src/HOL/Integ/IntDef.thy
author paulson
Fri Nov 19 17:31:49 2004 +0100 (2004-11-19)
changeset 15300 7dd5853a4812
parent 15251 bb6f072c8d10
child 15409 a063687d24eb
permissions -rw-r--r--
moved and renamed Integ/Equiv.thy
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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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lemma inj_on_Abs_Integ: "inj_on Abs_Integ Integ"
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apply (rule inj_on_inverseI)
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apply (erule Abs_Integ_inverse)
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done
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text{*This theorem reduces equality on abstractions to equality on 
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      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 inj_on_Abs_Integ [THEN inj_on_iff, simp]
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declare 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 Rep_Integ [of z, unfolded Integ_def, THEN quotientE])
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apply (drule arg_cong [where f=Abs_Integ])
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apply (auto simp add: Rep_Integ_inverse)
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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, arith)
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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 zmult_int: "(int m) * (int n) = int (m * n)"
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by (simp add: int_def mult)
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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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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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by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
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lemma zle_int [simp]: "(int m \<le> int n) = (m\<le>n)"
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by (simp add: linorder_not_less [symmetric])
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lemma zero_zle_int [simp]: "(0 \<le> int n)"
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by (simp add: Zero_int_def)
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lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)"
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by (simp add: Zero_int_def)
paulson@14378
   318
paulson@14378
   319
lemma int_0 [simp]: "int 0 = (0::int)"
paulson@14259
   320
by (simp add: Zero_int_def)
paulson@14259
   321
paulson@14378
   322
lemma int_1 [simp]: "int 1 = 1"
paulson@14378
   323
by (simp add: One_int_def)
paulson@14378
   324
paulson@14378
   325
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
paulson@14378
   326
by (simp add: One_int_def One_nat_def)
paulson@14378
   327
paulson@14479
   328
paulson@14378
   329
subsection{*Monotonicity results*}
paulson@14378
   330
paulson@14479
   331
lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
paulson@14479
   332
by (cases i, cases j, cases k, simp add: le add)
paulson@14378
   333
paulson@14479
   334
lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
paulson@14479
   335
apply (cases i, cases j, cases k)
paulson@14479
   336
apply (simp add: linorder_not_le [where 'a = int, symmetric]
paulson@14479
   337
                 linorder_not_le [where 'a = nat]  le add)
paulson@14378
   338
done
paulson@14378
   339
paulson@14378
   340
lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
paulson@14479
   341
by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
paulson@14378
   342
paulson@14378
   343
paulson@14378
   344
subsection{*Strict Monotonicity of Multiplication*}
paulson@14378
   345
paulson@14378
   346
text{*strict, in 1st argument; proof is by induction on k>0*}
paulson@15251
   347
lemma zmult_zless_mono2_lemma:
paulson@15251
   348
     "i<j ==> 0<k ==> int k * i < int k * j"
paulson@15251
   349
apply (induct "k", simp)
paulson@14378
   350
apply (simp add: int_Suc)
paulson@15251
   351
apply (case_tac "k=0")
paulson@14378
   352
apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
paulson@14378
   353
apply (simp add: zadd_zless_mono int_Suc0_eq_1 order_le_less)
paulson@14378
   354
done
paulson@14259
   355
paulson@14378
   356
lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
paulson@14479
   357
apply (cases k)
paulson@14479
   358
apply (auto simp add: le add int_def Zero_int_def)
paulson@14479
   359
apply (rule_tac x="x-y" in exI, simp)
paulson@14378
   360
done
paulson@14378
   361
paulson@14378
   362
lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
paulson@14479
   363
apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
paulson@14479
   364
apply (auto simp add: zmult_zless_mono2_lemma)
paulson@14378
   365
done
paulson@14378
   366
paulson@14378
   367
paulson@14378
   368
defs (overloaded)
paulson@14378
   369
    zabs_def:  "abs(i::int) == if i < 0 then -i else i"
paulson@14378
   370
paulson@14378
   371
nipkow@14740
   372
text{*The integers form an ordered @{text comm_ring_1}*}
obua@14738
   373
instance int :: ordered_idom
paulson@14378
   374
proof
paulson@14378
   375
  fix i j k :: int
paulson@14378
   376
  show "i \<le> j ==> k + i \<le> k + j" by (rule zadd_left_mono)
paulson@14378
   377
  show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2)
paulson@14378
   378
  show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def)
paulson@14378
   379
qed
paulson@14378
   380
paulson@14378
   381
paulson@14479
   382
lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
paulson@14479
   383
apply (cases w, cases z) 
paulson@14479
   384
apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)
paulson@14479
   385
done
paulson@14479
   386
paulson@14378
   387
subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
paulson@14378
   388
paulson@14378
   389
constdefs
paulson@14378
   390
   nat  :: "int => nat"
