author | huffman |
Sat, 09 Jun 2007 02:38:51 +0200 | |
changeset 23299 | 292b1cbd05dc |
parent 23282 | dfc459989d24 |
child 23303 | 6091e530ff77 |
permissions | -rw-r--r-- |
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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 Nat |
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begin |
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text {* the equivalence relation underlying the integers *} |
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definition |
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intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" |
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where |
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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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definition |
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int :: "nat \<Rightarrow> int" |
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where |
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[code func del]: "int m = Abs_Integ (intrel `` {(m, 0)})" |
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instance int :: zero |
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Zero_int_def: "0 \<equiv> Abs_Integ (intrel `` {(0, 0)})" .. |
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instance int :: one |
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One_int_def: "1 \<equiv> Abs_Integ (intrel `` {(1, 0)})" .. |
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instance int :: plus |
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add_int_def: "z + w \<equiv> Abs_Integ |
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(\<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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instance int :: minus |
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minus_int_def: |
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"- z \<equiv> Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})" |
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diff_int_def: "z - w \<equiv> z + (-w)" .. |
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instance int :: times |
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mult_int_def: "z * w \<equiv> Abs_Integ |
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(\<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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instance int :: ord |
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le_int_def: |
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"z \<le> w \<equiv> \<exists>x y u v. x+v \<le> u+y \<and> (x, y) \<in> Rep_Integ z \<and> (u, v) \<in> Rep_Integ w" |
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less_int_def: "z < w \<equiv> z \<le> w \<and> z \<noteq> w" .. |
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lemmas [code func del] = Zero_int_def One_int_def add_int_def |
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minus_int_def mult_int_def le_int_def less_int_def |
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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 [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I] |
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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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@{prop "\<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{*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: 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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(*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) |
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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: 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 zmult_1: "(1::int) * z = z" |
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by (cases z, simp add: One_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)" by (simp add: Zero_int_def One_int_def) |
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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 less: |
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"(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)" |
