src/HOL/Int.thy
author nipkow
Wed Jan 28 16:29:16 2009 +0100 (2009-01-28)
changeset 29667 53103fc8ffa3
parent 29046 773098b76201
child 29668 33ba3faeaa0e
permissions -rw-r--r--
Replaced group_ and ring_simps by algebra_simps;
removed compare_rls - use algebra_simps now
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(*  Title:      Int.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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                Tobias Nipkow, Florian Haftmann, TU Muenchen
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    Copyright   1994  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 Int
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imports Equiv_Relations Nat Wellfounded
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uses
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  ("Tools/numeral.ML")
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  ("Tools/numeral_syntax.ML")
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  ("~~/src/Provers/Arith/assoc_fold.ML")
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  "~~/src/Provers/Arith/cancel_numerals.ML"
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  "~~/src/Provers/Arith/combine_numerals.ML"
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  ("Tools/int_arith.ML")
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begin
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subsection {* The equivalence relation underlying the integers *}
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definition intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" where
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  [code del]: "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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instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"
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begin
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definition
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  Zero_int_def [code del]: "0 = Abs_Integ (intrel `` {(0, 0)})"
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definition
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  One_int_def [code del]: "1 = Abs_Integ (intrel `` {(1, 0)})"
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definition
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  add_int_def [code del]: "z + w = 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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definition
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  minus_int_def [code del]:
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    "- z = Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
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definition
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  diff_int_def [code del]:  "z - w = z + (-w \<Colon> int)"
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definition
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  mult_int_def [code del]: "z * w = 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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definition
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  le_int_def [code del]:
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   "z \<le> w \<longleftrightarrow> (\<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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definition
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  less_int_def [code del]: "(z\<Colon>int) < w \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
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definition
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  zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
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definition
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  zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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instance ..
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end
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subsection{*Construction of the Integers*}
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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,noatp]  Abs_Integ_inverse [simp,noatp]
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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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subsection {* Arithmetic Operations *}
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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 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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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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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)"
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    by (cases i, cases j, cases k) (simp add: add add_assoc)
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  show "i + j = j + i" 
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    by (cases i, cases j) (simp add: add_ac add)
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  show "0 + i = i"
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    by (cases i) (simp add: Zero_int_def add)
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  show "- i + i = 0"
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    by (cases i) (simp add: Zero_int_def minus add)
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  show "i - j = i + - j"
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    by (simp add: diff_int_def)
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  show "(i * j) * k = i * (j * k)"
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    by (cases i, cases j, cases k) (simp add: mult algebra_simps)
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  show "i * j = j * i"
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    by (cases i, cases j) (simp add: mult algebra_simps)
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  show "1 * i = i"
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    by (cases i) (simp add: One_int_def mult)
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  show "(i + j) * k = i * k + j * k"
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    by (cases i, cases j, cases k) (simp add: add mult algebra_simps)
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  show "0 \<noteq> (1::int)"
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    by (simp add: Zero_int_def One_int_def)
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qed
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lemma int_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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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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instance int :: linorder
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proof
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  fix i j k :: int
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  show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
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    by (cases i, cases j) (simp add: le)
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  show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
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    by (auto simp add: less_int_def dest: antisym) 
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  show "i \<le> i"
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    by (cases i) (simp add: le)
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  show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
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    by (cases i, cases j, cases k) (simp add: le)
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  show "i \<le> j \<or> j \<le> i"
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    by (cases i, cases j) (simp add: le linorder_linear)
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qed
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instantiation int :: distrib_lattice
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begin
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definition
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  "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by intro_classes
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    (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
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end
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instance int :: pordered_cancel_ab_semigroup_add
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proof
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  fix i j k :: int
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  show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
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    by (cases i, cases j, cases k) (simp add: le add)
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qed
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text{*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 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_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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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_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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lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
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apply (drule zero_less_imp_eq_int)
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apply (auto simp add: zmult_zless_mono2_lemma)
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done
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text{*The integers form an ordered integral domain*}
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instance int :: ordered_idom
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proof
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  fix i j k :: int
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  show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
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    by (rule zmult_zless_mono2)
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  show "\<bar>i\<bar> = (if i < 0 then -i else i)"
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    by (simp only: zabs_def)
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  show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
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    by (simp only: zsgn_def)
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qed
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instance int :: lordered_ring
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proof  
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  fix k :: int
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  show "abs k = sup k (- k)"
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    by (auto simp add: sup_int_def zabs_def max_def less_minus_self_iff [symmetric])
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qed
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lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
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apply (cases w, cases z) 
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apply (simp add: less le add One_int_def)
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done
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lemma zless_iff_Suc_zadd:
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  "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))"
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apply (cases z, cases w)
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apply (auto simp add: less add int_def)
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apply (rename_tac a b c d) 
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apply (rule_tac x="a+d - Suc(c+b)" in exI) 
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apply arith
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done
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lemmas int_distrib =
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  left_distrib [of "z1::int" "z2" "w", standard]
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  right_distrib [of "w::int" "z1" "z2", standard]
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  left_diff_distrib [of "z1::int" "z2" "w", standard]
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  right_diff_distrib [of "w::int" "z1" "z2", standard]
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subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
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context ring_1
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begin
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definition
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  of_int :: "int \<Rightarrow> 'a"
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where
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  [code del]: "of_int z = contents (\<Union>(i, j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
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lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
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proof -
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  have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
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    by (simp add: congruent_def algebra_simps of_nat_add [symmetric]
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            del: of_nat_add) 
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  thus ?thesis
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    by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
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qed
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lemma of_int_0 [simp]: "of_int 0 = 0"
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by (simp add: of_int Zero_int_def)
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lemma of_int_1 [simp]: "of_int 1 = 1"
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by (simp add: of_int One_int_def)
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lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
nipkow@29667
   317
by (cases w, cases z, simp add: algebra_simps of_int add)
haftmann@25919
   318
haftmann@25919
   319
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
nipkow@29667
   320
by (cases z, simp add: algebra_simps of_int minus)
haftmann@25919
   321
haftmann@25919
   322
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
nipkow@29667
   323
by (simp add: OrderedGroup.diff_minus diff_minus)
haftmann@25919
   324
haftmann@25919
   325
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
haftmann@25919
   326
apply (cases w, cases z)
nipkow@29667
   327
apply (simp add: algebra_simps of_int mult of_nat_mult)
haftmann@25919
   328
done
haftmann@25919
   329
haftmann@25919
   330
text{*Collapse nested embeddings*}
haftmann@25919
   331
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
nipkow@29667
   332
by (induct n) auto
haftmann@25919
   333
haftmann@25919
   334
end
haftmann@25919
   335
haftmann@25919
   336
context ordered_idom
haftmann@25919
   337
begin
haftmann@25919
   338
haftmann@25919
   339
lemma of_int_le_iff [simp]:
haftmann@25919
   340
  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
nipkow@29667
   341
by (cases w, cases z, simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add)
haftmann@25919
   342
haftmann@25919
   343
text{*Special cases where either operand is zero*}
haftmann@25919
   344
lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
haftmann@25919
   345
lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
haftmann@25919
   346
haftmann@25919
   347
lemma of_int_less_iff [simp]:
haftmann@25919
   348
  "of_int w < of_int z \<longleftrightarrow> w < z"
haftmann@25919
   349
  by (simp add: not_le [symmetric] linorder_not_le [symmetric])
haftmann@25919
   350
haftmann@25919
   351
text{*Special cases where either operand is zero*}
haftmann@25919
   352
lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
haftmann@25919
   353
lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
haftmann@25919
   354
haftmann@25919
   355
end
haftmann@25919
   356
haftmann@25919
   357
text{*Class for unital rings with characteristic zero.