paulson@14532
   391
    "nat z == contents (\<Union>(x,y) \<in> Rep_Integ z. { x-y })"
paulson@14479
   392
paulson@14479
   393
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
paulson@14479
   394
proof -
paulson@15169
   395
  have "(\<lambda>(x,y). {x-y}) respects intrel"
paulson@14479
   396
    by (simp add: congruent_def, arith) 
paulson@14479
   397
  thus ?thesis
paulson@14479
   398
    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
paulson@14479
   399
qed
paulson@14378
   400
paulson@14378
   401
lemma nat_int [simp]: "nat(int n) = n"
paulson@14479
   402
by (simp add: nat int_def) 
paulson@14378
   403
paulson@14378
   404
lemma nat_zero [simp]: "nat 0 = 0"
paulson@14479
   405
by (simp only: Zero_int_def nat_int)
paulson@14378
   406
paulson@14479
   407
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
paulson@14479
   408
by (cases z, simp add: nat le int_def Zero_int_def)
paulson@14479
   409
paulson@14479
   410
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
paulson@14479
   411
apply simp 
paulson@14259
   412
done
paulson@14259
   413
paulson@14378
   414
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
paulson@14479
   415
by (cases z, simp add: nat le int_def Zero_int_def)
paulson@14479
   416
paulson@14479
   417
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
paulson@14479
   418
apply (cases w, cases z) 
paulson@14479
   419
apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith)
paulson@14479
   420
done
paulson@14378
   421
paulson@14378
   422
text{*An alternative condition is @{term "0 \<le> w"} *}
paulson@14479
   423
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
paulson@14479
   424
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
paulson@14479
   425
paulson@14479
   426
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
paulson@14479
   427
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
paulson@14479
   428
paulson@14479
   429
lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
paulson@14479
   430
apply (cases w, cases z) 
paulson@14479
   431
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
paulson@14378
   432
done
paulson@14378
   433
paulson@14479
   434
lemma nonneg_eq_int: "[| 0 \<le> z;  !!m. z = int m ==> P |] ==> P"
paulson@14479
   435
by (blast dest: nat_0_le sym)
paulson@14479
   436
paulson@14479
   437
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
paulson@14479
   438
by (cases w, simp add: nat le int_def Zero_int_def, arith)
paulson@14479
   439
paulson@14479
   440
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
paulson@14479
   441
by (simp only: eq_commute [of m] nat_eq_iff) 
paulson@14479
   442
paulson@14479
   443
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
paulson@14479
   444
apply (cases w)
paulson@14479
   445
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
paulson@14378
   446
done
paulson@14378
   447
paulson@14479
   448
lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
paulson@14479
   449
by (auto simp add: nat_eq_iff2)
paulson@14479
   450
paulson@14479
   451
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
paulson@14479
   452
by (insert zless_nat_conj [of 0], auto)
paulson@14479
   453
paulson@14479
   454
paulson@14479
   455
lemma nat_add_distrib:
paulson@14479
   456
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
paulson@14479
   457
by (cases z, cases z', simp add: nat add le int_def Zero_int_def)
paulson@14479
   458
paulson@14479
   459
lemma nat_diff_distrib:
paulson@14479
   460
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
paulson@14479
   461
by (cases z, cases z', 
paulson@14479
   462
    simp add: nat add minus diff_minus le int_def Zero_int_def)
paulson@14479
   463
paulson@14479
   464
paulson@14479
   465
lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
paulson@14479
   466
by (simp add: int_def minus nat Zero_int_def) 
paulson@14479
   467
paulson@14479
   468
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
paulson@14479
   469
by (cases z, simp add: nat le int_def  linorder_not_le [symmetric], arith)
paulson@14479
   470
paulson@14378
   471
paulson@14378
   472
subsection{*Lemmas about the Function @{term int} and Orderings*}
paulson@14378
   473
paulson@14378
   474
lemma negative_zless_0: "- (int (Suc n)) < 0"
paulson@14479
   475
by (simp add: order_less_le)
paulson@14378
   476
paulson@14378
   477
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
paulson@14378
   478
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
paulson@14378
   479
paulson@14378
   480
lemma negative_zle_0: "- int n \<le> 0"
paulson@14378
   481
by (simp add: minus_le_iff)
paulson@14378
   482
paulson@14378
   483
lemma negative_zle [iff]: "- int n \<le> int m"
paulson@14378
   484
by (rule order_trans [OF negative_zle_0 zero_zle_int])
paulson@14378
   485
paulson@14378
   486
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
paulson@14378
   487
by (subst le_minus_iff, simp)
paulson@14378
   488
paulson@14378
   489
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
paulson@14479
   490
by (simp add: int_def le minus Zero_int_def) 
paulson@14259
   491
paulson@14378
   492
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
paulson@14378
   493
by (simp add: linorder_not_less)
paulson@14378
   494
paulson@14378
   495
lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
paulson@14378
   496
by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
paulson@14378
   497
paulson@14378
   498
lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
paulson@14479
   499
apply (cases w, cases z)
paulson@14479
   500
apply (auto simp add: le add int_def) 
paulson@14479
   501
apply (rename_tac a b c d) 
paulson@14479
   502
apply (rule_tac x="c+b - (a+d)" in exI) 
paulson@14479
   503
apply arith
paulson@14479
   504
done
paulson@14378
   505
paulson@14479
   506
lemma abs_int_eq [simp]: "abs (int m) = int m"
paulson@15003
   507
by (simp add: abs_if)
paulson@14378
   508
paulson@14378
   509
text{*This version is proved for all ordered rings, not just integers!