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by (simp add: less_int_def le order_less_le) |
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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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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 less_int_def [THEN meta_eq_to_obj_eq])+ |
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lemma zle_linear: "(z::int) \<le> w \<or> 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_0_less_1: "0 < (1::int)" |
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by (simp add: Zero_int_def One_int_def less) |
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lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)" |
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by (rule int_0_less_1 [THEN less_imp_neq]) |
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subsection{*Monotonicity results*} |
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instance int :: pordered_cancel_ab_semigroup_add |
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proof |
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fix a b c :: int |
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show "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
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by (cases a, cases b, cases c, simp add: le add) |
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qed |
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lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)" |
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by (rule add_left_mono) |
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lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)" |
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by (rule add_strict_right_mono) |
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lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)" |
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by (rule add_less_le_mono) |
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subsection{*Strict Monotonicity of Multiplication*} |
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text{*strict, in 1st argument; proof is by induction on k>0*} |
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lemma zmult_zless_mono2_lemma: |
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"(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j" |
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apply (induct "k", simp) |
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apply (simp add: left_distrib) |
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apply (case_tac "k=0") |
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apply (simp_all add: add_strict_mono) |
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done |
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lemma int_of_nat_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})" |
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by (induct m, simp_all add: Zero_int_def One_int_def add) |
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lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n" |
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apply (cases k) |
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apply (auto simp add: le add int_of_nat_def Zero_int_def) |
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apply (rule_tac x="x-y" in exI, simp) |
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done |
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305 |
|
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n" |
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apply (cases k) |
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apply (simp add: less int_of_nat_def Zero_int_def) |
23164 | 309 |
apply (rule_tac x="x-y" in exI, simp) |
310 |
done |
|
311 |
||
312 |
lemma zmult_zless_mono2: "[| i<j; (0::int) < k |] ==> k*i < k*j" |
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apply (drule zero_less_imp_eq_int) |
23164 | 314 |
apply (auto simp add: zmult_zless_mono2_lemma) |
315 |
done |
|
316 |
||
317 |
instance int :: minus |
|
318 |