haftmann@25919
   358
 Includes non-ordered rings like the complex numbers.*}
haftmann@25919
   359
class ring_char_0 = ring_1 + semiring_char_0
haftmann@25919
   360
begin
haftmann@25919
   361
haftmann@25919
   362
lemma of_int_eq_iff [simp]:
haftmann@25919
   363
   "of_int w = of_int z \<longleftrightarrow> w = z"
haftmann@25919
   364
apply (cases w, cases z, simp add: of_int)
haftmann@25919
   365
apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
haftmann@25919
   366
apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
haftmann@25919
   367
done
haftmann@25919
   368
haftmann@25919
   369
text{*Special cases where either operand is zero*}
haftmann@25919
   370
lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
haftmann@25919
   371
lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
haftmann@25919
   372
haftmann@25919
   373
end
haftmann@25919
   374
haftmann@25919
   375
text{*Every @{text ordered_idom} has characteristic zero.*}
haftmann@25919
   376
subclass (in ordered_idom) ring_char_0 by intro_locales
haftmann@25919
   377
haftmann@25919
   378
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   379
proof
haftmann@25919
   380
  fix z show "of_int z = id z"
haftmann@25919
   381
    by (cases z) (simp add: of_int add minus int_def diff_minus)
haftmann@25919
   382
qed
haftmann@25919
   383
haftmann@25919
   384
haftmann@25919
   385
subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
haftmann@25919
   386
haftmann@25919
   387
definition
haftmann@25919
   388
  nat :: "int \<Rightarrow> nat"
haftmann@25919
   389
where
haftmann@28562
   390
  [code del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
haftmann@25919
   391
haftmann@25919
   392
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
haftmann@25919
   393
proof -
haftmann@25919
   394
  have "(\<lambda>(x,y). {x-y}) respects intrel"
haftmann@25919
   395
    by (simp add: congruent_def) arith
haftmann@25919
   396
  thus ?thesis
haftmann@25919
   397
    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
haftmann@25919
   398
qed
haftmann@25919
   399
haftmann@25919
   400
lemma nat_int [simp]: "nat (of_nat n) = n"
haftmann@25919
   401
by (simp add: nat int_def)
haftmann@25919
   402
haftmann@25919
   403
lemma nat_zero [simp]: "nat 0 = 0"
haftmann@25919
   404
by (simp add: Zero_int_def nat)
haftmann@25919
   405
haftmann@25919
   406
lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
haftmann@25919
   407
by (cases z, simp add: nat le int_def Zero_int_def)
haftmann@25919
   408
haftmann@25919
   409
corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
haftmann@25919
   410
by simp
haftmann@25919
   411
haftmann@25919
   412
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
haftmann@25919
   413
by (cases z, simp add: nat le Zero_int_def)
haftmann@25919
   414
haftmann@25919
   415
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
haftmann@25919
   416
apply (cases w, cases z) 
haftmann@25919
   417
apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
haftmann@25919
   418
done
haftmann@25919
   419
haftmann@25919
   420
text{*An alternative condition is @{term "0 \<le> w"} *}
haftmann@25919
   421
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
haftmann@25919
   422
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919
   423
haftmann@25919
   424
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
haftmann@25919
   425
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919
   426
haftmann@25919
   427
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
haftmann@25919
   428
apply (cases w, cases z) 
haftmann@25919
   429
apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
haftmann@25919
   430
done
haftmann@25919
   431
haftmann@25919
   432
lemma nonneg_eq_int:
haftmann@25919
   433
  fixes z :: int
haftmann@25919
   434
  assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
haftmann@25919
   435
  shows P
haftmann@25919
   436
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   437
haftmann@25919
   438
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
haftmann@25919
   439
by (cases w, simp add: nat le int_def Zero_int_def, arith)
haftmann@25919
   440
haftmann@25919
   441
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
haftmann@25919
   442
by (simp only: eq_commute [of m] nat_eq_iff)
haftmann@25919
   443
haftmann@25919
   444
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
haftmann@25919
   445
apply (cases w)
haftmann@25919
   446
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
haftmann@25919
   447
done
haftmann@25919
   448
haftmann@25919
   449
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   450
by (auto simp add: nat_eq_iff2)
haftmann@25919
   451
haftmann@25919
   452
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   453
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   454
haftmann@25919
   455
lemma nat_add_distrib:
haftmann@25919
   456
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
haftmann@25919
   457
by (cases z, cases z', simp add: nat add le Zero_int_def)
haftmann@25919
   458
haftmann@25919
   459
lemma nat_diff_distrib:
haftmann@25919
   460
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
haftmann@25919
   461
by (cases z, cases z', 
haftmann@25919
   462
    simp add: nat add minus diff_minus le Zero_int_def)
haftmann@25919
   463
haftmann@25919
   464
lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
haftmann@25919
   465
by (simp add: int_def minus nat Zero_int_def) 
haftmann@25919
   466
haftmann@25919
   467
lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
haftmann@25919
   468
by (cases z, simp add: nat less int_def, arith)
haftmann@25919
   469
haftmann@25919
   470
context ring_1
haftmann@25919
   471
begin
haftmann@25919
   472
haftmann@25919
   473
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
haftmann@25919
   474
  by (cases z rule: eq_Abs_Integ)
haftmann@25919
   475
   (simp add: nat le of_int Zero_int_def of_nat_diff)
haftmann@25919
   476
haftmann@25919
   477
end
haftmann@25919
   478
haftmann@25919
   479
haftmann@25919
   480
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   481
haftmann@25919
   482
lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   483
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   484
haftmann@25919
   485
lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
haftmann@25919
   486
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   487
haftmann@25919
   488
lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
haftmann@25919
   489
by (simp add: minus_le_iff)
haftmann@25919
   490
haftmann@25919
   491
lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
haftmann@25919
   492
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   493
haftmann@25919
   494
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
haftmann@25919
   495
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   496
haftmann@25919
   497
lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   498
by (simp add: int_def le minus Zero_int_def)
haftmann@25919
   499
haftmann@25919
   500
lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
haftmann@25919
   501
by (simp add: linorder_not_less)
haftmann@25919
   502
haftmann@25919
   503
lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   504
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   505
haftmann@25919
   506
lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
haftmann@25919
   507
proof -
haftmann@25919
   508
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   509
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   510
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   511
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   512
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   513
    by (simp only: algebra_simps)
haftmann@25919
   514
  finally show ?thesis .
haftmann@25919
   515
qed
haftmann@25919
   516
haftmann@25919
   517
lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
haftmann@25919
   518
by simp
haftmann@25919
   519
haftmann@25919
   520
lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
haftmann@25919
   521
by simp
haftmann@25919
   522
haftmann@25919
   523
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   524
      It is proved here because attribute @{text arith_split} is not available
haftmann@25919
   525
      in theory @{text Ring_and_Field}.
haftmann@25919
   526
      But is it really better than just rewriting with @{text abs_if}?*}
haftmann@25919
   527
lemma abs_split [arith_split,noatp]:
haftmann@25919
   528
     "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   529
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   530
haftmann@25919
   531
lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
haftmann@25919
   532
apply (cases x)
haftmann@25919
   533
apply (auto simp add: le minus Zero_int_def int_def order_less_le)
haftmann@25919
   534
apply (rule_tac x="y - Suc x" in exI, arith)
haftmann@25919
   535
done
haftmann@25919
   536
haftmann@25919
   537
haftmann@25919
   538
subsection {* Cases and induction *}
haftmann@25919
   539
haftmann@25919
   540
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   541
whether an integer is negative or not.*}
haftmann@25919
   542
haftmann@25919
   543
theorem int_cases [cases type: int, case_names nonneg neg]:
haftmann@25919
   544
  "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"
haftmann@25919
   545
apply (cases "z < 0", blast dest!: negD)
haftmann@25919
   546
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   547
apply auto
haftmann@25919
   548
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   549
done
haftmann@25919
   550
haftmann@25919
   551
theorem int_induct [induct type: int, case_names nonneg neg]:
haftmann@25919
   552
     "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"
haftmann@25919
   553
  by (cases z rule: int_cases) auto
haftmann@25919
   554
haftmann@25919
   555
text{*Contributed by Brian Huffman*}
haftmann@25919
   556
theorem int_diff_cases:
haftmann@25919
   557
  obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
haftmann@25919
   558
apply (cases z rule: eq_Abs_Integ)
haftmann@25919
   559
apply (rule_tac m=x and n=y in diff)
haftmann@25919
   560
apply (simp add: int_def diff_def minus add)
haftmann@25919
   561
done
haftmann@25919
   562
haftmann@25919
   563
haftmann@25919
   564
subsection {* Binary representation *}
haftmann@25919
   565
haftmann@25919
   566
text {*
haftmann@25919
   567
  This formalization defines binary arithmetic in terms of the integers
haftmann@25919
   568
  rather than using a datatype. This avoids multiple representations (leading
haftmann@25919
   569
  zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
haftmann@25919
   570
  int_of_binary}, for the numerical interpretation.
haftmann@25919
   571
haftmann@25919
   572
  The representation expects that @{text "(m mod 2)"} is 0 or 1,
haftmann@25919
   573
  even if m is negative;
haftmann@25919
   574
  For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
haftmann@25919
   575
  @{text "-5 = (-3)*2 + 1"}.
haftmann@25919
   576
  
haftmann@25919
   577
  This two's complement binary representation derives from the paper 
haftmann@25919
   578
  "An Efficient Representation of Arithmetic for Term Rewriting" by
haftmann@25919
   579
  Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
haftmann@25919
   580
  Springer LNCS 488 (240-251), 1991.
haftmann@25919
   581
*}
haftmann@25919
   582
huffman@28958
   583
subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}
huffman@28958
   584
haftmann@25919
   585
definition
haftmann@25919
   586
  Pls :: int where
haftmann@28562
   587
  [code del]: "Pls = 0"
haftmann@25919
   588
haftmann@25919
   589
definition
haftmann@25919
   590
  Min :: int where
haftmann@28562
   591
  [code del]: "Min = - 1"
haftmann@25919
   592
haftmann@25919
   593
definition
huffman@26086
   594
  Bit0 :: "int \<Rightarrow> int" where
haftmann@28562
   595
  [code del]: "Bit0 k = k + k"
huffman@26086
   596
huffman@26086
   597
definition
huffman@26086
   598
  Bit1 :: "int \<Rightarrow> int" where
haftmann@28562
   599
  [code del]: "Bit1 k = 1 + k + k"
haftmann@25919
   600
haftmann@25919
   601
class number = type + -- {* for numeric types: nat, int, real, \dots *}
haftmann@25919
   602
  fixes number_of :: "int \<Rightarrow> 'a"
haftmann@25919
   603
haftmann@25919
   604
use "Tools/numeral.ML"
haftmann@25919
   605
haftmann@25919
   606
syntax
haftmann@25919
   607
  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
haftmann@25919
   608
haftmann@25919
   609
use "Tools/numeral_syntax.ML"
haftmann@25919
   610
setup NumeralSyntax.setup
haftmann@25919
   611
haftmann@25919
   612
abbreviation
haftmann@25919
   613
  "Numeral0 \<equiv> number_of Pls"
haftmann@25919
   614
haftmann@25919
   615
abbreviation
huffman@26086
   616
  "Numeral1 \<equiv> number_of (Bit1 Pls)"
haftmann@25919
   617
haftmann@25919
   618
lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
haftmann@25919
   619
  -- {* Unfold all @{text let}s involving constants *}
haftmann@25919
   620
  unfolding Let_def ..