paulson@14378
   510
      It is proved here because attribute @{text arith_split} is not available
paulson@14378
   511
      in theory @{text Ring_and_Field}.
paulson@14378
   512
      But is it really better than just rewriting with @{text abs_if}?*}
paulson@14378
   513
lemma abs_split [arith_split]:
obua@14738
   514
     "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
paulson@14378
   515
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
paulson@14378
   516
paulson@14378
   517
paulson@14378
   518
paulson@14378
   519
subsection{*The Constants @{term neg} and @{term iszero}*}
paulson@14378
   520
paulson@14378
   521
constdefs
paulson@14378
   522
obua@14738
   523
  neg   :: "'a::ordered_idom => bool"
paulson@14378
   524
  "neg(Z) == Z < 0"
paulson@14378
   525
paulson@14378
   526
  (*For simplifying equalities*)
obua@14738
   527
  iszero :: "'a::comm_semiring_1_cancel => bool"
paulson@14378
   528
  "iszero z == z = (0)"
paulson@14479
   529
paulson@14378
   530
paulson@14378
   531
lemma not_neg_int [simp]: "~ neg(int n)"
paulson@14378
   532
by (simp add: neg_def)
paulson@14378
   533
paulson@14378
   534
lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))"
paulson@14378
   535
by (simp add: neg_def neg_less_0_iff_less)
paulson@14378
   536
paulson@14378
   537
lemmas neg_eq_less_0 = neg_def
paulson@14378
   538
paulson@14378
   539
lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
paulson@14378
   540
by (simp add: neg_def linorder_not_less)
paulson@14378
   541
paulson@14479
   542
paulson@14378
   543
subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
paulson@14378
   544
paulson@14378
   545
lemma not_neg_0: "~ neg 0"
paulson@14378
   546
by (simp add: One_int_def neg_def)
paulson@14378
   547
paulson@14378
   548
lemma not_neg_1: "~ neg 1"
paulson@14479
   549
by (simp add: neg_def linorder_not_less zero_le_one)
paulson@14378
   550
paulson@14378
   551
lemma iszero_0: "iszero 0"
paulson@14378
   552
by (simp add: iszero_def)
paulson@14378
   553
paulson@14378
   554
lemma not_iszero_1: "~ iszero 1"
paulson@14479
   555
by (simp add: iszero_def eq_commute)
paulson@14378
   556
paulson@14378
   557
lemma neg_nat: "neg z ==> nat z = 0"
paulson@14479
   558
by (simp add: neg_def order_less_imp_le) 
paulson@14378
   559
paulson@14378
   560
lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
paulson@14378
   561
by (simp add: linorder_not_less neg_def)
paulson@14378
   562
paulson@14378
   563
nipkow@14740
   564
subsection{*Embedding of the Naturals into any @{text
nipkow@14740
   565
comm_semiring_1_cancel}: @{term of_nat}*}
paulson@14378
   566
obua@14738
   567
consts of_nat :: "nat => 'a::comm_semiring_1_cancel"
paulson@14378
   568
paulson@14378
   569
primrec
paulson@14378
   570
  of_nat_0:   "of_nat 0 = 0"
paulson@14378
   571
  of_nat_Suc: "of_nat (Suc m) = of_nat m + 1"
paulson@14378
   572
paulson@14378
   573
lemma of_nat_1 [simp]: "of_nat 1 = 1"
paulson@14378
   574
by simp
paulson@14378
   575
paulson@14378
   576
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
paulson@14378
   577
apply (induct m)
paulson@14479
   578
apply (simp_all add: add_ac)
paulson@14378
   579
done
paulson@14378
   580
paulson@14378
   581
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
paulson@14479
   582
apply (induct m)
paulson@14479
   583
apply (simp_all add: mult_ac add_ac right_distrib)
paulson@14378
   584
done
paulson@14378
   585
obua@14738
   586
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)"
paulson@14479
   587
apply (induct m, simp_all)
paulson@14479
   588
apply (erule order_trans)
paulson@14479
   589
apply (rule less_add_one [THEN order_less_imp_le])
paulson@14259
   590
done
paulson@14259
   591
paulson@14378
   592
lemma less_imp_of_nat_less:
obua@14738
   593
     "m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)"
paulson@14479
   594
apply (induct m n rule: diff_induct, simp_all)
paulson@14479
   595
apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
paulson@14378
   596
done
paulson@14378
   597
paulson@14378
   598
lemma of_nat_less_imp_less:
obua@14738
   599
     "of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n"
paulson@14479
   600
apply (induct m n rule: diff_induct, simp_all)
paulson@14479
   601
apply (insert zero_le_imp_of_nat)
paulson@14479
   602
apply (force simp add: linorder_not_less [symmetric])
paulson@14259
   603
done
paulson@14259
   604
paulson@14378
   605
lemma of_nat_less_iff [simp]:
obua@14738
   606
     "(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)"
paulson@14479
   607
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
paulson@14378
   608
paulson@14378
   609
text{*Special cases where either operand is zero*}
paulson@14378
   610
declare of_nat_less_iff [of 0, simplified, simp]
paulson@14378
   611
declare of_nat_less_iff [of _ 0, simplified, simp]
paulson@14378
   612
paulson@14378
   613
lemma of_nat_le_iff [simp]:
obua@14738
   614
     "(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)"
