z + (0::int) = z"
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by (rule trans [OF zadd_commute zadd_0])
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lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"
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by (cases z, simp add: int_def Zero_int_def minus add)
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subsection{*Integer Multiplication*}
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text{*Congruence property for multiplication*}
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lemma mult_congruent2:
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     "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
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      respects2 intrel"
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apply (rule equiv_intrel [THEN congruent2_commuteI])
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 apply (force simp add: mult_ac, clarify) 
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apply (simp add: congruent_def mult_ac)  
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apply (rename_tac u v w x y z)
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apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
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apply (simp add: mult_ac)
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apply (simp add: add_mult_distrib [symmetric])
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done
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lemma mult:
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     "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
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      Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
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by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
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              UN_equiv_class2 [OF equiv_intrel equiv_intrel])
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lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
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by (cases z, cases w, simp add: minus mult add_ac)
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lemma zmult_commute: "(z::int) * w = w * z"
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by (cases z, cases w, simp add: mult add_ac mult_ac)
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lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
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by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)
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lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
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by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)
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lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"
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by (simp add: zmult_commute [of w] zadd_zmult_distrib)
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lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"
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by (simp add: diff_int_def zadd_zmult_distrib zmult_zminus)
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lemma zdiff_zmult_distrib2: "(w::int) * (z1 - z2) = (w * z1) - (w * z2)"
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by (simp add: zmult_commute [of w] zdiff_zmult_distrib)
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lemmas int_distrib =
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  zadd_zmult_distrib zadd_zmult_distrib2
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  zdiff_zmult_distrib zdiff_zmult_distrib2
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lemma int_mult: "int (m * n) = (int m) * (int n)"
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by (simp add: int_def mult)
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text{*Compatibility binding*}
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lemmas zmult_int = int_mult [symmetric]
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lemma zmult_1: "(1::int) * z = z"
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by (cases z, simp add: One_int_def int_def mult)
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lemma zmult_1_right: "z * (1::int) = z"
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by (rule trans [OF zmult_commute zmult_1])
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text{*The integers form a @{text comm_ring_1}*}
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instance int :: comm_ring_1
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proof
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  fix i j k :: int
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  show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)
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  show "i + j = j + i" by (simp add: zadd_commute)
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  show "0 + i = i" by (rule zadd_0)
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  show "- i + i = 0" by (rule zadd_zminus_inverse2)
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  show "i - j = i + (-j)" by (simp add: diff_int_def)
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  show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)
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  show "i * j = j * i" by (rule zmult_commute)
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  show "1 * i = i" by (rule zmult_1) 
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  show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)
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  show "0 \<noteq> (1::int)"
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    by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
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qed
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subsection{*The @{text "\<le>"} Ordering*}
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lemma le:
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  "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
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by (force simp add: le_int_def)
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lemma zle_refl: "w \<le> (w::int)"
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by (cases w, simp add: le)
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lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
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by (cases i, cases j, cases k, simp add: le)
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lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
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by (cases w, cases z, simp add: le)
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(* Axiom 'order_less_le' of class 'order': *)
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lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"
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by (simp add: less_int_def)
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instance int :: order
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  by intro_classes
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    (assumption |
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      rule zle_refl zle_trans zle_anti_sym zless_le)+
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(* Axiom 'linorder_linear' of class 'linorder': *)
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lemma zle_linear: "(z::int) \<le> w | w \<le> z"
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by (cases z, cases w) (simp add: le linorder_linear)
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instance int :: linorder
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  by intro_classes (rule zle_linear)
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lemmas zless_linear = linorder_less_linear [where 'a = int]
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lemmas linorder_neqE_int = linorder_neqE[where 'a = int]
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lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
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by (simp add: Zero_int_def)
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lemma zless_int [simp]: "(int m < int n) = (m<n)"
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by (simp add: le add int_def linorder_not_le [symmetric]) 
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lemma int_less_0_conv [simp]: "~ (int k < 0)"
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by (simp add: Zero_int_def)
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lemma zero_less_int_conv [simp]: "(0 < int n) = (0 < n)"
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by (simp add: Zero_int_def)
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lemma int_0_less_1: "0 < (1::int)"
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by (simp only: Zero_int_def One_int_def One_nat_def zless_int)
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lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
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   303
by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   304
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   305
lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   306
by (simp add: linorder_not_less [symmetric])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   307
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   308
lemma zero_zle_int [simp]: "(0 \<le> int n)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   309
by (simp add: Zero_int_def)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   310
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   311
lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   312
by (simp add: Zero_int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   313
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   314
lemma int_0 [simp]: "int 0 = (0::int)"
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   315
by (simp add: Zero_int_def)
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   316
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   317
lemma int_1 [simp]: "int 1 = 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   318
by (simp add: One_int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   319
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   320
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   321
by (simp add: One_int_def One_nat_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   322
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   323
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   324
subsection{*Monotonicity results*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   325
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   326
lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   327
by (cases i, cases j, cases k, simp add: le add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   328
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   329
lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   330
apply (cases i, cases j, cases k)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   331
apply (simp add: linorder_not_le [where 'a = int, symmetric]
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   332
                 linorder_not_le [where 'a = nat]  le add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   333
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   334
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   335
lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   336
by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   337
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   338
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   339
subsection{*Strict Monotonicity of Multiplication*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   340
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   341
text{*strict, in 1st argument; proof is by induction on k>0*}
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15169
diff changeset
   342
lemma zmult_zless_mono2_lemma:
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15169
diff changeset
   343
     "i<j ==> 0<k ==> int k * i < int k * j"
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15169
diff changeset
   344
apply (induct "k", simp)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   345
apply (simp add: int_Suc)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15169
diff changeset
   346
apply (case_tac "k=0")
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   347
apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   348
apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   349
done
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   350
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   351
lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   352
apply (cases k)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   353
apply (auto simp add: le add int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   354
apply (rule_tac x="x-y" in exI, simp)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   355
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   356
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   357
lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   358
apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   359
apply (auto simp add: zmult_zless_mono2_lemma)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   360
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   361
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   362
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   363
defs (overloaded)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   364
    zabs_def:  "abs(i::int) == if i < 0 then -i else i"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   365
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   366
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   367
text{*The integers form an ordered @{text comm_ring_1}*}
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   368
instance int :: ordered_idom
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   369
proof
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   370
  fix i j k :: int
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   371
  show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   372
  show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   373
  show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   374
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   375
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   376
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   377
lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   378
apply (cases w, cases z) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   379
apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   380
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   381
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   382
subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   383
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   384
constdefs
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   385
   nat  :: "int => nat"
14532
paulson
parents: 14496
diff changeset
   386
    "nat z == contents (\<Union>(x,y) \<in> Rep_Integ z. { x-y })"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   387
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   388
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   389
proof -
15169
2b5da07a0b89 new "respects" syntax for quotienting
paulson
parents: 15140
diff changeset
   390
  have "(\<lambda>(x,y). {x-y}) respects intrel"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   391
    by (simp add: congruent_def, arith) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   392
  thus ?thesis
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   393
    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   394
qed
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   395
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   396
lemma nat_int [simp]: "nat(int n) = n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   397
by (simp add: nat int_def) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   398
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   399
lemma nat_zero [simp]: "nat 0 = 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   400
by (simp only: Zero_int_def nat_int)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   401
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   402
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   403
by (cases z, simp add: nat le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   404
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   405
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
15413
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   406
by simp
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   407
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   408
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   409
by (cases z, simp add: nat le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   410
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   411
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   412
apply (cases w, cases z) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   413
apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   414
done
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   415
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   416
text{*An alternative condition is @{term "0 \<le> w"} *}
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   417
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   418
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   419
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   420
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   421
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   422
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   423
lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   424
apply (cases w, cases z) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   425
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   426
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   427
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   428
lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   429
by (blast dest: nat_0_le sym)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   430
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   431
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   432
by (cases w, simp add: nat le int_def Zero_int_def, arith)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   433