haftmann@25919
   621
haftmann@25919
   622
definition
haftmann@25919
   623
  succ :: "int \<Rightarrow> int" where
haftmann@28562
   624
  [code del]: "succ k = k + 1"
haftmann@25919
   625
haftmann@25919
   626
definition
haftmann@25919
   627
  pred :: "int \<Rightarrow> int" where
haftmann@28562
   628
  [code del]: "pred k = k - 1"
haftmann@25919
   629
haftmann@25919
   630
lemmas
haftmann@25919
   631
  max_number_of [simp] = max_def
haftmann@25919
   632
    [of "number_of u" "number_of v", standard, simp]
haftmann@25919
   633
and
haftmann@25919
   634
  min_number_of [simp] = min_def 
haftmann@25919
   635
    [of "number_of u" "number_of v", standard, simp]
haftmann@25919
   636
  -- {* unfolding @{text minx} and @{text max} on numerals *}
haftmann@25919
   637
haftmann@25919
   638
lemmas numeral_simps = 
huffman@26086
   639
  succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
haftmann@25919
   640
haftmann@25919
   641
text {* Removal of leading zeroes *}
haftmann@25919
   642
huffman@26086
   643
lemma Bit0_Pls [simp, code post]:
huffman@26086
   644
  "Bit0 Pls = Pls"
haftmann@25919
   645
  unfolding numeral_simps by simp
haftmann@25919
   646
huffman@26086
   647
lemma Bit1_Min [simp, code post]:
huffman@26086
   648
  "Bit1 Min = Min"
haftmann@25919
   649
  unfolding numeral_simps by simp
haftmann@25919
   650
huffman@26075
   651
lemmas normalize_bin_simps =
huffman@26086
   652
  Bit0_Pls Bit1_Min
huffman@26075
   653
haftmann@25919
   654
huffman@28958
   655
subsubsection {* Successor and predecessor functions *}
huffman@28958
   656
huffman@28958
   657
text {* Successor *}
huffman@28958
   658
huffman@28958
   659
lemma succ_Pls:
huffman@26086
   660
  "succ Pls = Bit1 Pls"
haftmann@25919
   661
  unfolding numeral_simps by simp
haftmann@25919
   662
huffman@28958
   663
lemma succ_Min:
haftmann@25919
   664
  "succ Min = Pls"
haftmann@25919
   665
  unfolding numeral_simps by simp
haftmann@25919
   666
huffman@28958
   667
lemma succ_Bit0:
huffman@26086
   668
  "succ (Bit0 k) = Bit1 k"
haftmann@25919
   669
  unfolding numeral_simps by simp
haftmann@25919
   670
huffman@28958
   671
lemma succ_Bit1:
huffman@26086
   672
  "succ (Bit1 k) = Bit0 (succ k)"
haftmann@25919
   673
  unfolding numeral_simps by simp
haftmann@25919
   674
huffman@28958
   675
lemmas succ_bin_simps [simp] =
huffman@26086
   676
  succ_Pls succ_Min succ_Bit0 succ_Bit1
huffman@26075
   677
huffman@28958
   678
text {* Predecessor *}
huffman@28958
   679
huffman@28958
   680
lemma pred_Pls:
haftmann@25919
   681
  "pred Pls = Min"
haftmann@25919
   682
  unfolding numeral_simps by simp
haftmann@25919
   683
huffman@28958
   684
lemma pred_Min:
huffman@26086
   685
  "pred Min = Bit0 Min"
haftmann@25919
   686
  unfolding numeral_simps by simp
haftmann@25919
   687
huffman@28958
   688
lemma pred_Bit0:
huffman@26086
   689
  "pred (Bit0 k) = Bit1 (pred k)"
haftmann@25919
   690
  unfolding numeral_simps by simp 
haftmann@25919
   691
huffman@28958
   692
lemma pred_Bit1:
huffman@26086
   693
  "pred (Bit1 k) = Bit0 k"
huffman@26086
   694
  unfolding numeral_simps by simp
huffman@26086
   695
huffman@28958
   696
lemmas pred_bin_simps [simp] =
huffman@26086
   697
  pred_Pls pred_Min pred_Bit0 pred_Bit1
huffman@26075
   698
huffman@28958
   699
huffman@28958
   700
subsubsection {* Binary arithmetic *}
huffman@28958
   701
huffman@28958
   702
text {* Addition *}
huffman@28958
   703
huffman@28958
   704
lemma add_Pls:
huffman@28958
   705
  "Pls + k = k"
huffman@28958
   706
  unfolding numeral_simps by simp
huffman@28958
   707
huffman@28958
   708
lemma add_Min:
huffman@28958
   709
  "Min + k = pred k"
huffman@28958
   710
  unfolding numeral_simps by simp
huffman@28958
   711
huffman@28958
   712
lemma add_Bit0_Bit0:
huffman@28958
   713
  "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"
huffman@28958
   714
  unfolding numeral_simps by simp
huffman@28958
   715
huffman@28958
   716
lemma add_Bit0_Bit1:
huffman@28958
   717
  "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"
huffman@28958
   718
  unfolding numeral_simps by simp
huffman@28958
   719
huffman@28958
   720
lemma add_Bit1_Bit0:
huffman@28958
   721
  "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"
huffman@28958
   722
  unfolding numeral_simps by simp
huffman@28958
   723
huffman@28958
   724
lemma add_Bit1_Bit1:
huffman@28958
   725
  "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"
huffman@28958
   726
  unfolding numeral_simps by simp
huffman@28958
   727
huffman@28958
   728
lemma add_Pls_right:
huffman@28958
   729
  "k + Pls = k"
huffman@28958
   730
  unfolding numeral_simps by simp
huffman@28958
   731
huffman@28958
   732
lemma add_Min_right:
huffman@28958
   733
  "k + Min = pred k"
huffman@28958
   734
  unfolding numeral_simps by simp
huffman@28958
   735
huffman@28958
   736
lemmas add_bin_simps [simp] =
huffman@28958
   737
  add_Pls add_Min add_Pls_right add_Min_right
huffman@28958
   738
  add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1
huffman@28958
   739
huffman@28958
   740
text {* Negation *}
huffman@28958
   741
huffman@28958
   742
lemma minus_Pls:
haftmann@25919
   743
  "- Pls = Pls"
huffman@28958
   744
  unfolding numeral_simps by simp
huffman@28958
   745
huffman@28958
   746
lemma minus_Min:
huffman@26086
   747
  "- Min = Bit1 Pls"
huffman@28958
   748
  unfolding numeral_simps by simp
huffman@28958
   749
huffman@28958
   750
lemma minus_Bit0:
huffman@26086
   751
  "- (Bit0 k) = Bit0 (- k)"
huffman@28958
   752
  unfolding numeral_simps by simp
huffman@28958
   753
huffman@28958
   754
lemma minus_Bit1:
huffman@26086
   755
  "- (Bit1 k) = Bit1 (pred (- k))"
huffman@26086
   756
  unfolding numeral_simps by simp
haftmann@25919
   757
huffman@28958
   758
lemmas minus_bin_simps [simp] =
huffman@26086
   759
  minus_Pls minus_Min minus_Bit0 minus_Bit1
huffman@26075
   760
huffman@28958
   761
text {* Subtraction *}
huffman@28958
   762
huffman@29046
   763
lemma diff_bin_simps [simp]:
huffman@29046
   764
  "k - Pls = k"
huffman@29046
   765
  "k - Min = succ k"
huffman@29046
   766
  "Pls - (Bit0 l) = Bit0 (Pls - l)"
huffman@29046
   767
  "Pls - (Bit1 l) = Bit1 (Min - l)"
huffman@29046
   768
  "Min - (Bit0 l) = Bit1 (Min - l)"
huffman@29046
   769
  "Min - (Bit1 l) = Bit0 (Min - l)"
huffman@28958
   770
  "(Bit0 k) - (Bit0 l) = Bit0 (k - l)"
huffman@28958
   771
  "(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)"
huffman@28958
   772
  "(Bit1 k) - (Bit0 l) = Bit1 (k - l)"
huffman@28958
   773
  "(Bit1 k) - (Bit1 l) = Bit0 (k - l)"
huffman@29046
   774
  unfolding numeral_simps by simp_all
huffman@28958
   775
huffman@28958
   776
text {* Multiplication *}
huffman@28958
   777
huffman@28958
   778
lemma mult_Pls:
huffman@28958
   779
  "Pls * w = Pls"
huffman@26086
   780
  unfolding numeral_simps by simp
haftmann@25919
   781
huffman@28958
   782
lemma mult_Min:
haftmann@25919
   783
  "Min * k = - k"
haftmann@25919
   784
  unfolding numeral_simps by simp
haftmann@25919
   785
huffman@28958
   786
lemma mult_Bit0:
huffman@26086
   787
  "(Bit0 k) * l = Bit0 (k * l)"
huffman@26086
   788
  unfolding numeral_simps int_distrib by simp
haftmann@25919
   789
huffman@28958
   790
lemma mult_Bit1:
huffman@26086
   791
  "(Bit1 k) * l = (Bit0 (k * l)) + l"
huffman@28958
   792
  unfolding numeral_simps int_distrib by simp
huffman@28958
   793
huffman@28958
   794
lemmas mult_bin_simps [simp] =
huffman@26086
   795
  mult_Pls mult_Min mult_Bit0 mult_Bit1
huffman@26075
   796
haftmann@25919
   797
huffman@28958
   798
subsubsection {* Binary comparisons *}
huffman@28958
   799
huffman@28958
   800
text {* Preliminaries *}
huffman@28958
   801
huffman@28958
   802
lemma even_less_0_iff:
huffman@28958
   803
  "a + a < 0 \<longleftrightarrow> a < (0::'a::ordered_idom)"
huffman@28958
   804
proof -
huffman@28958
   805
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)
huffman@28958
   806
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958
   807
    by (simp add: mult_less_0_iff zero_less_two 
huffman@28958
   808
                  order_less_not_sym [OF zero_less_two])
huffman@28958
   809
  finally show ?thesis .