paulson@14479
   615
by (simp add: linorder_not_less [symmetric])
paulson@14378
   616
paulson@14378
   617
text{*Special cases where either operand is zero*}
paulson@14378
   618
declare of_nat_le_iff [of 0, simplified, simp]
paulson@14378
   619
declare of_nat_le_iff [of _ 0, simplified, simp]
paulson@14378
   620
nipkow@14740
   621
text{*The ordering on the @{text comm_semiring_1_cancel} is necessary
nipkow@14740
   622
to exclude the possibility of a finite field, which indeed wraps back to
nipkow@14740
   623
zero.*}
paulson@14378
   624
lemma of_nat_eq_iff [simp]:
obua@14738
   625
     "(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)"
paulson@14479
   626
by (simp add: order_eq_iff)
paulson@14378
   627
paulson@14378
   628
text{*Special cases where either operand is zero*}
paulson@14378
   629
declare of_nat_eq_iff [of 0, simplified, simp]
paulson@14378
   630
declare of_nat_eq_iff [of _ 0, simplified, simp]
paulson@14378
   631
paulson@14378
   632
lemma of_nat_diff [simp]:
obua@14738
   633
     "n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::comm_ring_1)"
paulson@14378
   634
by (simp del: of_nat_add
paulson@14479
   635
	 add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
paulson@14378
   636
paulson@14378
   637
paulson@14378
   638
subsection{*The Set of Natural Numbers*}
paulson@14378
   639
paulson@14378
   640
constdefs
obua@14738
   641
   Nats  :: "'a::comm_semiring_1_cancel set"
paulson@14378
   642
    "Nats == range of_nat"
paulson@14378
   643
paulson@14378
   644
syntax (xsymbols)    Nats :: "'a set"   ("\<nat>")
paulson@14378
   645
paulson@14378
   646
lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
paulson@14479
   647
by (simp add: Nats_def)
paulson@14378
   648
paulson@14378
   649
lemma Nats_0 [simp]: "0 \<in> Nats"
paulson@14479
   650
apply (simp add: Nats_def)
paulson@14479
   651
apply (rule range_eqI)
paulson@14378
   652
apply (rule of_nat_0 [symmetric])
paulson@14378
   653
done
paulson@14378
   654
paulson@14378
   655
lemma Nats_1 [simp]: "1 \<in> Nats"
paulson@14479
   656
apply (simp add: Nats_def)
paulson@14479
   657
apply (rule range_eqI)
paulson@14378
   658
apply (rule of_nat_1 [symmetric])
paulson@14378
   659
done
paulson@14378
   660
paulson@14378
   661
lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
paulson@14479
   662
apply (auto simp add: Nats_def)
paulson@14479
   663
apply (rule range_eqI)
paulson@14378
   664
apply (rule of_nat_add [symmetric])
paulson@14378
   665
done
paulson@14378
   666
paulson@14378
   667
lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
paulson@14479
   668
apply (auto simp add: Nats_def)
paulson@14479
   669
apply (rule range_eqI)
paulson@14378
   670
apply (rule of_nat_mult [symmetric])
paulson@14259
   671
done
paulson@14259
   672
paulson@14378
   673
text{*Agreement with the specific embedding for the integers*}
paulson@14378
   674
lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
paulson@14378
   675
proof
paulson@14378
   676
  fix n
paulson@14479
   677
  show "int n = of_nat n"  by (induct n, simp_all add: int_Suc add_ac)
paulson@14378
   678
qed
paulson@14378
   679
paulson@14496
   680
lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)"
paulson@14496
   681
proof
paulson@14496
   682
  fix n
paulson@14496
   683
  show "of_nat n = id n"  by (induct n, simp_all)
paulson@14496
   684
qed
paulson@14496
   685
paulson@14378
   686
nipkow@14740
   687
subsection{*Embedding of the Integers into any @{text comm_ring_1}:
nipkow@14740
   688
@{term of_int}*}
paulson@14378
   689
paulson@14378
   690
constdefs
obua@14738
   691
   of_int :: "int => 'a::comm_ring_1"
paulson@14532
   692
   "of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
paulson@14378
   693
paulson@14378
   694
paulson@14378
   695
lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
paulson@14496
   696
proof -
paulson@15169
   697
  have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
paulson@14496
   698
    by (simp add: congruent_def compare_rls of_nat_add [symmetric]
paulson@14496
   699
            del: of_nat_add) 
paulson@14496
   700
  thus ?thesis
paulson@14496
   701
    by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
paulson@14496
   702
qed
paulson@14378
   703
paulson@14378
   704
lemma of_int_0 [simp]: "of_int 0 = 0"
paulson@14378
   705
by (simp add: of_int Zero_int_def int_def)
paulson@14378
   706
paulson@14378
   707
lemma of_int_1 [simp]: "of_int 1 = 1"
paulson@14378
   708
by (simp add: of_int One_int_def int_def)
paulson@14378
   709
paulson@14378
   710
lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
paulson@14479
   711
by (cases w, cases z, simp add: compare_rls of_int add)
paulson@14378
   712
paulson@14378
   713
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
paulson@14479
   714
by (cases z, simp add: compare_rls of_int minus)
paulson@14259
   715
paulson@14378
   716
lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
paulson@14378
   717