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   434
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   435
by (simp only: eq_commute [of m] nat_eq_iff) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   436
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   437
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   438
apply (cases w)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   439
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   440
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   441
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   442
lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   443
by (auto simp add: nat_eq_iff2)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   444
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   445
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   446
by (insert zless_nat_conj [of 0], auto)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   447
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   448
lemma nat_add_distrib:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   449
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   450
by (cases z, cases z', simp add: nat add le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   451
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   452
lemma nat_diff_distrib:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   453
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   454
by (cases z, cases z', 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   455
    simp add: nat add minus diff_minus le int_def Zero_int_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   456
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   457
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   458
lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   459
by (simp add: int_def minus nat Zero_int_def) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   460
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   461
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   462
by (cases z, simp add: nat le int_def  linorder_not_le [symmetric], arith)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   463
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   464
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   465
subsection{*Lemmas about the Function @{term int} and Orderings*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   466
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   467
lemma negative_zless_0: "- (int (Suc n)) < 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   468
by (simp add: order_less_le)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   469
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   470
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   471
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   472
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   473
lemma negative_zle_0: "- int n \<le> 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   474
by (simp add: minus_le_iff)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   475
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   476
lemma negative_zle [iff]: "- int n \<le> int m"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   477
by (rule order_trans [OF negative_zle_0 zero_zle_int])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   478
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   479
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   480
by (subst le_minus_iff, simp)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   481
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   482
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   483
by (simp add: int_def le minus Zero_int_def) 
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   484
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   485
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   486
by (simp add: linorder_not_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   487
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   488
lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   489
by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   490
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   491
lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
15413
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   492
proof (cases w, cases z, simp add: le add int_def)
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   493
  fix a b c d
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   494
  assume "w = Abs_Integ (intrel `` {(a,b)})" "z = Abs_Integ (intrel `` {(c,d)})"
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   495
  show "(a+d \<le> c+b) = (\<exists>n. c+b = a+n+d)"
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   496
  proof
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   497
    assume "a + d \<le> c + b" 
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   498
    thus "\<exists>n. c + b = a + n + d" 
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   499
      by (auto intro!: exI [where x="c+b - (a+d)"])
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   500
  next    
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   501
    assume "\<exists>n. c + b = a + n + d" 
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   502
    then obtain n where "c + b = a + n + d" ..
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   503
    thus "a + d \<le> c + b" by arith
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   504
  qed
901d1bfedf09 removal of archaic Abs/Rep proofs
paulson
parents: 15409
diff changeset
   505
qed
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   506
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   507
lemma abs_int_eq [simp]: "abs (int m) = int m"
15003
6145dd7538d7 replaced monomorphic abs definitions by abs_if
paulson
parents: 14740
diff changeset
   508
by (simp add: abs_if)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   509
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   510
text{*This version is proved for all ordered rings, not just integers!
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   511
      It is proved here because attribute @{text arith_split} is not available
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   512
      in theory @{text Ring_and_Field}.
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   513
      But is it really better than just rewriting with @{text abs_if}?*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   514
lemma abs_split [arith_split]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   515
     "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   516
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   517
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   518
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   519
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   520
subsection{*The Constants @{term neg} and @{term iszero}*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   521
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   522
constdefs
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   523
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   524
  neg   :: "'a::ordered_idom => bool"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   525
  "neg(Z) == Z < 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   526
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   527
  (*For simplifying equalities*)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   528
  iszero :: "'a::comm_semiring_1_cancel => bool"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   529
  "iszero z == z = (0)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   530
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   531
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   532
lemma not_neg_int [simp]: "~ neg(int n)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   533
by (simp add: neg_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   534
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   535
lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   536
by (simp add: neg_def neg_less_0_iff_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   537
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   538
lemmas neg_eq_less_0 = neg_def
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   539
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   540
lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   541
by (simp add: neg_def linorder_not_less)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   542
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   543
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   544
subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   545
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   546
lemma not_neg_0: "~ neg 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   547
by (simp add: One_int_def neg_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   548
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   549
lemma not_neg_1: "~ neg 1"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   550
by (simp add: neg_def linorder_not_less zero_le_one)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   551
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   552