huffman@28958
   810
qed
huffman@28958
   811
huffman@28958
   812
lemma le_imp_0_less: 
huffman@28958
   813
  assumes le: "0 \<le> z"
huffman@28958
   814
  shows "(0::int) < 1 + z"
huffman@28958
   815
proof -
huffman@28958
   816
  have "0 \<le> z" by fact
huffman@28958
   817
  also have "... < z + 1" by (rule less_add_one) 
huffman@28958
   818
  also have "... = 1 + z" by (simp add: add_ac)
huffman@28958
   819
  finally show "0 < 1 + z" .
huffman@28958
   820
qed
huffman@28958
   821
huffman@28958
   822
lemma odd_less_0_iff:
huffman@28958
   823
  "(1 + z + z < 0) = (z < (0::int))"
huffman@28958
   824
proof (cases z rule: int_cases)
huffman@28958
   825
  case (nonneg n)
huffman@28958
   826
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
huffman@28958
   827
                             le_imp_0_less [THEN order_less_imp_le])  
huffman@28958
   828
next
huffman@28958
   829
  case (neg n)
huffman@28958
   830
  thus ?thesis by (simp del: of_nat_Suc of_nat_add
nipkow@29667
   831
    add: algebra_simps of_nat_1 [symmetric] of_nat_add [symmetric])
huffman@28958
   832
qed
huffman@28958
   833
huffman@28985
   834
lemma bin_less_0_simps:
huffman@28958
   835
  "Pls < 0 \<longleftrightarrow> False"
huffman@28958
   836
  "Min < 0 \<longleftrightarrow> True"
huffman@28958
   837
  "Bit0 w < 0 \<longleftrightarrow> w < 0"
huffman@28958
   838
  "Bit1 w < 0 \<longleftrightarrow> w < 0"
huffman@28958
   839
  unfolding numeral_simps
huffman@28958
   840
  by (simp_all add: even_less_0_iff odd_less_0_iff)
huffman@28958
   841
huffman@28958
   842
lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)"
huffman@28958
   843
  by simp
huffman@28958
   844
huffman@28958
   845
lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l"
huffman@28958
   846
  unfolding numeral_simps
huffman@28958
   847
  proof
huffman@28958
   848
    have "k - 1 < k" by simp
huffman@28958
   849
    also assume "k \<le> l"
huffman@28958
   850
    finally show "k - 1 < l" .
huffman@28958
   851
  next
huffman@28958
   852
    assume "k - 1 < l"
huffman@28958
   853
    hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle)
huffman@28958
   854
    thus "k \<le> l" by simp
huffman@28958
   855
  qed
huffman@28958
   856
huffman@28958
   857
lemma succ_pred: "succ (pred x) = x"
huffman@28958
   858
  unfolding numeral_simps by simp
huffman@28958
   859
huffman@28958
   860
text {* Less-than *}
huffman@28958
   861
huffman@28958
   862
lemma less_bin_simps [simp]:
huffman@28958
   863
  "Pls < Pls \<longleftrightarrow> False"
huffman@28958
   864
  "Pls < Min \<longleftrightarrow> False"
huffman@28958
   865
  "Pls < Bit0 k \<longleftrightarrow> Pls < k"
huffman@28958
   866
  "Pls < Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   867
  "Min < Pls \<longleftrightarrow> True"
huffman@28958
   868
  "Min < Min \<longleftrightarrow> False"
huffman@28958
   869
  "Min < Bit0 k \<longleftrightarrow> Min < k"
huffman@28958
   870
  "Min < Bit1 k \<longleftrightarrow> Min < k"
huffman@28958
   871
  "Bit0 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958
   872
  "Bit0 k < Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   873
  "Bit1 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958
   874
  "Bit1 k < Min \<longleftrightarrow> k < Min"
huffman@28958
   875
  "Bit0 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   876
  "Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   877
  "Bit1 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   878
  "Bit1 k < Bit1 l \<longleftrightarrow> k < l"
huffman@28958
   879
  unfolding le_iff_pred_less
huffman@28958
   880
    less_bin_lemma [of Pls]
huffman@28958
   881
    less_bin_lemma [of Min]
huffman@28958
   882
    less_bin_lemma [of "k"]
huffman@28958
   883
    less_bin_lemma [of "Bit0 k"]
huffman@28958
   884
    less_bin_lemma [of "Bit1 k"]
huffman@28958
   885
    less_bin_lemma [of "pred Pls"]
huffman@28958
   886
    less_bin_lemma [of "pred k"]
huffman@28985
   887
  by (simp_all add: bin_less_0_simps succ_pred)
huffman@28958
   888
huffman@28958
   889
text {* Less-than-or-equal *}
huffman@28958
   890
huffman@28958
   891
lemma le_bin_simps [simp]:
huffman@28958
   892
  "Pls \<le> Pls \<longleftrightarrow> True"
huffman@28958
   893
  "Pls \<le> Min \<longleftrightarrow> False"
huffman@28958
   894
  "Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   895
  "Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   896
  "Min \<le> Pls \<longleftrightarrow> True"
huffman@28958
   897
  "Min \<le> Min \<longleftrightarrow> True"
huffman@28958
   898
  "Min \<le> Bit0 k \<longleftrightarrow> Min < k"
huffman@28958
   899
  "Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k"
huffman@28958
   900
  "Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls"
huffman@28958
   901
  "Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   902
  "Bit1 k \<le> Pls \<longleftrightarrow> k < Pls"
huffman@28958
   903
  "Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   904
  "Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l"
huffman@28958
   905
  "Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   906
  "Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   907
  "Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   908
  unfolding not_less [symmetric]
huffman@28958
   909
  by (simp_all add: not_le)
huffman@28958
   910
huffman@28958
   911
text {* Equality *}
huffman@28958
   912
huffman@28958
   913
lemma eq_bin_simps [simp]:
huffman@28958
   914
  "Pls = Pls \<longleftrightarrow> True"
huffman@28958
   915
  "Pls = Min \<longleftrightarrow> False"
huffman@28958
   916
  "Pls = Bit0 l \<longleftrightarrow> Pls = l"
huffman@28958
   917
  "Pls = Bit1 l \<longleftrightarrow> False"
huffman@28958
   918
  "Min = Pls \<longleftrightarrow> False"
huffman@28958
   919
  "Min = Min \<longleftrightarrow> True"
huffman@28958
   920
  "Min = Bit0 l \<longleftrightarrow> False"
huffman@28958
   921
  "Min = Bit1 l \<longleftrightarrow> Min = l"
huffman@28958
   922
  "Bit0 k = Pls \<longleftrightarrow> k = Pls"
huffman@28958
   923
  "Bit0 k = Min \<longleftrightarrow> False"
huffman@28958
   924
  "Bit1 k = Pls \<longleftrightarrow> False"
huffman@28958
   925
  "Bit1 k = Min \<longleftrightarrow> k = Min"
huffman@28958
   926
  "Bit0 k = Bit0 l \<longleftrightarrow> k = l"
huffman@28958
   927
  "Bit0 k = Bit1 l \<longleftrightarrow> False"
huffman@28958
   928
  "Bit1 k = Bit0 l \<longleftrightarrow> False"
huffman@28958
   929
  "Bit1 k = Bit1 l \<longleftrightarrow> k = l"
huffman@28958
   930
  unfolding order_eq_iff [where 'a=int]
huffman@28958
   931
  by (simp_all add: not_less)
huffman@28958
   932
huffman@28958
   933
haftmann@25919
   934
subsection {* Converting Numerals to Rings: @{term number_of} *}
haftmann@25919
   935
haftmann@25919
   936
class number_ring = number + comm_ring_1 +
haftmann@25919
   937
  assumes number_of_eq: "number_of k = of_int k"
haftmann@25919
   938
haftmann@25919
   939
text {* self-embedding of the integers *}
haftmann@25919
   940
haftmann@25919
   941
instantiation int :: number_ring
haftmann@25919
   942
begin
haftmann@25919
   943
haftmann@28724
   944
definition int_number_of_def [code del]:
haftmann@28724
   945
  "number_of w = (of_int w \<Colon> int)"
haftmann@25919
   946
haftmann@28724
   947
instance proof
haftmann@28724
   948
qed (simp only: int_number_of_def)
haftmann@25919
   949
haftmann@25919
   950
end
haftmann@25919
   951
haftmann@25919
   952
lemma number_of_is_id:
haftmann@25919
   953
  "number_of (k::int) = k"
haftmann@25919
   954
  unfolding int_number_of_def by simp
haftmann@25919
   955
haftmann@25919
   956
lemma number_of_succ:
haftmann@25919
   957
  "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
haftmann@25919
   958
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   959
haftmann@25919
   960
lemma number_of_pred:
haftmann@25919
   961
  "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
haftmann@25919
   962
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   963
haftmann@25919
   964
lemma number_of_minus:
haftmann@25919
   965
  "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
huffman@28958
   966
  unfolding number_of_eq by (rule of_int_minus)
haftmann@25919
   967
haftmann@25919
   968
lemma number_of_add:
haftmann@25919
   969
  "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
huffman@28958
   970
  unfolding number_of_eq by (rule of_int_add)
huffman@28958
   971
huffman@28958
   972
lemma number_of_diff:
huffman@28958
   973
  "number_of (v - w) = (number_of v - number_of w::'a::number_ring)"
huffman@28958
   974
  unfolding number_of_eq by (rule of_int_diff)
haftmann@25919
   975
haftmann@25919
   976
lemma number_of_mult:
haftmann@25919
   977
  "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
huffman@28958
   978
  unfolding number_of_eq by (rule of_int_mult)
haftmann@25919
   979
haftmann@25919
   980
text {*
haftmann@25919
   981
  The correctness of shifting.
haftmann@25919
   982
  But it doesn't seem to give a measurable speed-up.
haftmann@25919
   983
*}
haftmann@25919
   984
huffman@26086
   985
lemma double_number_of_Bit0:
huffman@26086
   986
  "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"
haftmann@25919
   987
  unfolding number_of_eq numeral_simps left_distrib by simp
haftmann@25919
   988
haftmann@25919
   989
text {*
haftmann@25919
   990
  Converting numerals 0 and 1 to their abstract versions.