by (simp add: diff_minus)
paulson@14378
   718
paulson@14378
   719
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
paulson@14479
   720
apply (cases w, cases z)
paulson@14479
   721
apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
paulson@14479
   722
                 mult add_ac)
paulson@14378
   723
done
paulson@14378
   724
paulson@14378
   725
lemma of_int_le_iff [simp]:
obua@14738
   726
     "(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)"
paulson@14479
   727
apply (cases w)
paulson@14479
   728
apply (cases z)
paulson@14479
   729
apply (simp add: compare_rls of_int le diff_int_def add minus
paulson@14479
   730
                 of_nat_add [symmetric]   del: of_nat_add)
paulson@14378
   731
done
paulson@14378
   732
paulson@14378
   733
text{*Special cases where either operand is zero*}
paulson@14378
   734
declare of_int_le_iff [of 0, simplified, simp]
paulson@14378
   735
declare of_int_le_iff [of _ 0, simplified, simp]
paulson@14259
   736
paulson@14378
   737
lemma of_int_less_iff [simp]:
obua@14738
   738
     "(of_int w < (of_int z::'a::ordered_idom)) = (w < z)"
paulson@14378
   739
by (simp add: linorder_not_le [symmetric])
paulson@14378
   740
paulson@14378
   741
text{*Special cases where either operand is zero*}
paulson@14378
   742
declare of_int_less_iff [of 0, simplified, simp]
paulson@14378
   743
declare of_int_less_iff [of _ 0, simplified, simp]
paulson@14378
   744
nipkow@14740
   745
text{*The ordering on the @{text comm_ring_1} is necessary.
nipkow@14740
   746
 See @{text of_nat_eq_iff} above.*}
paulson@14378
   747
lemma of_int_eq_iff [simp]:
obua@14738
   748
     "(of_int w = (of_int z::'a::ordered_idom)) = (w = z)"
paulson@14479
   749
by (simp add: order_eq_iff)
paulson@14378
   750
paulson@14378
   751
text{*Special cases where either operand is zero*}
paulson@14378
   752
declare of_int_eq_iff [of 0, simplified, simp]
paulson@14378
   753
declare of_int_eq_iff [of _ 0, simplified, simp]
paulson@14378
   754
paulson@14496
   755
lemma of_int_eq_id [simp]: "of_int = (id :: int => int)"
paulson@14496
   756
proof
paulson@14496
   757
 fix z
paulson@14496
   758
 show "of_int z = id z"  
paulson@14496
   759
  by (cases z,
paulson@14496
   760
      simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus)
paulson@14496
   761
qed
paulson@14496
   762
paulson@14378
   763
paulson@14378
   764
subsection{*The Set of Integers*}
paulson@14378
   765
paulson@14378
   766
constdefs
obua@14738
   767
   Ints  :: "'a::comm_ring_1 set"
paulson@14378
   768
    "Ints == range of_int"
paulson@14271
   769
paulson@14259
   770
paulson@14378
   771
syntax (xsymbols)
paulson@14378
   772
  Ints      :: "'a set"                   ("\<int>")
paulson@14378
   773
paulson@14378
   774
lemma Ints_0 [simp]: "0 \<in> Ints"
paulson@14479
   775
apply (simp add: Ints_def)
paulson@14479
   776
apply (rule range_eqI)
paulson@14378
   777
apply (rule of_int_0 [symmetric])
paulson@14378
   778
done
paulson@14378
   779
paulson@14378
   780
lemma Ints_1 [simp]: "1 \<in> Ints"
paulson@14479
   781
apply (simp add: Ints_def)
paulson@14479
   782
apply (rule range_eqI)
paulson@14378
   783
apply (rule of_int_1 [symmetric])
paulson@14378
   784
done
paulson@14378
   785
paulson@14378
   786
lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
paulson@14479
   787
apply (auto simp add: Ints_def)
paulson@14479
   788
apply (rule range_eqI)
paulson@14378
   789
apply (rule of_int_add [symmetric])
paulson@14378
   790
done
paulson@14378
   791
paulson@14378
   792
lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
paulson@14479
   793
apply (auto simp add: Ints_def)
paulson@14479
   794
apply (rule range_eqI)
paulson@14378
   795
apply (rule of_int_minus [symmetric])
paulson@14378
   796
done
paulson@14378
   797
paulson@14378
   798
lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
paulson@14479
   799
apply (auto simp add: Ints_def)
paulson@14479
   800
apply (rule range_eqI)
paulson@14378
   801
apply (rule of_int_diff [symmetric])
paulson@14378
   802
done
paulson@14378
   803
paulson@14378
   804
lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
paulson@14479
   805
apply (auto simp add: Ints_def)
paulson@14479
   806
apply (rule range_eqI)
paulson@14378
   807
apply (rule of_int_mult [symmetric])
paulson@14378
   808
done
paulson@14378
   809
paulson@14378
   810
text{*Collapse nested embeddings*}
paulson@14378
   811
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
paulson@14479
   812
by (induct n, auto)
paulson@14378
   813
paulson@15013
   814
lemma of_int_int_eq [simp]: "of_int (int n) = of_nat n"
paulson@14479
   815
by (simp add: int_eq_of_nat)
paulson@14341
   816
paulson@14378
   817
lemma Ints_cases [case_names of_int, cases set: Ints]:
paulson@14378
   818
  "q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
paulson@14479
   819
proof (simp add: Ints_def)
paulson@14378
   820
  assume "!!z. q = of_int z ==> C"
paulson@14378
   821
  assume "q \<in> range of_int" thus C ..