zabs_def: "\<bar>i\<Colon>int\<bar> \<equiv> if i < 0 then - i else i" .. |
|
319 |
||
320 |
instance int :: distrib_lattice |
|
321 |
"inf \<equiv> min" |
|
322 |
"sup \<equiv> max" |
|
323 |
by intro_classes |
|
324 |
(auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1) |
|
325 |
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text{*The integers form an ordered integral domain*} |
23164 | 327 |
instance int :: ordered_idom |
328 |
proof |
|
329 |
fix i j k :: int |
|
330 |
show "i < j ==> 0 < k ==> k * i < k * j" by (rule zmult_zless_mono2) |
|
331 |
show "\<bar>i\<bar> = (if i < 0 then -i else i)" by (simp only: zabs_def) |
|
332 |
qed |
|
333 |
||
334 |
lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z" |
|
335 |
apply (cases w, cases z) |
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apply (simp add: less le add One_int_def) |
23164 | 337 |
done |
338 |
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339 |
|
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subsection{*@{term int}: Embedding the Naturals into the Integers*} |
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341 |
|
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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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344 |
|
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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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347 |
|
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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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350 |
|
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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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353 |
|
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lemma int_mult: "int (m * n) = (int m) * (int n)" |
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355 |
by (simp add: int_def mult) |
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356 |
|
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357 |
text{*Compatibility binding*} |
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lemmas zmult_int = int_mult [symmetric] |
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359 |
|
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lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)" |
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361 |
by (simp add: Zero_int_def [folded int_def]) |
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362 |
|
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363 |
lemma zless_int [simp]: "(int m < int n) = (m<n)" |
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364 |
by (simp add: le add int_def linorder_not_le [symmetric]) |
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365 |
|
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366 |
lemma int_less_0_conv [simp]: "~ (int k < 0)" |
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367 |
by (simp add: Zero_int_def [folded int_def]) |
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368 |
|
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lemma zero_less_int_conv [simp]: "(0 < int n) = (0 < n)" |
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370 |
by (simp add: Zero_int_def [folded int_def]) |
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371 |
|
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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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374 |
|
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lemma zero_zle_int [simp]: "(0 \<le> int n)" |
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376 |
by (simp add: Zero_int_def [folded int_def]) |
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377 |
|
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lemma int_le_0_conv [simp]: "(int n \<le> 0) = (n = 0)" |
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379 |
by (simp add: Zero_int_def [folded int_def]) |
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380 |
|
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381 |
lemma int_0 [simp]: "int 0 = (0::int)" |
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382 |