lemma iszero_0: "iszero 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   553
by (simp add: iszero_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   554
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   555
lemma not_iszero_1: "~ iszero 1"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   556
by (simp add: iszero_def eq_commute)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   557
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   558
lemma neg_nat: "neg z ==> nat z = 0"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   559
by (simp add: neg_def order_less_imp_le) 
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   560
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   561
lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   562
by (simp add: linorder_not_less neg_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   563
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   564
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   565
subsection{*The Set of Natural Numbers*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   566
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   567
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   568
   Nats  :: "'a::comm_semiring_1_cancel set"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   569
    "Nats == range of_nat"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   570
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   571
syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   572
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   573
lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   574
by (simp add: Nats_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   575
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   576
lemma Nats_0 [simp]: "0 \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   577
apply (simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   578
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   579
apply (rule of_nat_0 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   580
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   581
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   582
lemma Nats_1 [simp]: "1 \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   583
apply (simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   584
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   585
apply (rule of_nat_1 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   586
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   587
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   588
lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   589
apply (auto simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   590
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   591
apply (rule of_nat_add [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   592
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   593
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   594
lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   595
apply (auto simp add: Nats_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   596
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   597
apply (rule of_nat_mult [symmetric])
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   598
done
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   599
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   600
text{*Agreement with the specific embedding for the integers*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   601
lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   602
proof
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   603
  fix n
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   604
  show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   605
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   606
14496
paulson
parents: 14485
diff changeset
   607
lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
paulson
parents: 14485
diff changeset
   608
proof
paulson
parents: 14485
diff changeset
   609
  fix n
paulson
parents: 14485
diff changeset
   610
  show "of_nat n = id n"  by (induct n, simp_all)
paulson
parents: 14485
diff changeset
   611
qed
paulson
parents: 14485
diff changeset
   612
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   613
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   614
subsection{*Embedding of the Integers into any @{text comm_ring_1}:
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   615
@{term of_int}*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   616
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   617
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   618
   of_int :: "int => 'a::comm_ring_1"
14532
paulson
parents: 14496
diff changeset
   619
   "of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   620
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   621
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   622
lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
14496
paulson
parents: 14485
diff changeset
   623
proof -
15169
2b5da07a0b89 new "respects" syntax for quotienting
paulson
parents: 15140
diff changeset
   624
  have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
14496
paulson
parents: 14485
diff changeset
   625
    by (simp add: congruent_def compare_rls of_nat_add [symmetric]
paulson
parents: 14485
diff changeset
   626
            del: of_nat_add) 
paulson
parents: 14485
diff changeset
   627
  thus ?thesis
paulson
parents: 14485
diff changeset
   628
    by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
paulson
parents: 14485
diff changeset
   629
qed
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   630
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   631
lemma of_int_0 [simp]: "of_int 0 = 0"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   632
by (simp add: of_int Zero_int_def int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   633
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   634
lemma of_int_1 [simp]: "of_int 1 = 1"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   635
by (simp add: of_int One_int_def int_def)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   636
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   637
lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   638
by (cases w, cases z, simp add: compare_rls of_int add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   639
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   640
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   641
by (cases z, simp add: compare_rls of_int minus)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   642
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   643
lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   644
by (simp add: diff_minus)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   645
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   646
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   647
apply (cases w, cases z)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   648
apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   649
                 mult add_ac)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   650
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   651
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   652
lemma of_int_le_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   653
     "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   654
apply (cases w)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   655
apply (cases z)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   656
apply (simp add: compare_rls of_int le diff_int_def add minus
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   657
                 of_nat_add [symmetric]   del: of_nat_add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   658
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   659
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   660
text{*Special cases where either operand is zero*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   661
lemmas of_int_0_le_iff = of_int_le_iff [of 0, simplified]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   662
lemmas of_int_le_0_iff = of_int_le_iff [of _ 0, simplified]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   663
declare of_int_0_le_iff [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   664
declare of_int_le_0_iff [simp]
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   665
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   666
lemma of_int_less_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   667
     "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   668
by (simp add: linorder_not_le [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   669
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   670
text{*Special cases where either operand is zero*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   671
lemmas of_int_0_less_iff = of_int_less_iff [of 0, simplified]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   672
lemmas of_int_less_0_iff = of_int_less_iff [of _ 0, simplified]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   673
declare of_int_0_less_iff [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   674
declare of_int_less_0_iff [simp]
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   675
14740
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   676
text{*The ordering on the @{text comm_ring_1} is necessary.