paulson@14378
   822
qed
paulson@14378
   823
paulson@14378
   824
lemma Ints_induct [case_names of_int, induct set: Ints]:
paulson@14378
   825
  "q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q"
paulson@14378
   826
  by (rule Ints_cases) auto
paulson@14378
   827
paulson@14378
   828
paulson@14387
   829
(* int (Suc n) = 1 + int n *)
paulson@14387
   830
declare int_Suc [simp]
paulson@14387
   831
paulson@14387
   832
text{*Simplification of @{term "x-y < 0"}, etc.*}
paulson@14387
   833
declare less_iff_diff_less_0 [symmetric, simp]
paulson@14387
   834
declare eq_iff_diff_eq_0 [symmetric, simp]
paulson@14387
   835
declare le_iff_diff_le_0 [symmetric, simp]
paulson@14387
   836
paulson@14378
   837
paulson@14430
   838
subsection{*More Properties of @{term setsum} and  @{term setprod}*}
paulson@14430
   839
paulson@14430
   840
text{*By Jeremy Avigad*}
paulson@14430
   841
paulson@14430
   842
paulson@14430
   843
lemma setsum_of_nat: "of_nat (setsum f A) = setsum (of_nat \<circ> f) A"
paulson@14430
   844
  apply (case_tac "finite A")
paulson@14430
   845
  apply (erule finite_induct, auto)
paulson@14430
   846
  apply (simp add: setsum_def)
paulson@14430
   847
  done
paulson@14430
   848
paulson@14430
   849
lemma setsum_of_int: "of_int (setsum f A) = setsum (of_int \<circ> f) A"
paulson@14430
   850
  apply (case_tac "finite A")
paulson@14430
   851
  apply (erule finite_induct, auto)
paulson@14430
   852
  apply (simp add: setsum_def)
paulson@14430
   853
  done
paulson@14430
   854
paulson@14430
   855
lemma int_setsum: "int (setsum f A) = setsum (int \<circ> f) A"
paulson@14430
   856
  by (subst int_eq_of_nat, rule setsum_of_nat)
paulson@14430
   857
paulson@14430
   858
lemma setprod_of_nat: "of_nat (setprod f A) = setprod (of_nat \<circ> f) A"
paulson@14430
   859
  apply (case_tac "finite A")
paulson@14430
   860
  apply (erule finite_induct, auto)
paulson@14430
   861
  apply (simp add: setprod_def)
paulson@14430
   862
  done
paulson@14430
   863
paulson@14430
   864
lemma setprod_of_int: "of_int (setprod f A) = setprod (of_int \<circ> f) A"
paulson@14430
   865
  apply (case_tac "finite A")
paulson@14430
   866
  apply (erule finite_induct, auto)
paulson@14430
   867
  apply (simp add: setprod_def)
paulson@14430
   868
  done
paulson@14430
   869
paulson@14430
   870
lemma int_setprod: "int (setprod f A) = setprod (int \<circ> f) A"
paulson@14430
   871
  by (subst int_eq_of_nat, rule setprod_of_nat)
paulson@14430
   872
paulson@15047
   873
lemma setsum_constant [simp]: "finite A ==> (\<Sum>x \<in> A. y) = of_nat(card A) * y"
paulson@14430
   874
  apply (erule finite_induct)
paulson@14430
   875
  apply (auto simp add: ring_distrib add_ac)
paulson@14430
   876
  done
paulson@14430
   877
paulson@14430
   878
lemma setprod_nonzero_nat:
paulson@14430
   879
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
paulson@14430
   880
  by (rule setprod_nonzero, auto)
paulson@14430
   881
paulson@14430
   882
lemma setprod_zero_eq_nat:
paulson@14430
   883
    "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
paulson@14430
   884
  by (rule setprod_zero_eq, auto)
paulson@14430
   885
paulson@14430
   886
lemma setprod_nonzero_int:
paulson@14430
   887
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
paulson@14430
   888
  by (rule setprod_nonzero, auto)
paulson@14430
   889
paulson@14430
   890
lemma setprod_zero_eq_int:
paulson@14430
   891
    "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
paulson@14430
   892
  by (rule setprod_zero_eq, auto)
paulson@14430
   893
paulson@14430
   894
paulson@14479
   895
text{*Now we replace the case analysis rule by a more conventional one:
paulson@14479
   896
whether an integer is negative or not.*}
paulson@14479
   897
paulson@14479
   898
lemma zless_iff_Suc_zadd:
paulson@14479
   899
    "(w < z) = (\<exists>n. z = w + int(Suc n))"
paulson@14479
   900
apply (cases z, cases w)
paulson@14479
   901
apply (auto simp add: le add int_def linorder_not_le [symmetric]) 