by (simp add: Zero_int_def [folded int_def]) |
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383 |
|
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lemma int_1 [simp]: "int 1 = 1" |
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by (simp add: One_int_def [folded int_def]) |
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386 |
|
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lemma int_Suc0_eq_1: "int (Suc 0) = 1" |
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388 |
by (simp add: One_int_def [folded int_def]) |
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389 |
|
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lemma int_Suc: "int (Suc m) = 1 + (int m)" |
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391 |
by (simp add: One_int_def [folded int_def] zadd_int) |
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392 |
|
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393 |
|
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394 |
subsection{*Magnitude of an Integer, as a Natural Number: @{term nat}*} |
23164 | 395 |
|
396 |
definition |
|
397 |
nat :: "int \<Rightarrow> nat" |
|
398 |
where |
|
399 |
[code func del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})" |
|
400 |
||
401 |
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y" |
|
402 |
proof - |
|
403 |
have "(\<lambda>(x,y). {x-y}) respects intrel" |
|
404 |
by (simp add: congruent_def) arith |
|
405 |
thus ?thesis |
|
406 |
by (simp add: nat_def UN_equiv_class [OF equiv_intrel]) |
|
407 |
qed |
|
408 |
||
409 |
lemma nat_int [simp]: "nat(int n) = n" |
|
410 |
by (simp add: nat int_def) |
|
411 |
||
412 |
lemma nat_zero [simp]: "nat 0 = 0" |
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by (simp only: Zero_int_def [folded int_def] nat_int) |
23164 | 414 |
|
415 |
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)" |
|
416 |
by (cases z, simp add: nat le int_def Zero_int_def) |
|
417 |
||
418 |
corollary nat_0_le: "0 \<le> z ==> int (nat z) = z" |
|
419 |
by simp |
|
420 |
||
421 |
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0" |
|
422 |
by (cases z, simp add: nat le int_def Zero_int_def) |
|
423 |
||
424 |
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)" |
|
425 |
apply (cases w, cases z) |
|
426 |
apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith) |
|
427 |
done |
|
428 |
||
429 |
text{*An alternative condition is @{term "0 \<le> w"} *} |
|
430 |
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)" |
|
431 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
|
432 |
||
433 |
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)" |
|
434 |
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) |
|
435 |
||
436 |
lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)" |
|
437 |
apply (cases w, cases z) |
|
438 |
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith) |
|
439 |
done |
|
440 |
||
441 |
lemma nonneg_eq_int: "[| 0 \<le> z; !!m. z = int m ==> P |] ==> P" |
|
442 |
by (blast dest: nat_0_le sym) |
|
443 |
||
444 |
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)" |
|
445 |
by (cases w, simp add: nat le int_def Zero_int_def, arith) |
|
446 |
||
447 |
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)" |
|
448 |
by (simp only: eq_commute [of m] nat_eq_iff) |
|
449 |
||
450 |
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)" |
|
451 |
apply (cases w) |
|
452 |
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith) |
|
453 |
done |
|
454 |
||
455 |
lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)" |
|
456 |
by (auto simp add: nat_eq_iff2) |
|
457 |
||
458 |
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)" |
|
459 |
by (insert zless_nat_conj [of 0], auto) |
|
460 |
||
461 |
lemma nat_add_distrib: |
|
462 |
"[| (0::int) \<le> z; 0 \<le> z' |] ==> nat (z+z') = nat z + nat z'" |
|
463 |
by (cases z, cases z', simp add: nat add le int_def Zero_int_def) |
|
464 |
||
465 |
lemma nat_diff_distrib: |
|
466 |
"[| (0::int) \<le> z'; z' \<le> z |] ==> nat (z-z') = nat z - nat z'" |
|
467 |
by (cases z, cases z', |
|
468 |
simp add: nat add minus diff_minus le int_def Zero_int_def) |
|
469 |
||
470 |
||
471 |
lemma nat_zminus_int [simp]: "nat (- (int n)) = 0" |
|