c8e1937110c2 fixed latex problems
nipkow
parents: 14738
diff changeset
   677
 See @{text of_nat_eq_iff} above.*}
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   678
lemma of_int_eq_iff [simp]:
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   679
     "(of_int w = (of_int z::'a::ordered_idom)) = (w = z)"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   680
by (simp add: order_eq_iff)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   681
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   682
text{*Special cases where either operand is zero*}
17085
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   683
lemmas of_int_0_eq_iff = of_int_eq_iff [of 0, simplified]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   684
lemmas of_int_eq_0_iff = of_int_eq_iff [of _ 0, simplified]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   685
declare of_int_0_eq_iff [simp]
5b57f995a179 more simprules now have names
paulson
parents: 16770
diff changeset
   686
declare of_int_eq_0_iff [simp]
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   687
14496
paulson
parents: 14485
diff changeset
   688
lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"
paulson
parents: 14485
diff changeset
   689
proof
paulson
parents: 14485
diff changeset
   690
 fix z
paulson
parents: 14485
diff changeset
   691
 show "of_int z = id z"  
paulson
parents: 14485
diff changeset
   692
  by (cases z,
paulson
parents: 14485
diff changeset
   693
      simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus)
paulson
parents: 14485
diff changeset
   694
qed
paulson
parents: 14485
diff changeset
   695
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   696
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   697
subsection{*The Set of Integers*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   698
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   699
constdefs
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14691
diff changeset
   700
   Ints  :: "'a::comm_ring_1 set"
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   701
    "Ints == range of_int"
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
   702
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   703
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   704
syntax (xsymbols)
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   705
  Ints      :: "'a set"                   ("\<int>")
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   706
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   707
lemma Ints_0 [simp]: "0 \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   708
apply (simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   709
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   710
apply (rule of_int_0 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   711
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   712
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   713
lemma Ints_1 [simp]: "1 \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   714
apply (simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   715
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   716
apply (rule of_int_1 [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   717
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   718
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   719
lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   720
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   721
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   722
apply (rule of_int_add [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   723
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   724
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   725
lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   726
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   727
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   728
apply (rule of_int_minus [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   729
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   730
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   731
lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   732
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   733
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   734
apply (rule of_int_diff [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   735
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   736
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   737
lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   738
apply (auto simp add: Ints_def)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   739
apply (rule range_eqI)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   740
apply (rule of_int_mult [symmetric])
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   741
done
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   742
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   743
text{*Collapse nested embeddings*}
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   744
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   745
by (induct n, auto)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   746
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   747
lemma of_int_int_eq [simp]: "of_int (int n) = of_nat n"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   748
by (simp add: int_eq_of_nat)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14271
diff changeset
   749
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   750
lemma Ints_cases [case_names of_int, cases set: Ints]:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   751
  "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   752
proof (simp add: Ints_def)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   753
  assume "!!z. q = of_int z ==> C"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   754
  assume "q \<in> range of_int" thus C ..
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   755
qed
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   756
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   757
lemma Ints_induct [case_names of_int, induct set: Ints]:
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   758
  "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   759
  by (rule Ints_cases) auto
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   760
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   761
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   762
(* int (Suc n) = 1 + int n *)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   763
declare int_Suc [simp]
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   764
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   765
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   766
subsection{*More Properties of @{term setsum} and  @{term setprod}*}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   767
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   768
text{*By Jeremy Avigad*}
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   769
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   770
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   771
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   772
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   773
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   774
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   775
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   776
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   777
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   778
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   779
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   780
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   781
lemma int_setsum: "int (setsum f A) = (\<Sum>x\<in>A. int(f x))"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   782
  by (simp add: int_eq_of_nat of_nat_setsum)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   783
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   784
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   785
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   786
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   787
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   788
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   789
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   790
  apply (case_tac "finite A")
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   791
  apply (erule finite_induct, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   792
  done
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   793
15554
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   794
lemma int_setprod: "int (setprod f A) = (\<Prod>x\<in>A. int(f x))"
03d4347b071d integrated Jeremy's FiniteLib
nipkow
parents: 15540
diff changeset
   795
  by (simp add: int_eq_of_nat of_nat_setprod)
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   796
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   797
lemma setprod_nonzero_nat:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   798
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   799