paulson@14479
   902
apply (rename_tac a b c d) 
paulson@14479
   903
apply (rule_tac x="a+d - Suc(c+b)" in exI) 
paulson@14479
   904
apply arith
paulson@14479
   905
done
paulson@14479
   906
paulson@14479
   907
lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
paulson@14479
   908
apply (cases x)
paulson@14479
   909
apply (auto simp add: le minus Zero_int_def int_def order_less_le) 
paulson@14496
   910
apply (rule_tac x="y - Suc x" in exI, arith)
paulson@14479
   911
done
paulson@14479
   912
paulson@14479
   913
theorem int_cases [cases type: int, case_names nonneg neg]:
paulson@14479
   914
     "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
paulson@14479
   915
apply (case_tac "z < 0", blast dest!: negD)
paulson@14479
   916
apply (simp add: linorder_not_less)
paulson@14479
   917
apply (blast dest: nat_0_le [THEN sym])
paulson@14479
   918
done
paulson@14479
   919
paulson@14479
   920
theorem int_induct [induct type: int, case_names nonneg neg]:
paulson@14479
   921
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
paulson@14479
   922
  by (cases z) auto
paulson@14479
   923
paulson@14479
   924
paulson@15013
   925
lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z"
paulson@15013
   926
apply (cases z)
paulson@15013
   927
apply (simp_all add: not_zle_0_negative del: int_Suc)
paulson@15013
   928
done
paulson@15013
   929
paulson@15013
   930
paulson@14378
   931
(*Legacy ML bindings, but no longer the structure Int.*)
paulson@14259
   932
ML
paulson@14259
   933
{*
paulson@14378
   934
val zabs_def = thm "zabs_def"
paulson@14378
   935
paulson@14378
   936
val int_0 = thm "int_0";
paulson@14378
   937
val int_1 = thm "int_1";
paulson@14378
   938
val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
paulson@14378
   939
val neg_eq_less_0 = thm "neg_eq_less_0";
paulson@14378
   940
val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
paulson@14378
   941
val not_neg_0 = thm "not_neg_0";
paulson@14378
   942
val not_neg_1 = thm "not_neg_1";
paulson@14378
   943
val iszero_0 = thm "iszero_0";
paulson@14378
   944
val not_iszero_1 = thm "not_iszero_1";
paulson@14378
   945
val int_0_less_1 = thm "int_0_less_1";
paulson@14378
   946
val int_0_neq_1 = thm "int_0_neq_1";
paulson@14378
   947
val negative_zless = thm "negative_zless";
paulson@14378
   948
val negative_zle = thm "negative_zle";
paulson@14378
   949
val not_zle_0_negative = thm "not_zle_0_negative";
paulson@14378
   950
val not_int_zless_negative = thm "not_int_zless_negative";
paulson@14378
   951
val negative_eq_positive = thm "negative_eq_positive";
paulson@14378
   952
val zle_iff_zadd = thm "zle_iff_zadd";
paulson@14378
   953
val abs_int_eq = thm "abs_int_eq";
paulson@14378
   954
val abs_split = thm"abs_split";
paulson@14378
   955
val nat_int = thm "nat_int";
paulson@14378
   956
val nat_zminus_int = thm "nat_zminus_int";
paulson@14378
   957
val nat_zero = thm "nat_zero";
paulson@14378
   958
val not_neg_nat = thm "not_neg_nat";
paulson@14378
   959
val neg_nat = thm "neg_nat";
paulson@14378
   960
val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
paulson@14378
   961
val nat_0_le = thm "nat_0_le";
paulson@14378
   962
val nat_le_0 = thm "nat_le_0";
paulson@14378
   963
val zless_nat_conj = thm "zless_nat_conj";
paulson@14378
   964
val int_cases = thm "int_cases";
paulson@14378
   965
paulson@14259
   966
val int_def = thm "int_def";
paulson@14259
   967
val Zero_int_def = thm "Zero_int_def";
paulson@14259
   968
val One_int_def = thm "One_int_def";
paulson@14479
   969
val diff_int_def = thm "diff_int_def";
paulson@14259
   970
paulson@14259
   971
val inj_int = thm "inj_int";
paulson@14259
   972
val zminus_zminus = thm "zminus_zminus";
paulson@14259
   973
val zminus_0 = thm "zminus_0";
paulson@14259
   974
val zminus_zadd_distrib = thm "zminus_zadd_distrib";
paulson@14259
   975
val zadd_commute = thm "zadd_commute";
paulson@14259
   976
val zadd_assoc = thm "zadd_assoc";
paulson@14259
   977
val zadd_left_commute = thm "zadd_left_commute";