haftmann@25919
   991
*}
haftmann@25919
   992
haftmann@25919
   993
lemma numeral_0_eq_0 [simp]:
haftmann@25919
   994
  "Numeral0 = (0::'a::number_ring)"
haftmann@25919
   995
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   996
haftmann@25919
   997
lemma numeral_1_eq_1 [simp]:
haftmann@25919
   998
  "Numeral1 = (1::'a::number_ring)"
haftmann@25919
   999
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1000
haftmann@25919
  1001
text {*
haftmann@25919
  1002
  Special-case simplification for small constants.
haftmann@25919
  1003
*}
haftmann@25919
  1004
haftmann@25919
  1005
text{*
haftmann@25919
  1006
  Unary minus for the abstract constant 1. Cannot be inserted
haftmann@25919
  1007
  as a simprule until later: it is @{text number_of_Min} re-oriented!
haftmann@25919
  1008
*}
haftmann@25919
  1009
haftmann@25919
  1010
lemma numeral_m1_eq_minus_1:
haftmann@25919
  1011
  "(-1::'a::number_ring) = - 1"
haftmann@25919
  1012
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1013
haftmann@25919
  1014
lemma mult_minus1 [simp]:
haftmann@25919
  1015
  "-1 * z = -(z::'a::number_ring)"
haftmann@25919
  1016
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1017
haftmann@25919
  1018
lemma mult_minus1_right [simp]:
haftmann@25919
  1019
  "z * -1 = -(z::'a::number_ring)"
haftmann@25919
  1020
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1021
haftmann@25919
  1022
(*Negation of a coefficient*)
haftmann@25919
  1023
lemma minus_number_of_mult [simp]:
haftmann@25919
  1024
   "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
haftmann@25919
  1025
   unfolding number_of_eq by simp
haftmann@25919
  1026
haftmann@25919
  1027
text {* Subtraction *}
haftmann@25919
  1028
haftmann@25919
  1029
lemma diff_number_of_eq:
haftmann@25919
  1030
  "number_of v - number_of w =
haftmann@25919
  1031
    (number_of (v + uminus w)::'a::number_ring)"
haftmann@25919
  1032
  unfolding number_of_eq by simp
haftmann@25919
  1033
haftmann@25919
  1034
lemma number_of_Pls:
haftmann@25919
  1035
  "number_of Pls = (0::'a::number_ring)"
haftmann@25919
  1036
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1037
haftmann@25919
  1038
lemma number_of_Min:
haftmann@25919
  1039
  "number_of Min = (- 1::'a::number_ring)"
haftmann@25919
  1040
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1041
huffman@26086
  1042
lemma number_of_Bit0:
huffman@26086
  1043
  "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086
  1044
  unfolding number_of_eq numeral_simps by simp
huffman@26086
  1045
huffman@26086
  1046
lemma number_of_Bit1:
huffman@26086
  1047
  "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086
  1048
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1049
haftmann@25919
  1050
huffman@28958
  1051
subsubsection {* Equality of Binary Numbers *}
haftmann@25919
  1052
haftmann@25919
  1053
text {* First version by Norbert Voelker *}
haftmann@25919
  1054
haftmann@25919
  1055
definition (*for simplifying equalities*)
haftmann@25919
  1056
  iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
haftmann@25919
  1057
where
haftmann@25919
  1058
  "iszero z \<longleftrightarrow> z = 0"
haftmann@25919
  1059
haftmann@25919
  1060
lemma iszero_0: "iszero 0"
haftmann@25919
  1061
by (simp add: iszero_def)
haftmann@25919
  1062
haftmann@25919
  1063
lemma not_iszero_1: "~ iszero 1"
haftmann@25919
  1064
by (simp add: iszero_def eq_commute)
haftmann@25919
  1065
haftmann@25919
  1066
lemma eq_number_of_eq:
haftmann@25919
  1067
  "((number_of x::'a::number_ring) = number_of y) =
haftmann@25919
  1068
   iszero (number_of (x + uminus y) :: 'a)"
nipkow@29667
  1069
unfolding iszero_def number_of_add number_of_minus
nipkow@29667
  1070
by (simp add: algebra_simps)
haftmann@25919
  1071
haftmann@25919
  1072
lemma iszero_number_of_Pls:
haftmann@25919
  1073
  "iszero ((number_of Pls)::'a::number_ring)"
nipkow@29667
  1074
unfolding iszero_def numeral_0_eq_0 ..
haftmann@25919
  1075
haftmann@25919
  1076
lemma nonzero_number_of_Min:
haftmann@25919
  1077
  "~ iszero ((number_of Min)::'a::number_ring)"
nipkow@29667
  1078
unfolding iszero_def numeral_m1_eq_minus_1 by simp
haftmann@25919
  1079
haftmann@25919
  1080
huffman@28958
  1081
subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919
  1082
haftmann@25919
  1083
lemmas double_eq_0_iff = double_zero
haftmann@25919
  1084
haftmann@25919
  1085
lemma odd_nonzero:
haftmann@25919
  1086
  "1 + z + z \<noteq> (0::int)";
haftmann@25919
  1087
proof (cases z rule: int_cases)
haftmann@25919
  1088
  case (nonneg n)
haftmann@25919
  1089
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
  1090
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919
  1091
    by (auto simp add: add_assoc) 
haftmann@25919
  1092
next
haftmann@25919
  1093
  case (neg n)
haftmann@25919
  1094
  show ?thesis
haftmann@25919
  1095
  proof
haftmann@25919
  1096
    assume eq: "1 + z + z = 0"
haftmann@25919
  1097
    have "(0::int) < 1 + (of_nat n + of_nat n)"
haftmann@25919
  1098
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
  1099
    also have "... = - (1 + z + z)" 
haftmann@25919
  1100
      by (simp add: neg add_assoc [symmetric]) 
haftmann@25919
  1101
    also have "... = 0" by (simp add: eq) 
haftmann@25919
  1102
    finally have "0<0" ..
haftmann@25919
  1103
    thus False by blast
haftmann@25919
  1104
  qed
haftmann@25919
  1105
qed
haftmann@25919
  1106
huffman@26086
  1107
lemma iszero_number_of_Bit0:
huffman@26086
  1108
  "iszero (number_of (Bit0 w)::'a) = 
huffman@26086
  1109
   iszero (number_of w::'a::{ring_char_0,number_ring})"
haftmann@25919
  1110
proof -
haftmann@25919
  1111
  have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
haftmann@25919
  1112
  proof -
haftmann@25919
  1113
    assume eq: "of_int w + of_int w = (0::'a)"
haftmann@25919
  1114
    then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
haftmann@25919
  1115
    then have "w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1116
    then show "w = 0" by (simp only: double_eq_0_iff)
haftmann@25919
  1117
  qed
huffman@26086
  1118
  thus ?thesis
huffman@26086
  1119
    by (auto simp add: iszero_def number_of_eq numeral_simps)
huffman@26086
  1120
qed
huffman@26086
  1121
huffman@26086
  1122
lemma iszero_number_of_Bit1:
huffman@26086
  1123
  "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"
huffman@26086
  1124
proof -
huffman@26086
  1125
  have "1 + of_int w + of_int w \<noteq> (0::'a)"
haftmann@25919
  1126
  proof
haftmann@25919
  1127
    assume eq: "1 + of_int w + of_int w = (0::'a)"
haftmann@25919
  1128
    hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp 
haftmann@25919
  1129
    hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1130
    with odd_nonzero show False by blast
haftmann@25919
  1131
  qed
huffman@26086
  1132
  thus ?thesis
huffman@26086
  1133
    by (auto simp add: iszero_def number_of_eq numeral_simps)
haftmann@25919
  1134
qed
haftmann@25919
  1135
huffman@28985
  1136
lemmas iszero_simps =
huffman@28985
  1137
  iszero_0 not_iszero_1
huffman@28985
  1138
  iszero_number_of_Pls nonzero_number_of_Min
huffman@28985
  1139
  iszero_number_of_Bit0 iszero_number_of_Bit1
huffman@28985
  1140
(* iszero_number_of_Pls would never normally be used
huffman@28985
  1141
   because its lhs simplifies to "iszero 0" *)
haftmann@25919
  1142
huffman@28958
  1143
subsubsection {* The Less-Than Relation *}
haftmann@25919
  1144
haftmann@25919
  1145
lemma double_less_0_iff:
haftmann@25919
  1146
  "(a + a < 0) = (a < (0::'a::ordered_idom))"
haftmann@25919
  1147
proof -
haftmann@25919
  1148
  have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
haftmann@25919
  1149
  also have "... = (a < 0)"
haftmann@25919
  1150
    by (simp add: mult_less_0_iff zero_less_two 
haftmann@25919
  1151
                  order_less_not_sym [OF zero_less_two]) 
haftmann@25919
  1152
  finally show ?thesis .