472 |
by (simp add: int_def minus nat Zero_int_def) |
|
473 |
||
474 |
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)" |
|
475 |
by (cases z, simp add: nat le int_def linorder_not_le [symmetric], arith) |
|
476 |
||
477 |
||
478 |
subsection{*Lemmas about the Function @{term int} and Orderings*} |
|
479 |
||
480 |
lemma negative_zless_0: "- (int (Suc n)) < 0" |
|
481 |
by (simp add: order_less_le) |
|
482 |
||
483 |
lemma negative_zless [iff]: "- (int (Suc n)) < int m" |
|
484 |
by (rule negative_zless_0 [THEN order_less_le_trans], simp) |
|
485 |
||
486 |
lemma negative_zle_0: "- int n \<le> 0" |
|
487 |
by (simp add: minus_le_iff) |
|
488 |
||
489 |
lemma negative_zle [iff]: "- int n \<le> int m" |
|
490 |
by (rule order_trans [OF negative_zle_0 zero_zle_int]) |
|
491 |
||
492 |
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))" |
|
493 |
by (subst le_minus_iff, simp) |
|
494 |
||
495 |
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)" |
|
496 |
by (simp add: int_def le minus Zero_int_def) |
|
497 |
||
498 |
lemma not_int_zless_negative [simp]: "~ (int n < - int m)" |
|
499 |
by (simp add: linorder_not_less) |
|
500 |
||
501 |
lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)" |
|
502 |
by (force simp add: order_eq_iff [of "- int n"] int_zle_neg) |
|
503 |
||
504 |
lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)" |
|
505 |
proof (cases w, cases z, simp add: le add int_def) |
|
506 |
fix a b c d |
|
507 |
assume "w = Abs_Integ (intrel `` {(a,b)})" "z = Abs_Integ (intrel `` {(c,d)})" |
|
508 |
show "(a+d \<le> c+b) = (\<exists>n. c+b = a+n+d)" |
|
509 |
proof |
|
510 |
assume "a + d \<le> c + b" |
|
511 |
thus "\<exists>n. c + b = a + n + d" |
|
512 |
by (auto intro!: exI [where x="c+b - (a+d)"]) |
|
513 |
next |
|
514 |
assume "\<exists>n. c + b = a + n + d" |
|
515 |
then obtain n where "c + b = a + n + d" .. |
|
516 |
thus "a + d \<le> c + b" by arith |
|
517 |
qed |
|
518 |
qed |
|
519 |
||
520 |
lemma abs_int_eq [simp]: "abs (int m) = int m" |
|
521 |
by (simp add: abs_if) |
|
522 |
||
523 |
text{*This version is proved for all ordered rings, not just integers! |
|
524 |
It is proved here because attribute @{text arith_split} is not available |
|
525 |
in theory @{text Ring_and_Field}. |
|
526 |
But is it really better than just rewriting with @{text abs_if}?*} |
|
527 |
lemma abs_split [arith_split]: |
|
528 |
"P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))" |
|
529 |
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less) |
|
530 |
||
531 |
||
532 |
subsection {* Constants @{term neg} and @{term iszero} *} |
|
533 |
||
534 |
definition |
|
535 |
neg :: "'a\<Colon>ordered_idom \<Rightarrow> bool" |
|
536 |
where |
|
537 |
[code inline]: "neg Z \<longleftrightarrow> Z < 0" |
|
538 |
||
539 |
definition (*for simplifying equalities*) |
|
23276
a70934b63910
generalize of_nat and related constants to class semiring_1
huffman
parents:
23164
diff
changeset
|
540 |
iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool" |
23164 | 541 |
where |
542 |
"iszero z \<longleftrightarrow> z = 0" |
|
543 |
||
544 |
lemma not_neg_int [simp]: "~ neg(int n)" |
|
545 |
by (simp add: neg_def) |
|
546 |
||
547 |
lemma neg_zminus_int [simp]: "neg(- (int (Suc n)))" |
|
548 |
by (simp add: neg_def neg_less_0_iff_less) |
|
549 |
||
550 |
lemmas neg_eq_less_0 = neg_def |
|
551 |
||
552 |
lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)" |
|
553 |
by (simp add: neg_def linorder_not_less) |
|
554 |
||
555 |
||
556 |
subsection{*To simplify inequalities when Numeral1 can get simplified to 1*} |
|
557 |
||
558 |
lemma not_neg_0: "~ neg 0" |
|
559 |
by (simp add: One_int_def neg_def) |
|
560 |
||
561 |
lemma not_neg_1: "~ neg 1" |
|
562 |
by (simp add: neg_def linorder_not_less zero_le_one) |
|
563 |
||
564 |
lemma iszero_0: "iszero 0" |
|
565 |
by (simp add: iszero_def) |
|
566 |
||
567 |
lemma not_iszero_1: "~ iszero 1" |
|
568 |
by (simp add: iszero_def eq_commute) |
|
569 |
||
570 |
lemma neg_nat: "neg z ==> nat z = 0" |
|
571 |
by (simp add: neg_def order_less_imp_le) |
|
572 |
||
573 |
lemma not_neg_nat: "~ neg z ==> int (nat z) = z" |
|
574 |
by (simp add: linorder_not_less neg_def) |
|
575 |
||
576 |
||
577 |
subsection{*The Set of Natural Numbers*} |