paulson@14259
   978
val zadd_ac = thms "zadd_ac";
paulson@14271
   979
val zmult_ac = thms "zmult_ac";
paulson@14259
   980
val zadd_int = thm "zadd_int";
paulson@14259
   981
val zadd_int_left = thm "zadd_int_left";
paulson@14259
   982
val int_Suc = thm "int_Suc";
paulson@14259
   983
val zadd_0 = thm "zadd_0";
paulson@14259
   984
val zadd_0_right = thm "zadd_0_right";
paulson@14259
   985
val zmult_zminus = thm "zmult_zminus";
paulson@14259
   986
val zmult_commute = thm "zmult_commute";
paulson@14259
   987
val zmult_assoc = thm "zmult_assoc";
paulson@14259
   988
val zadd_zmult_distrib = thm "zadd_zmult_distrib";
paulson@14259
   989
val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2";
paulson@14259
   990
val zdiff_zmult_distrib = thm "zdiff_zmult_distrib";
paulson@14259
   991
val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2";
paulson@14259
   992
val int_distrib = thms "int_distrib";
paulson@14259
   993
val zmult_int = thm "zmult_int";
paulson@14259
   994
val zmult_1 = thm "zmult_1";
paulson@14259
   995
val zmult_1_right = thm "zmult_1_right";
paulson@14259
   996
val int_int_eq = thm "int_int_eq";
paulson@14259
   997
val int_eq_0_conv = thm "int_eq_0_conv";
paulson@14259
   998
val zless_int = thm "zless_int";
paulson@14259
   999
val int_less_0_conv = thm "int_less_0_conv";
paulson@14259
  1000
val zero_less_int_conv = thm "zero_less_int_conv";
paulson@14259
  1001
val zle_int = thm "zle_int";
paulson@14259
  1002
val zero_zle_int = thm "zero_zle_int";
paulson@14259
  1003
val int_le_0_conv = thm "int_le_0_conv";
paulson@14259
  1004
val zle_refl = thm "zle_refl";
paulson@14259
  1005
val zle_linear = thm "zle_linear";
paulson@14259
  1006
val zle_trans = thm "zle_trans";
paulson@14259
  1007
val zle_anti_sym = thm "zle_anti_sym";
paulson@14378
  1008
paulson@14378
  1009
val Ints_def = thm "Ints_def";
paulson@14378
  1010
val Nats_def = thm "Nats_def";
paulson@14378
  1011
paulson@14378
  1012
val of_nat_0 = thm "of_nat_0";
paulson@14378
  1013
val of_nat_Suc = thm "of_nat_Suc";
paulson@14378
  1014
val of_nat_1 = thm "of_nat_1";
paulson@14378
  1015
val of_nat_add = thm "of_nat_add";
paulson@14378
  1016
val of_nat_mult = thm "of_nat_mult";
paulson@14378
  1017
val zero_le_imp_of_nat = thm "zero_le_imp_of_nat";
paulson@14378
  1018
val less_imp_of_nat_less = thm "less_imp_of_nat_less";
paulson@14378
  1019
val of_nat_less_imp_less = thm "of_nat_less_imp_less";
paulson@14378
  1020
val of_nat_less_iff = thm "of_nat_less_iff";
paulson@14378
  1021
val of_nat_le_iff = thm "of_nat_le_iff";
paulson@14378
  1022
val of_nat_eq_iff = thm "of_nat_eq_iff";
paulson@14378
  1023
val Nats_0 = thm "Nats_0";
paulson@14378
  1024
val Nats_1 = thm "Nats_1";
paulson@14378
  1025
val Nats_add = thm "Nats_add";
paulson@14378
  1026
val Nats_mult = thm "Nats_mult";
paulson@14387
  1027
val int_eq_of_nat = thm"int_eq_of_nat";
paulson@14378
  1028
val of_int = thm "of_int";
paulson@14378
  1029
val of_int_0 = thm "of_int_0";
paulson@14378
  1030
val of_int_1 = thm "of_int_1";
paulson@14378
  1031
val of_int_add = thm "of_int_add";
paulson@14378
  1032
val of_int_minus = thm "of_int_minus";
paulson@14378
  1033
val of_int_diff = thm "of_int_diff";
paulson@14378
  1034
val of_int_mult = thm "of_int_mult";
paulson@14378
  1035
val of_int_le_iff = thm "of_int_le_iff";
paulson@14378
  1036
val of_int_less_iff = thm "of_int_less_iff";
paulson@14378
  1037
val of_int_eq_iff = thm "of_int_eq_iff";
paulson@14378
  1038
val Ints_0 = thm "Ints_0";
paulson@14378
  1039
val Ints_1 = thm "Ints_1";
paulson@14378
  1040
val Ints_add = thm "Ints_add";
paulson@14378
  1041
val Ints_minus = thm "Ints_minus";
paulson@14378
  1042
val Ints_diff = thm "Ints_diff";
paulson@14378
  1043
val Ints_mult = thm "Ints_mult";
paulson@14378
  1044
val of_int_of_nat_eq = thm"of_int_of_nat_eq";
paulson@14378
  1045
val Ints_cases = thm "Ints_cases";
paulson@14378
  1046
val Ints_induct = thm "Ints_induct";
paulson@14259
  1047
*}
paulson@14259
  1048
paulson@5508
  1049
end