haftmann@25919
  1153
qed
haftmann@25919
  1154
haftmann@25919
  1155
lemma odd_less_0:
haftmann@25919
  1156
  "(1 + z + z < 0) = (z < (0::int))";
haftmann@25919
  1157
proof (cases z rule: int_cases)
haftmann@25919
  1158
  case (nonneg n)
haftmann@25919
  1159
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
haftmann@25919
  1160
                             le_imp_0_less [THEN order_less_imp_le])  
haftmann@25919
  1161
next
haftmann@25919
  1162
  case (neg n)
haftmann@25919
  1163
  thus ?thesis by (simp del: of_nat_Suc of_nat_add
nipkow@29667
  1164
    add: algebra_simps of_nat_1 [symmetric] of_nat_add [symmetric])
haftmann@25919
  1165
qed
haftmann@25919
  1166
haftmann@25919
  1167
text {* Less-Than or Equals *}
haftmann@25919
  1168
haftmann@25919
  1169
text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
haftmann@25919
  1170
haftmann@25919
  1171
lemmas le_number_of_eq_not_less =
haftmann@25919
  1172
  linorder_not_less [of "number_of w" "number_of v", symmetric, 
haftmann@25919
  1173
  standard]
haftmann@25919
  1174
haftmann@25919
  1175
haftmann@25919
  1176
text {* Absolute value (@{term abs}) *}
haftmann@25919
  1177
haftmann@25919
  1178
lemma abs_number_of:
haftmann@25919
  1179
  "abs(number_of x::'a::{ordered_idom,number_ring}) =
haftmann@25919
  1180
   (if number_of x < (0::'a) then -number_of x else number_of x)"
haftmann@25919
  1181
  by (simp add: abs_if)
haftmann@25919
  1182
haftmann@25919
  1183
haftmann@25919
  1184
text {* Re-orientation of the equation nnn=x *}
haftmann@25919
  1185
haftmann@25919
  1186
lemma number_of_reorient:
haftmann@25919
  1187
  "(number_of w = x) = (x = number_of w)"
haftmann@25919
  1188
  by auto
haftmann@25919
  1189
haftmann@25919
  1190
huffman@28958
  1191
subsubsection {* Simplification of arithmetic operations on integer constants. *}
haftmann@25919
  1192
haftmann@25919
  1193
lemmas arith_extra_simps [standard, simp] =
haftmann@25919
  1194
  number_of_add [symmetric]
huffman@28958
  1195
  number_of_minus [symmetric]
huffman@28958
  1196
  numeral_m1_eq_minus_1 [symmetric]
haftmann@25919
  1197
  number_of_mult [symmetric]
haftmann@25919
  1198
  diff_number_of_eq abs_number_of 
haftmann@25919
  1199
haftmann@25919
  1200
text {*
haftmann@25919
  1201
  For making a minimal simpset, one must include these default simprules.
haftmann@25919
  1202
  Also include @{text simp_thms}.
haftmann@25919
  1203
*}
haftmann@25919
  1204
haftmann@25919
  1205
lemmas arith_simps = 
huffman@26075
  1206
  normalize_bin_simps pred_bin_simps succ_bin_simps
huffman@26075
  1207
  add_bin_simps minus_bin_simps mult_bin_simps
haftmann@25919
  1208
  abs_zero abs_one arith_extra_simps
haftmann@25919
  1209
haftmann@25919
  1210
text {* Simplification of relational operations *}
haftmann@25919
  1211
huffman@28962
  1212
lemma less_number_of [simp]:
huffman@28962
  1213
  "(number_of x::'a::{ordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
huffman@28962
  1214
  unfolding number_of_eq by (rule of_int_less_iff)
huffman@28962
  1215
huffman@28962
  1216
lemma le_number_of [simp]:
huffman@28962
  1217
  "(number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
huffman@28962
  1218
  unfolding number_of_eq by (rule of_int_le_iff)
huffman@28962
  1219
huffman@28967
  1220
lemma eq_number_of [simp]:
huffman@28967
  1221
  "(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y"
huffman@28967
  1222
  unfolding number_of_eq by (rule of_int_eq_iff)
huffman@28967
  1223
haftmann@25919
  1224
lemmas rel_simps [simp] = 
huffman@28962
  1225
  less_number_of less_bin_simps
huffman@28962
  1226
  le_number_of le_bin_simps
huffman@28988
  1227
  eq_number_of_eq eq_bin_simps
huffman@29039
  1228
  iszero_simps
haftmann@25919
  1229
haftmann@25919
  1230
huffman@28958
  1231
subsubsection {* Simplification of arithmetic when nested to the right. *}
haftmann@25919
  1232
haftmann@25919
  1233
lemma add_number_of_left [simp]:
haftmann@25919
  1234
  "number_of v + (number_of w + z) =
haftmann@25919
  1235
   (number_of(v + w) + z::'a::number_ring)"
haftmann@25919
  1236
  by (simp add: add_assoc [symmetric])
haftmann@25919
  1237
haftmann@25919
  1238
lemma mult_number_of_left [simp]:
haftmann@25919
  1239
  "number_of v * (number_of w * z) =
haftmann@25919
  1240
   (number_of(v * w) * z::'a::number_ring)"
haftmann@25919
  1241
  by (simp add: mult_assoc [symmetric])
haftmann@25919
  1242
haftmann@25919
  1243
lemma add_number_of_diff1:
haftmann@25919
  1244
  "number_of v + (number_of w - c) = 
haftmann@25919
  1245
  number_of(v + w) - (c::'a::number_ring)"
haftmann@25919
  1246
  by (simp add: diff_minus add_number_of_left)
haftmann@25919
  1247
haftmann@25919
  1248
lemma add_number_of_diff2 [simp]:
haftmann@25919
  1249
  "number_of v + (c - number_of w) =
haftmann@25919
  1250
   number_of (v + uminus w) + (c::'a::number_ring)"
nipkow@29667
  1251
by (simp add: algebra_simps diff_number_of_eq [symmetric])
haftmann@25919
  1252
haftmann@25919
  1253
haftmann@25919
  1254
subsection {* The Set of Integers *}
haftmann@25919
  1255
haftmann@25919
  1256
context ring_1
haftmann@25919
  1257
begin
haftmann@25919
  1258
haftmann@25919
  1259
definition
haftmann@25919
  1260
  Ints  :: "'a set"
haftmann@25919
  1261
where
haftmann@28562
  1262
  [code del]: "Ints = range of_int"
haftmann@25919
  1263
haftmann@25919
  1264
end
haftmann@25919
  1265
haftmann@25919
  1266
notation (xsymbols)
haftmann@25919
  1267
  Ints  ("\<int>")
haftmann@25919
  1268
haftmann@25919
  1269
context ring_1
haftmann@25919
  1270
begin
haftmann@25919
  1271
haftmann@25919
  1272
lemma Ints_0 [simp]: "0 \<in> \<int>"
haftmann@25919
  1273
apply (simp add: Ints_def)
haftmann@25919
  1274
apply (rule range_eqI)
haftmann@25919
  1275
apply (rule of_int_0 [symmetric])
haftmann@25919
  1276
done
haftmann@25919
  1277
haftmann@25919
  1278
lemma Ints_1 [simp]: "1 \<in> \<int>"
haftmann@25919
  1279
apply (simp add: Ints_def)
haftmann@25919
  1280
apply (rule range_eqI)
haftmann@25919
  1281
apply (rule of_int_1 [symmetric])
haftmann@25919
  1282
done
haftmann@25919
  1283
haftmann@25919
  1284
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
  1285
apply (auto simp add: Ints_def)
haftmann@25919
  1286
apply (rule range_eqI)
haftmann@25919
  1287
apply (rule of_int_add [symmetric])
haftmann@25919
  1288
done
haftmann@25919
  1289
haftmann@25919
  1290
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
  1291
apply (auto simp add: Ints_def)
haftmann@25919
  1292
apply (rule range_eqI)
haftmann@25919
  1293
apply (rule of_int_minus [symmetric])
haftmann@25919
  1294
done
haftmann@25919
  1295
haftmann@25919
  1296
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
  1297
apply (auto simp add: Ints_def)
haftmann@25919
  1298
apply (rule range_eqI)
haftmann@25919
  1299
apply (rule of_int_mult [symmetric])
haftmann@25919
  1300
done
haftmann@25919
  1301
haftmann@25919
  1302
lemma Ints_cases [cases set: Ints]:
haftmann@25919
  1303
  assumes "q \<in> \<int>"
haftmann@25919
  1304
  obtains (of_int) z where "q = of_int z"
haftmann@25919
  1305
  unfolding Ints_def
haftmann@25919
  1306
proof -
haftmann@25919
  1307
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
  1308
  then obtain z where "q = of_int z" ..
haftmann@25919
  1309
  then show thesis ..
haftmann@25919
  1310
qed
haftmann@25919
  1311
haftmann@25919
  1312
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
  1313
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
  1314
  by (rule Ints_cases) auto
haftmann@25919
  1315
haftmann@25919
  1316
end
haftmann@25919
  1317
haftmann@25919
  1318
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a-b \<in> \<int>"
haftmann@25919
  1319
apply (auto simp add: Ints_def)
haftmann@25919
  1320
apply (rule range_eqI)
haftmann@25919
  1321
apply (rule of_int_diff [symmetric])
haftmann@25919
  1322
done
haftmann@25919
  1323
haftmann@25919
  1324
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
  1325
haftmann@25919
  1326
lemma Ints_double_eq_0_iff:
haftmann@25919
  1327
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1328
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
  1329
proof -
haftmann@25919
  1330
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1331
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1332
  show ?thesis
haftmann@25919
  1333
  proof
haftmann@25919
  1334
    assume "a = 0"
haftmann@25919
  1335
    thus "a + a = 0" by simp
haftmann@25919
  1336
  next
haftmann@25919
  1337
    assume eq: "a + a = 0"
haftmann@25919
  1338
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1339
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1340
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
  1341
    thus "a = 0" by (simp add: a)
haftmann@25919
  1342
  qed
haftmann@25919
  1343
qed
haftmann@25919
  1344
haftmann@25919
  1345
lemma Ints_odd_nonzero:
haftmann@25919
  1346
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1347
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
  1348
proof -
haftmann@25919
  1349
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1350
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1351
  show ?thesis
haftmann@25919
  1352
  proof
haftmann@25919
  1353
    assume eq: "1 + a + a = 0"
haftmann@25919
  1354
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1355
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1356
    with odd_nonzero show False by blast
haftmann@25919
  1357
  qed
haftmann@25919
  1358
qed 
haftmann@25919
  1359
haftmann@25919
  1360
lemma Ints_number_of:
haftmann@25919
  1361
  "(number_of w :: 'a::number_ring) \<in> Ints"
haftmann@25919
  1362
  unfolding number_of_eq Ints_def by simp
haftmann@25919
  1363
haftmann@25919
  1364
lemma Ints_odd_less_0: 
haftmann@25919
  1365
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1366
  shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))";
haftmann@25919
  1367
proof -
haftmann@25919
  1368
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1369
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1370
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
  1371
    by (simp add: a)
haftmann@25919
  1372
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
haftmann@25919
  1373
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
  1374
  finally show ?thesis .