|
578 |
||
579 |
constdefs |
|
23276
a70934b63910
generalize of_nat and related constants to class semiring_1
huffman
parents:
23164
diff
changeset
|
580 |
Nats :: "'a::semiring_1 set" |
23164 | 581 |
"Nats == range of_nat" |
582 |
||
583 |
notation (xsymbols) |
|
584 |
Nats ("\<nat>") |
|
585 |
||
586 |
lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats" |
|
587 |
by (simp add: Nats_def) |
|
588 |
||
589 |
lemma Nats_0 [simp]: "0 \<in> Nats" |
|
590 |
apply (simp add: Nats_def) |
|
591 |
apply (rule range_eqI) |
|
592 |
apply (rule of_nat_0 [symmetric]) |
|
593 |
done |
|
594 |
||
595 |
lemma Nats_1 [simp]: "1 \<in> Nats" |
|
596 |
apply (simp add: Nats_def) |
|
597 |
apply (rule range_eqI) |
|
598 |
apply (rule of_nat_1 [symmetric]) |
|
599 |
done |
|
600 |
||
601 |
lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats" |
|
602 |
apply (auto simp add: Nats_def) |
|
603 |
apply (rule range_eqI) |
|
604 |
apply (rule of_nat_add [symmetric]) |
|
605 |
done |
|
606 |
||
607 |
lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats" |
|
608 |
apply (auto simp add: Nats_def) |
|
609 |
apply (rule range_eqI) |
|
610 |
apply (rule of_nat_mult [symmetric]) |
|
611 |
done |
|
612 |
||
613 |
text{*Agreement with the specific embedding for the integers*} |
|
614 |
lemma int_eq_of_nat: "int = (of_nat :: nat => int)" |
|
615 |
proof |
|
616 |
fix n |
|
617 |
show "int n = of_nat n" by (induct n, simp_all add: int_Suc add_ac) |
|
618 |
qed |
|
619 |
||
620 |
lemma of_nat_eq_id [simp]: "of_nat = (id :: nat => nat)" |
|
621 |
proof |
|
622 |
fix n |
|
623 |
show "of_nat n = id n" by (induct n, simp_all) |
|
624 |
qed |
|
625 |
||
626 |
||
627 |
subsection{*Embedding of the Integers into any @{text ring_1}: |
|
628 |
@{term of_int}*} |
|
629 |
||
630 |
constdefs |
|
631 |
of_int :: "int => 'a::ring_1" |
|
632 |
"of_int z == contents (\<Union>(i,j) \<in> Rep_Integ z. { of_nat i - of_nat j })" |
|
633 |
||
634 |
||
635 |
lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j" |
|
636 |
proof - |
|
637 |
have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel" |
|
638 |
by (simp add: congruent_def compare_rls of_nat_add [symmetric] |
|
639 |
del: of_nat_add) |
|
640 |
thus ?thesis |
|
641 |
by (simp add: of_int_def UN_equiv_class [OF equiv_intrel]) |
|
642 |
qed |
|
643 |
||
644 |
lemma of_int_0 [simp]: "of_int 0 = 0" |
|
645 |
by (simp add: of_int Zero_int_def int_def) |
|
646 |
||
647 |
lemma of_int_1 [simp]: "of_int 1 = 1" |
|
648 |
by (simp add: of_int One_int_def int_def) |
|
649 |
||
650 |
lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z" |
|
651 |
by (cases w, cases z, simp add: compare_rls of_int add) |
|
652 |
||
653 |
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)" |
|
654 |
by (cases z, simp add: compare_rls of_int minus) |
|
655 |
||
656 |
lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z" |
|
657 |
by (simp add: diff_minus) |
|
658 |
||
659 |
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z" |
|
660 |
apply (cases w, cases z) |
|
661 |
apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib |
|
662 |
mult add_ac) |
|
663 |
done |
|
664 |
||
665 |
lemma of_int_le_iff [simp]: |
|
666 |
"(of_int w \<le> (of_int z::'a::ordered_idom)) = (w \<le> z)" |
|
667 |
apply (cases w) |
|
668 |
apply (cases z) |
|
669 |
apply (simp add: compare_rls of_int le diff_int_def add minus |
|
670 |
of_nat_add [symmetric] del: of_nat_add) |
|
671 |
done |
|
672 |
||
673 |
text{*Special cases where either operand is zero*} |
|
674 |
lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified] |
|
675 |
lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified] |
|
676 |
||
677 |
||
678 |
lemma of_int_less_iff [simp]: |
|
679 |
"(of_int w < (of_int z::'a::ordered_idom)) = (w < z)" |
|
680 |
by (simp add: linorder_not_le [symmetric]) |
|
681 |
||
682 |
text{*Special cases where either operand is zero*} |
|
683 |
lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified] |
|
684 |
lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified] |
|
685 |
||
686 |
text{*Class for unital rings with characteristic zero. |
|
687 |
Includes non-ordered rings like the complex numbers.*} |
|
23282
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23276
diff
changeset
|