  by (rule setprod_nonzero, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   800
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   801
lemma setprod_zero_eq_nat:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   802
    "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   803
  by (rule setprod_zero_eq, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   804
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   805
lemma setprod_nonzero_int:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   806
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   807
  by (rule setprod_nonzero, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   808
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   809
lemma setprod_zero_eq_int:
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   810
    "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   811
  by (rule setprod_zero_eq, auto)
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   812
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14421
diff changeset
   813
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   814
text{*Now we replace the case analysis rule by a more conventional one:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   815
whether an integer is negative or not.*}
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   816
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   817
lemma zless_iff_Suc_zadd:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   818
    "(w < z) = (\<exists>n. z = w + int(Suc n))"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   819
apply (cases z, cases w)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   820
apply (auto simp add: le add int_def linorder_not_le [symmetric]) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   821
apply (rename_tac a b c d) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   822
apply (rule_tac x="a+d - Suc(c+b)" in exI) 
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   823
apply arith
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   824
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   825
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   826
lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   827
apply (cases x)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   828
apply (auto simp add: le minus Zero_int_def int_def order_less_le) 
14496
paulson
parents: 14485
diff changeset
   829
apply (rule_tac x="y - Suc x" in exI, arith)
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   830
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   831
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   832
theorem int_cases [cases type: int, case_names nonneg neg]:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   833
     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   834
apply (case_tac "z < 0", blast dest!: negD)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   835
apply (simp add: linorder_not_less)
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   836
apply (blast dest: nat_0_le [THEN sym])
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   837
done
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   838
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   839
theorem int_induct [induct type: int, case_names nonneg neg]:
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   840
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   841
  by (cases z) auto
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   842
15558
f5f4f89a3b84 new lemmas int_diff_cases
paulson
parents: 15554
diff changeset
   843
text{*Contributed by Brian Huffman*}
f5f4f89a3b84 new lemmas int_diff_cases
paulson
parents: 15554
diff changeset
   844
theorem int_diff_cases [case_names diff]:
f5f4f89a3b84 new lemmas int_diff_cases
paulson
parents: 15554
diff changeset
   845
assumes prem: "!!m n. z = int m - int n ==> P" shows "P"
f5f4f89a3b84 new lemmas int_diff_cases
paulson
parents: 15554
diff changeset
   846
 apply (rule_tac z=z in int_cases)
f5f4f89a3b84 new lemmas int_diff_cases
paulson
parents: 15554
diff changeset
   847
  apply (rule_tac m=n and n=0 in prem, simp)
f5f4f89a3b84 new lemmas int_diff_cases
paulson
parents: 15554
diff changeset
   848
 apply (rule_tac m=0 and n="Suc n" in prem, simp)
f5f4f89a3b84 new lemmas int_diff_cases
paulson
parents: 15554
diff changeset
   849
done
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   850
15013
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   851
lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z"
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   852
apply (cases z)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   853
apply (simp_all add: not_zle_0_negative del: int_Suc)
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   854
done
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   855
34264f5e4691 new treatment of binary numerals
paulson
parents: 15003
diff changeset
   856
16642
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   857
subsection {* Configuration of the code generator *}
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   858
16770
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   859
(*FIXME: the IntInf.fromInt below hides a dependence on fixed-precision ints!*)
16642
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   860
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   861
types_code
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   862
  "int" ("int")
16770
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   863
attach (term_of) {*
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   864
val term_of_int = HOLogic.mk_int o IntInf.fromInt;
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   865
*}
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   866
attach (test) {*
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   867
fun gen_int i = one_of [~1, 1] * random_range 0 i;
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   868
*}
16642
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   869
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   870
constdefs
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   871
  int_aux :: "int \<Rightarrow> nat \<Rightarrow> int"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   872
  "int_aux i n == (i + int n)"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   873
  nat_aux :: "nat \<Rightarrow> int \<Rightarrow> nat"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   874
  "nat_aux n i == (n + nat i)"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   875
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   876
lemma [code]:
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   877
  "int_aux i 0 = i"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   878
  "int_aux i (Suc n) = int_aux (i + 1) n" -- {* tail recursive *}
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   879
  "int n = int_aux 0 n"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   880
  by (simp add: int_aux_def)+
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   881
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   882
lemma [code]: "nat_aux n i = (if i <= 0 then n else nat_aux (Suc n) (i - 1))"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   883
  -- {* tail recursive *}
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   884
  by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   885
    dest: zless_imp_add1_zle)
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   886
lemma [code]: "nat i = nat_aux 0 i"
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   887
  by (simp add: nat_aux_def)
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   888
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   889
consts_code
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   890
  "0" :: "int"                  ("0")
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   891
  "1" :: "int"                  ("1")
16770
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   892
  "uminus" :: "int => int"      ("~")
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   893
  "op +" :: "int => int => int" ("(_ +/ _)")
1f1b1fae30e4 Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents: 16733
diff changeset
   894
  "op *" :: "int => int => int" ("(_ */ _)")
16642
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   895
  "op <" :: "int => int => bool" ("(_ </ _)")
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   896
  "op <=" :: "int => int => bool" ("(_ <=/ _)")
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   897
  "neg"                         ("(_ < 0)")
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   898
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   899
ML {*
17551
2a747fc49a8c Simplified code generator for numerals.
berghofe
parents: 17464
diff changeset
   900
fun number_of_codegen thy defs gr dep module b (Const ("Numeral.number_of",
2a747fc49a8c Simplified code generator for numerals.
berghofe
parents: 17464
diff changeset
   901
      Type ("fun", [_, T as Type ("IntDef.int", [])])) $ bin) =
2a747fc49a8c Simplified code generator for numerals.