haftmann@25919
  1375
qed
haftmann@25919
  1376
haftmann@25919
  1377
haftmann@25919
  1378
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
  1379
haftmann@25919
  1380
text {*By Jeremy Avigad*}
haftmann@25919
  1381
haftmann@25919
  1382
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
  1383
  apply (cases "finite A")
haftmann@25919
  1384
  apply (erule finite_induct, auto)
haftmann@25919
  1385
  done
haftmann@25919
  1386
haftmann@25919
  1387
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
  1388
  apply (cases "finite A")
haftmann@25919
  1389
  apply (erule finite_induct, auto)
haftmann@25919
  1390
  done
haftmann@25919
  1391
haftmann@25919
  1392
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
  1393
  apply (cases "finite A")
haftmann@25919
  1394
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
  1395
  done
haftmann@25919
  1396
haftmann@25919
  1397
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
  1398
  apply (cases "finite A")
haftmann@25919
  1399
  apply (erule finite_induct, auto)
haftmann@25919
  1400
  done
haftmann@25919
  1401
haftmann@25919
  1402
lemma setprod_nonzero_nat:
haftmann@25919
  1403
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
haftmann@25919
  1404
  by (rule setprod_nonzero, auto)
haftmann@25919
  1405
haftmann@25919
  1406
lemma setprod_zero_eq_nat:
haftmann@25919
  1407
    "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
haftmann@25919
  1408
  by (rule setprod_zero_eq, auto)
haftmann@25919
  1409
haftmann@25919
  1410
lemma setprod_nonzero_int:
haftmann@25919
  1411
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
haftmann@25919
  1412
  by (rule setprod_nonzero, auto)
haftmann@25919
  1413
haftmann@25919
  1414
lemma setprod_zero_eq_int:
haftmann@25919
  1415
    "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
haftmann@25919
  1416
  by (rule setprod_zero_eq, auto)
haftmann@25919
  1417
haftmann@25919
  1418
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
  1419
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
  1420
haftmann@25919
  1421
haftmann@25919
  1422
subsection{*Inequality Reasoning for the Arithmetic Simproc*}
haftmann@25919
  1423
haftmann@25919
  1424
lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
haftmann@25919
  1425
by simp 
haftmann@25919
  1426
haftmann@25919
  1427
lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
haftmann@25919
  1428
by simp
haftmann@25919
  1429
haftmann@25919
  1430
lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
haftmann@25919
  1431
by simp 
haftmann@25919
  1432
haftmann@25919
  1433
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
haftmann@25919
  1434
by simp
haftmann@25919
  1435
haftmann@25919
  1436
lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
haftmann@25919
  1437
by simp
haftmann@25919
  1438
haftmann@25919
  1439
lemma inverse_numeral_1:
haftmann@25919
  1440
  "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
haftmann@25919
  1441
by simp
haftmann@25919
  1442
haftmann@25919
  1443
text{*Theorem lists for the cancellation simprocs. The use of binary numerals
haftmann@25919
  1444
for 0 and 1 reduces the number of special cases.*}
haftmann@25919
  1445
haftmann@25919
  1446
lemmas add_0s = add_numeral_0 add_numeral_0_right
haftmann@25919
  1447
lemmas mult_1s = mult_numeral_1 mult_numeral_1_right 
haftmann@25919
  1448
                 mult_minus1 mult_minus1_right
haftmann@25919
  1449
haftmann@25919
  1450
haftmann@25919
  1451
subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
haftmann@25919
  1452
haftmann@25919
  1453
text{*Arithmetic computations are defined for binary literals, which leaves 0
haftmann@25919
  1454
and 1 as special cases. Addition already has rules for 0, but not 1.
haftmann@25919
  1455
Multiplication and unary minus already have rules for both 0 and 1.*}
haftmann@25919
  1456
haftmann@25919
  1457
haftmann@25919
  1458
lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
haftmann@25919
  1459
by simp
haftmann@25919
  1460
haftmann@25919
  1461
haftmann@25919
  1462
lemmas add_number_of_eq = number_of_add [symmetric]
haftmann@25919
  1463
haftmann@25919
  1464
text{*Allow 1 on either or both sides*}
haftmann@25919
  1465
lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
haftmann@25919
  1466
by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric] add_number_of_eq)
haftmann@25919
  1467
haftmann@25919
  1468
lemmas add_special =
haftmann@25919
  1469
    one_add_one_is_two
haftmann@25919
  1470
    binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1471
    binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1472
haftmann@25919
  1473
text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
haftmann@25919
  1474
lemmas diff_special =
haftmann@25919
  1475
    binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1476
    binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1477
haftmann@25919
  1478
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1479
lemmas eq_special =
haftmann@25919
  1480
    binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
haftmann@25919
  1481
    binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1482
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
haftmann@25919
  1483
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1484
haftmann@25919
  1485
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1486
lemmas less_special =
huffman@28984
  1487
  binop_eq [of "op <", OF less_number_of numeral_0_eq_0 refl, standard]
huffman@28984
  1488
  binop_eq [of "op <", OF less_number_of numeral_1_eq_1 refl, standard]
huffman@28984
  1489
  binop_eq [of "op <", OF less_number_of refl numeral_0_eq_0, standard]
huffman@28984
  1490
  binop_eq [of "op <", OF less_number_of refl numeral_1_eq_1, standard]
haftmann@25919
  1491
haftmann@25919
  1492
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1493
lemmas le_special =
huffman@28984
  1494
    binop_eq [of "op \<le>", OF le_number_of numeral_0_eq_0 refl, standard]
huffman@28984
  1495
    binop_eq [of "op \<le>", OF le_number_of numeral_1_eq_1 refl, standard]
huffman@28984
  1496
    binop_eq [of "op \<le>", OF le_number_of refl numeral_0_eq_0, standard]
huffman@28984
  1497
    binop_eq [of "op \<le>", OF le_number_of refl numeral_1_eq_1, standard]
haftmann@25919
  1498
haftmann@25919
  1499
lemmas arith_special[simp] = 
haftmann@25919
  1500
       add_special diff_special eq_special less_special le_special
haftmann@25919
  1501
haftmann@25919
  1502
haftmann@25919
  1503
lemma min_max_01: "min (0::int) 1 = 0 & min (1::int) 0 = 0 &
haftmann@25919
  1504
                   max (0::int) 1 = 1 & max (1::int) 0 = 1"
haftmann@25919
  1505
by(simp add:min_def max_def)
haftmann@25919
  1506
haftmann@25919
  1507
lemmas min_max_special[simp] =
haftmann@25919
  1508
 min_max_01
haftmann@25919
  1509
 max_def[of "0::int" "number_of v", standard, simp]
haftmann@25919
  1510
 min_def[of "0::int" "number_of v", standard, simp]
haftmann@25919
  1511
 max_def[of "number_of u" "0::int", standard, simp]
haftmann@25919
  1512
 min_def[of "number_of u" "0::int", standard, simp]
haftmann@25919
  1513
 max_def[of "1::int" "number_of v", standard, simp]
haftmann@25919
  1514
 min_def[of "1::int" "number_of v", standard, simp]
haftmann@25919
  1515
 max_def[of "number_of u" "1::int", standard, simp]
haftmann@25919
  1516
 min_def[of "number_of u" "1::int", standard, simp]
haftmann@25919
  1517
haftmann@25919
  1518
text {* Legacy theorems *}
haftmann@25919
  1519
haftmann@25919
  1520
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
  1521
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@25919
  1522
haftmann@25919
  1523
use "~~/src/Provers/Arith/assoc_fold.ML"
haftmann@28952
  1524
use "Tools/int_arith.ML"
haftmann@25919
  1525
declaration {* K int_arith_setup *}
haftmann@25919
  1526
haftmann@25919
  1527
haftmann@25919
  1528
subsection{*Lemmas About Small Numerals*}
haftmann@25919
  1529
haftmann@25919
  1530
lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
haftmann@25919
  1531
proof -
haftmann@25919
  1532
  have "(of_int -1 :: 'a) = of_int (- 1)" by simp
haftmann@25919
  1533
  also have "... = - of_int 1" by (simp only: of_int_minus)
haftmann@25919
  1534
  also have "... = -1" by simp
haftmann@25919
  1535
  finally show ?thesis .
haftmann@25919
  1536
qed
haftmann@25919
  1537
haftmann@25919
  1538
lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
haftmann@25919
  1539
by (simp add: abs_if)
haftmann@25919
  1540
haftmann@25919
  1541
lemma abs_power_minus_one [simp]:
haftmann@25919
  1542
     "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
haftmann@25919
  1543
by (simp add: power_abs)
haftmann@25919
  1544
haftmann@25919
  1545
lemma of_int_number_of_eq:
haftmann@25919
  1546
     "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
haftmann@25919
  1547
by (simp add: number_of_eq) 
haftmann@25919
  1548
haftmann@25919
  1549
text{*Lemmas for specialist use, NOT as default simprules*}
haftmann@25919
  1550
lemma mult_2: "2 * z = (z+z::'a::number_ring)"
haftmann@25919
  1551
proof -
haftmann@25919
  1552
  have "2*z = (1 + 1)*z" by simp
haftmann@25919
  1553
  also have "... = z+z" by (simp add: left_distrib)
haftmann@25919
  1554
  finally show ?thesis .