688 |
axclass ring_char_0 < ring_1, semiring_char_0 |
23164 | 689 |
|
690 |
lemma of_int_eq_iff [simp]: |
|
691 |
"(of_int w = (of_int z::'a::ring_char_0)) = (w = z)" |
|
23282
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23276
diff
changeset
|
692 |
apply (cases w, cases z, simp add: of_int) |
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23276
diff
changeset
|
693 |
apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq) |
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23276
diff
changeset
|
694 |
apply (simp only: of_nat_add [symmetric] of_nat_eq_iff) |
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23276
diff
changeset
|
695 |
done |
23164 | 696 |
|
697 |
text{*Every @{text ordered_idom} has characteristic zero.*} |
|
23282
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23276
diff
changeset
|
698 |
instance ordered_idom < ring_char_0 .. |
23164 | 699 |
|
700 |
text{*Special cases where either operand is zero*} |
|
701 |
lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified] |
|
702 |
lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified] |
|
703 |
||
704 |
lemma of_int_eq_id [simp]: "of_int = (id :: int => int)" |
|
705 |
proof |
|
706 |
fix z |
|
23299
292b1cbd05dc
remove dependencies of proofs on constant int::nat=>int, preparing to remove it
huffman
parents:
23282
diff
changeset
|
707 |
show "of_int z = id z" |
23164 | 708 |
by (cases z) |
709 |
(simp add: of_int add minus int_eq_of_nat [symmetric] int_def diff_minus) |
|
710 |
qed |
|
711 |
||
712 |
||
713 |
subsection{*The Set of Integers*} |
|
714 |
||
715 |
constdefs |
|
716 |
Ints :: "'a::ring_1 set" |
|
717 |
"Ints == range of_int" |
|
718 |
||
719 |
notation (xsymbols) |
|
720 |
Ints ("\<int>") |
|
721 |
||
722 |
lemma Ints_0 [simp]: "0 \<in> Ints" |
|
723 |
apply (simp add: Ints_def) |
|
724 |
apply (rule range_eqI) |
|
725 |
apply (rule of_int_0 [symmetric]) |
|
726 |
done |
|
727 |
||
728 |
lemma Ints_1 [simp]: "1 \<in> Ints" |
|
729 |
apply (simp add: Ints_def) |
|
730 |
apply (rule range_eqI) |
|
731 |
apply (rule of_int_1 [symmetric]) |
|
732 |
done |
|
733 |
||
734 |
lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints" |
|
735 |
apply (auto simp add: Ints_def) |
|
736 |
apply (rule range_eqI) |
|
737 |
apply (rule of_int_add [symmetric]) |
|
738 |
done |
|
739 |
||
740 |
lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints" |
|
741 |
apply (auto simp add: Ints_def) |
|
742 |
apply (rule range_eqI) |
|
743 |
apply (rule of_int_minus [symmetric]) |
|
744 |
done |
|
745 |
||
746 |
lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints" |
|
747 |
apply (auto simp add: Ints_def) |
|
748 |
apply (rule range_eqI) |
|
749 |
apply (rule of_int_diff [symmetric]) |
|
750 |
done |
|
751 |
||
752 |
lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints" |
|
753 |
apply (auto simp add: Ints_def) |
|
754 |
apply (rule range_eqI) |
|
755 |
apply (rule of_int_mult [symmetric]) |
|
756 |
done |
|
757 |
||
758 |
text{*Collapse nested embeddings*} |
|
759 |
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n" |
|
760 |
by (induct n, auto) |
|
761 |
||
762 |
lemma of_int_int_eq [simp]: "of_int (int n) = of_nat n" |
|
763 |
by (simp add: int_eq_of_nat) |
|
764 |
||
765 |
lemma Ints_cases [cases set: Ints]: |
|
766 |
assumes "q \<in> \<int>" |
|
767 |
obtains (of_int) z where "q = of_int z" |
|
768 |
unfolding Ints_def |
|
769 |
proof - |
|
770 |
from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def . |
|
771 |
then obtain z where "q = of_int z" .. |
|
772 |
then show thesis .. |
|
773 |
qed |
|
774 |
||
775 |
lemma Ints_induct [case_names of_int, induct set: Ints]: |
|
776 |
"q \<in> \<int> ==> (!!z. P (of_int z)) ==> P q" |
|
777 |
by (rule Ints_cases) auto |
|
778 |
||
779 |
||
780 |
(* int (Suc n) = 1 + int n *) |
|
781 |
||
782 |
||
783 |
||
784 |
subsection{*More Properties of @{term setsum} and @{term setprod}*} |
|
785 |
||
786 |
text{*By Jeremy Avigad*} |
|
787 |
||
788 |
||
789 |
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))" |
|
790 |
apply (cases "finite A") |
|
791 |
apply (erule finite_induct, auto) |
|
792 |
done |
|
793 |
||
794 |
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))" |
|
795 |
apply (cases "finite A") |
|
796 |