berghofe
parents: 17464
diff changeset
   902
        (SOME (fst (Codegen.invoke_tycodegen thy defs dep module false (gr, T)),
2a747fc49a8c Simplified code generator for numerals.
berghofe
parents: 17464
diff changeset
   903
           Pretty.str (IntInf.toString (HOLogic.dest_binum bin))) handle TERM _ => NONE)
17667
2beb71c0f92e Inserted clause for nat in number_of_codegen again ("code unfold" turned
berghofe
parents: 17551
diff changeset
   904
  | number_of_codegen thy defs gr s thyname b (Const ("Numeral.number_of",
2beb71c0f92e Inserted clause for nat in number_of_codegen again ("code unfold" turned
berghofe
parents: 17551
diff changeset
   905
      Type ("fun", [_, Type ("nat", [])])) $ bin) =
2beb71c0f92e Inserted clause for nat in number_of_codegen again ("code unfold" turned
berghofe
parents: 17551
diff changeset
   906
        SOME (Codegen.invoke_codegen thy defs s thyname b (gr,
2beb71c0f92e Inserted clause for nat in number_of_codegen again ("code unfold" turned
berghofe
parents: 17551
diff changeset
   907
          Const ("IntDef.nat", HOLogic.intT --> HOLogic.natT) $
2beb71c0f92e Inserted clause for nat in number_of_codegen again ("code unfold" turned
berghofe
parents: 17551
diff changeset
   908
            (Const ("Numeral.number_of", HOLogic.binT --> HOLogic.intT) $ bin)))
16642
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   909
  | number_of_codegen _ _ _ _ _ _ _ = NONE;
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   910
*}
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   911
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   912
setup {* [Codegen.add_codegen "number_of_codegen" number_of_codegen] *}
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   913
17464
a4090ccf14a8 added quickcheck_params (from Main.thy);
wenzelm
parents: 17085
diff changeset
   914
quickcheck_params [default_type = int]
a4090ccf14a8 added quickcheck_params (from Main.thy);
wenzelm
parents: 17085
diff changeset
   915
16642
849ec3962b55 Moved code generator setup from NatBin to IntDef.
berghofe
parents: 16413
diff changeset
   916
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   917
(*Legacy ML bindings, but no longer the structure Int.*)
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   918
ML
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   919
{*
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   920
val zabs_def = thm "zabs_def"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   921
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   922
val int_0 = thm "int_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   923
val int_1 = thm "int_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   924
val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   925
val neg_eq_less_0 = thm "neg_eq_less_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   926
val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   927
val not_neg_0 = thm "not_neg_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   928
val not_neg_1 = thm "not_neg_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   929
val iszero_0 = thm "iszero_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   930
val not_iszero_1 = thm "not_iszero_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   931
val int_0_less_1 = thm "int_0_less_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   932
val int_0_neq_1 = thm "int_0_neq_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   933
val negative_zless = thm "negative_zless";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   934
val negative_zle = thm "negative_zle";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   935
val not_zle_0_negative = thm "not_zle_0_negative";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   936
val not_int_zless_negative = thm "not_int_zless_negative";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   937
val negative_eq_positive = thm "negative_eq_positive";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   938
val zle_iff_zadd = thm "zle_iff_zadd";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   939
val abs_int_eq = thm "abs_int_eq";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   940
val abs_split = thm"abs_split";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   941
val nat_int = thm "nat_int";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   942
val nat_zminus_int = thm "nat_zminus_int";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   943
val nat_zero = thm "nat_zero";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   944
val not_neg_nat = thm "not_neg_nat";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   945
val neg_nat = thm "neg_nat";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   946
val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   947
val nat_0_le = thm "nat_0_le";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   948
val nat_le_0 = thm "nat_le_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   949
val zless_nat_conj = thm "zless_nat_conj";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   950
val int_cases = thm "int_cases";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   951
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   952
val int_def = thm "int_def";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   953
val Zero_int_def = thm "Zero_int_def";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   954
val One_int_def = thm "One_int_def";
14479
0eca4aabf371 streamlined treatment of quotients for the integers
paulson
parents: 14430
diff changeset
   955
val diff_int_def = thm "diff_int_def";
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   956
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   957
val inj_int = thm "inj_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   958
val zminus_zminus = thm "zminus_zminus";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   959
val zminus_0 = thm "zminus_0";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   960
val zminus_zadd_distrib = thm "zminus_zadd_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   961
val zadd_commute = thm "zadd_commute";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   962
val zadd_assoc = thm "zadd_assoc";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   963
val zadd_left_commute = thm "zadd_left_commute";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   964
val zadd_ac = thms "zadd_ac";
14271
8ed6989228bb Simplification of the development of Integers
paulson