apply (erule finite_induct, auto) |
|
797 |
done |
|
798 |
||
799 |
lemma int_setsum: "int (setsum f A) = (\<Sum>x\<in>A. int(f x))" |
|
800 |
by (simp add: int_eq_of_nat of_nat_setsum) |
|
801 |
||
802 |
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))" |
|
803 |
apply (cases "finite A") |
|
804 |
apply (erule finite_induct, auto) |
|
805 |
done |
|
806 |
||
807 |
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))" |
|
808 |
apply (cases "finite A") |
|
809 |
apply (erule finite_induct, auto) |
|
810 |
done |
|
811 |
||
812 |
lemma int_setprod: "int (setprod f A) = (\<Prod>x\<in>A. int(f x))" |
|
813 |
by (simp add: int_eq_of_nat of_nat_setprod) |
|
814 |
||
815 |
lemma setprod_nonzero_nat: |
|
816 |
"finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0" |
|
817 |
by (rule setprod_nonzero, auto) |
|
818 |
||
819 |
lemma setprod_zero_eq_nat: |
|
820 |
"finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)" |
|
821 |
by (rule setprod_zero_eq, auto) |
|
822 |
||
823 |
lemma setprod_nonzero_int: |
|
824 |
"finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0" |
|
825 |
by (rule setprod_nonzero, auto) |
|
826 |
||
827 |
lemma setprod_zero_eq_int: |
|
828 |
"finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)" |
|
829 |
by (rule setprod_zero_eq, auto) |
|
830 |
||
831 |
||
832 |
subsection {* Further properties *} |
|
833 |
||
834 |
text{*Now we replace the case analysis rule by a more conventional one: |
|
835 |
whether an integer is negative or not.*} |
|
836 |
||
837 |
lemma zless_iff_Suc_zadd: |
|
838 |
"(w < z) = (\<exists>n. z = w + int(Suc n))" |
|
839 |
apply (cases z, cases w) |
|
840 |
apply (auto simp add: le add int_def linorder_not_le [symmetric]) |
|
841 |
apply (rename_tac a b c d) |
|
842 |
apply (rule_tac x="a+d - Suc(c+b)" in exI) |
|
843 |
apply arith |
|
844 |
done |
|
845 |
||
846 |
lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))" |
|
847 |
apply (cases x) |
|
848 |
apply (auto simp add: le minus Zero_int_def int_def order_less_le) |
|
849 |
apply (rule_tac x="y - Suc x" in exI, arith) |
|
850 |
done |
|
851 |
||
852 |
theorem int_cases [cases type: int, case_names nonneg neg]: |
|
853 |
"[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P" |
|
854 |
apply (cases "z < 0", blast dest!: negD) |
|
855 |
apply (simp add: linorder_not_less) |
|
856 |
apply (blast dest: nat_0_le [THEN sym]) |
|
857 |
done |
|
858 |
||
859 |
theorem int_induct [induct type: int, case_names nonneg neg]: |
|
860 |
"[|!! n. P (int n); !!n. P (- (int (Suc n))) |] ==> P z" |
|
861 |
by (cases z) auto |
|
862 |
||
863 |
text{*Contributed by Brian Huffman*} |
|
864 |
theorem int_diff_cases [case_names diff]: |
|
865 |
assumes prem: "!!m n. z = int m - int n ==> P" shows "P" |
|
866 |
apply (rule_tac z=z in int_cases) |
|
867 |
apply (rule_tac m=n and n=0 in prem, simp) |
|
868 |
apply (rule_tac m=0 and n="Suc n" in prem, simp) |
|
869 |
done |
|
870 |
||
871 |
lemma of_nat_nat: "0 \<le> z ==> of_nat (nat z) = of_int z" |
|
872 |
apply (cases z) |
|
873 |
apply (simp_all add: not_zle_0_negative del: int_Suc) |
|
874 |
done |
|
875 |
||
876 |
lemmas zless_le = less_int_def [THEN meta_eq_to_obj_eq] |
|
877 |
||
878 |
lemmas [simp] = int_Suc |
|
879 |
||
880 |
||
881 |
subsection {* Legacy ML bindings *} |
|
882 |
||
883 |
ML {* |
|
884 |
val of_nat_0 = @{thm of_nat_0}; |
|
885 |
val of_nat_1 = @{thm of_nat_1}; |
|
886 |
val of_nat_Suc = @{thm of_nat_Suc}; |
|
887 |
val of_nat_add = @{thm of_nat_add}; |
|
888 |
val of_nat_mult = @{thm of_nat_mult}; |
|
889 |
val of_int_0 = @{thm of_int_0}; |
|
890 |
val of_int_1 = @{thm of_int_1}; |
|
891 |
val of_int_add = @{thm of_int_add}; |
|
892 |
val of_int_mult = @{thm of_int_mult}; |
|
893 |
val int_eq_of_nat = @{thm int_eq_of_nat}; |
|
894 |
val zle_int = @{thm zle_int}; |
|
895 |
val int_int_eq = @{thm int_int_eq}; |
|
896 |
val diff_int_def = @{thm diff_int_def}; |
|
897 |
val zadd_ac = @{thms zadd_ac}; |
|
898 |
val zless_int = @{thm zless_int}; |
|
899 |
val zadd_int = @{thm zadd_int}; |
|
900 |
val zmult_int = @{thm zmult_int}; |
|
901 |
val nat_0_le = @{thm nat_0_le}; |
|
902 |
val int_0 = @{thm int_0}; |
|
903 |
val int_1 = @{thm int_1}; |
|
904 |
val abs_split = @{thm abs_split}; |
|
905 |
*} |
|
906 |
||
907 |
end |