parents: 14269
diff changeset
   965
val zmult_ac = thms "zmult_ac";
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   966
val zadd_int = thm "zadd_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   967
val zadd_int_left = thm "zadd_int_left";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   968
val int_Suc = thm "int_Suc";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   969
val zadd_0 = thm "zadd_0";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   970
val zadd_0_right = thm "zadd_0_right";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   971
val zmult_zminus = thm "zmult_zminus";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   972
val zmult_commute = thm "zmult_commute";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   973
val zmult_assoc = thm "zmult_assoc";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   974
val zadd_zmult_distrib = thm "zadd_zmult_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   975
val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   976
val zdiff_zmult_distrib = thm "zdiff_zmult_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   977
val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   978
val int_distrib = thms "int_distrib";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   979
val zmult_int = thm "zmult_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   980
val zmult_1 = thm "zmult_1";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   981
val zmult_1_right = thm "zmult_1_right";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   982
val int_int_eq = thm "int_int_eq";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   983
val int_eq_0_conv = thm "int_eq_0_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   984
val zless_int = thm "zless_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   985
val int_less_0_conv = thm "int_less_0_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   986
val zero_less_int_conv = thm "zero_less_int_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   987
val zle_int = thm "zle_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   988
val zero_zle_int = thm "zero_zle_int";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   989
val int_le_0_conv = thm "int_le_0_conv";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   990
val zle_refl = thm "zle_refl";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   991
val zle_linear = thm "zle_linear";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   992
val zle_trans = thm "zle_trans";
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
   993
val zle_anti_sym = thm "zle_anti_sym";
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   994
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   995
val Ints_def = thm "Ints_def";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   996
val Nats_def = thm "Nats_def";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   997
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   998
val of_nat_0 = thm "of_nat_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
   999
val of_nat_Suc = thm "of_nat_Suc";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1000
val of_nat_1 = thm "of_nat_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1001
val of_nat_add = thm "of_nat_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1002
val of_nat_mult = thm "of_nat_mult";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1003
val zero_le_imp_of_nat = thm "zero_le_imp_of_nat";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1004
val less_imp_of_nat_less = thm "less_imp_of_nat_less";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1005
val of_nat_less_imp_less = thm "of_nat_less_imp_less";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1006
val of_nat_less_iff = thm "of_nat_less_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1007
val of_nat_le_iff = thm "of_nat_le_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1008
val of_nat_eq_iff = thm "of_nat_eq_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1009
val Nats_0 = thm "Nats_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1010
val Nats_1 = thm "Nats_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1011
val Nats_add = thm "Nats_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1012
val Nats_mult = thm "Nats_mult";
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1013
val int_eq_of_nat = thm"int_eq_of_nat";
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1014
val of_int = thm "of_int";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1015
val of_int_0 = thm "of_int_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1016
val of_int_1 = thm "of_int_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1017
val of_int_add = thm "of_int_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1018
val of_int_minus = thm "of_int_minus";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1019
val of_int_diff = thm "of_int_diff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1020
val of_int_mult = thm "of_int_mult";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1021
val of_int_le_iff = thm "of_int_le_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1022
val of_int_less_iff = thm "of_int_less_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1023
val of_int_eq_iff = thm "of_int_eq_iff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1024
val Ints_0 = thm "Ints_0";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1025
val Ints_1 = thm "Ints_1";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1026
val Ints_add = thm "Ints_add";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1027
val Ints_minus = thm "Ints_minus";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1028
val Ints_diff = thm "Ints_diff";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1029
val Ints_mult = thm "Ints_mult";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1030
val of_int_of_nat_eq = thm"of_int_of_nat_eq";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1031
val Ints_cases = thm "Ints_cases";
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14348
diff changeset
  1032
val Ints_induct = thm "Ints_induct";
14259
79f7d3451b1e conversion of ML to Isar scripts
paulson
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diff changeset
  1033
*}
79f7d3451b1e conversion of ML to Isar scripts
paulson
parents: 13487
diff changeset
  1034
5508
691c70898518 new files in Integ
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parents:
diff changeset
  1035
end