haftmann@25919
  1555
qed
haftmann@25919
  1556
haftmann@25919
  1557
lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
haftmann@25919
  1558
by (subst mult_commute, rule mult_2)
haftmann@25919
  1559
haftmann@25919
  1560
haftmann@25919
  1561
subsection{*More Inequality Reasoning*}
haftmann@25919
  1562
haftmann@25919
  1563
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
  1564
by arith
haftmann@25919
  1565
haftmann@25919
  1566
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
  1567
by arith
haftmann@25919
  1568
haftmann@25919
  1569
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
  1570
by arith
haftmann@25919
  1571
haftmann@25919
  1572
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
  1573
by arith
haftmann@25919
  1574
haftmann@25919
  1575
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
  1576
by arith
haftmann@25919
  1577
haftmann@25919
  1578
huffman@28958
  1579
subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919
  1580
haftmann@25919
  1581
text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
haftmann@25919
  1582
  @{term "w + - z"}*}
haftmann@25919
  1583
declare Zero_int_def [symmetric, simp]
haftmann@25919
  1584
declare One_int_def [symmetric, simp]
haftmann@25919
  1585
haftmann@25919
  1586
lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
haftmann@25919
  1587
haftmann@25919
  1588
lemma nat_0: "nat 0 = 0"
haftmann@25919
  1589
by (simp add: nat_eq_iff)
haftmann@25919
  1590
haftmann@25919
  1591
lemma nat_1: "nat 1 = Suc 0"
haftmann@25919
  1592
by (subst nat_eq_iff, simp)
haftmann@25919
  1593
haftmann@25919
  1594
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@25919
  1595
by (subst nat_eq_iff, simp)
haftmann@25919
  1596
haftmann@25919
  1597
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
  1598
apply (insert zless_nat_conj [of 1 z])
haftmann@25919
  1599
apply (auto simp add: nat_1)
haftmann@25919
  1600
done
haftmann@25919
  1601
haftmann@25919
  1602
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
  1603
      z is an integer literal.*}
haftmann@25919
  1604
lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
haftmann@25919
  1605
haftmann@25919
  1606
lemma split_nat [arith_split]:
haftmann@25919
  1607
  "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
  1608
  (is "?P = (?L & ?R)")
haftmann@25919
  1609
proof (cases "i < 0")
haftmann@25919
  1610
  case True thus ?thesis by auto
haftmann@25919
  1611
next
haftmann@25919
  1612
  case False
haftmann@25919
  1613
  have "?P = ?L"
haftmann@25919
  1614
  proof
haftmann@25919
  1615
    assume ?P thus ?L using False by clarsimp
haftmann@25919
  1616
  next
haftmann@25919
  1617
    assume ?L thus ?P using False by simp
haftmann@25919
  1618
  qed
haftmann@25919
  1619
  with False show ?thesis by simp
haftmann@25919
  1620
qed
haftmann@25919
  1621
haftmann@25919
  1622
context ring_1
haftmann@25919
  1623
begin
haftmann@25919
  1624
haftmann@25919
  1625
lemma of_int_of_nat:
haftmann@25919
  1626
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1627
proof (cases "k < 0")
haftmann@25919
  1628
  case True then have "0 \<le> - k" by simp
haftmann@25919
  1629
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1630
  with True show ?thesis by simp
haftmann@25919
  1631
next
haftmann@25919
  1632
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
  1633
qed
haftmann@25919
  1634
haftmann@25919
  1635
end
haftmann@25919
  1636
haftmann@25919
  1637
lemma nat_mult_distrib:
haftmann@25919
  1638
  fixes z z' :: int
haftmann@25919
  1639
  assumes "0 \<le> z"
haftmann@25919
  1640
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1641
proof (cases "0 \<le> z'")
haftmann@25919
  1642
  case False with assms have "z * z' \<le> 0"
haftmann@25919
  1643
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1644
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1645
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1646
  ultimately show ?thesis by simp
haftmann@25919
  1647
next
haftmann@25919
  1648
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1649
  show ?thesis
haftmann@25919
  1650
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1651
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1652
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1653
qed
haftmann@25919
  1654
haftmann@25919
  1655
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
  1656
apply (rule trans)
haftmann@25919
  1657
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
  1658
done
haftmann@25919
  1659
haftmann@25919
  1660
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
  1661
apply (cases "z=0 | w=0")
haftmann@25919
  1662
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
  1663
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1664
done
haftmann@25919
  1665
haftmann@25919
  1666
haftmann@25919
  1667
subsection "Induction principles for int"
haftmann@25919
  1668
haftmann@25919
  1669
text{*Well-founded segments of the integers*}
haftmann@25919
  1670
haftmann@25919
  1671
definition
haftmann@25919
  1672
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
  1673
where
haftmann@25919
  1674
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
  1675
haftmann@25919
  1676
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1677
proof -
haftmann@25919
  1678
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
  1679
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
  1680
  thus ?thesis 
haftmann@25919
  1681
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1682
qed
haftmann@25919
  1683
haftmann@25919
  1684
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
  1685
by RankFinder.*}
haftmann@25919
  1686
haftmann@25919
  1687
definition
haftmann@25919
  1688
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
  1689
where
haftmann@25919
  1690
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
  1691
haftmann@25919
  1692
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1693
proof -
haftmann@25919
  1694
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
  1695
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
  1696
  thus ?thesis 
haftmann@25919
  1697
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1698
qed
haftmann@25919
  1699
haftmann@25919
  1700
abbreviation
haftmann@25919
  1701
  int :: "nat \<Rightarrow> int"
haftmann@25919
  1702
where
haftmann@25919
  1703
  "int \<equiv> of_nat"
haftmann@25919
  1704
haftmann@25919
  1705
(* `set:int': dummy construction *)
haftmann@25919
  1706
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1707
  fixes i :: int
haftmann@25919
  1708
  assumes ge: "k \<le> i" and
haftmann@25919
  1709
    base: "P k" and
haftmann@25919
  1710
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1711
  shows "P i"
haftmann@25919
  1712
proof -
haftmann@25919
  1713
  { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
  1714
    proof (induct n)
haftmann@25919
  1715
      case 0
haftmann@25919
  1716
      hence "i = k" by arith
haftmann@25919
  1717
      thus "P i" using base by simp
haftmann@25919
  1718
    next
haftmann@25919
  1719
      case (Suc n)
haftmann@25919
  1720
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
  1721
      moreover
haftmann@25919
  1722
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
  1723
      ultimately
haftmann@25919
  1724
      have "P(i - 1)" by(rule Suc.hyps)
haftmann@25919
  1725
      from step[OF ki1 this] show ?case by simp
haftmann@25919
  1726
    qed
haftmann@25919
  1727
  }
haftmann@25919
  1728
  with ge show ?thesis by fast
haftmann@25919
  1729
qed
haftmann@25919
  1730
haftmann@25928
  1731
(* `set:int': dummy construction *)
haftmann@25928
  1732
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
  1733
  assumes gr: "k < (i::int)" and
haftmann@25919
  1734
        base: "P(k+1)" and
haftmann@25919
  1735
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
  1736
  shows "P i"
haftmann@25919
  1737
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
  1738
  using gr apply arith
haftmann@25919
  1739
 apply(rule base)
haftmann@25919
  1740
apply (rule step, simp+)
haftmann@25919
  1741
done
haftmann@25919
  1742
haftmann@25919
  1743
theorem int_le_induct[consumes 1,case_names base step]:
haftmann@25919
  1744
  assumes le: "i \<le> (k::int)" and
haftmann@25919
  1745
        base: "P(k)" and
haftmann@25919
  1746
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1747
  shows "P i"
haftmann@25919
  1748
proof -
haftmann@25919
  1749
  { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
  1750
    proof (induct n)
haftmann@25919
  1751
      case 0
haftmann@25919
  1752
      hence "i = k" by arith
haftmann@25919
  1753
      thus "P i" using base by simp
haftmann@25919
  1754
    next
haftmann@25919
  1755
      case (Suc n)
haftmann@25919
  1756
      hence "n = nat(k - (i+1))" by arith
haftmann@25919
  1757
      moreover
haftmann@25919
  1758
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
  1759
      ultimately
haftmann@25919
  1760
      have "P(i+1)" by(rule Suc.hyps)
haftmann@25919
  1761
      from step[OF ki1 this] show ?case by simp
haftmann@25919
  1762
    qed
haftmann@25919
  1763
  }
haftmann@25919
  1764
  with le show ?thesis by fast
haftmann@25919
  1765
qed
haftmann@25919
  1766
haftmann@25919
  1767
theorem int_less_induct [consumes 1,case_names base step]:
haftmann@25919
  1768
  assumes less: "(i::int) < k" and
haftmann@25919
  1769
        base: "P(k - 1)" and
haftmann@25919
  1770
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1771
  shows "P i"
haftmann@25919
  1772
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1773
  using less apply arith
haftmann@25919
  1774
 apply(rule base)
haftmann@25919
  1775
apply (rule step, simp+)
haftmann@25919
  1776
done
haftmann@25919
  1777
haftmann@25919
  1778
subsection{*Intermediate value theorems*}
haftmann@25919
  1779
haftmann@25919
  1780
lemma int_val_lemma:
haftmann@25919
  1781
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1782
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
haftmann@27106
  1783
apply (induct n, simp)
haftmann@25919
  1784
apply (intro strip)
haftmann@25919
  1785
apply (erule impE, simp)
haftmann@25919
  1786
apply (erule_tac x = n in allE, simp)
haftmann@25919
  1787
apply (case_tac "k = f (n+1) ")
haftmann@27106
  1788
apply force
haftmann@25919
  1789
apply (erule impE)
haftmann@25919
  1790
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1791
apply (blast intro: le_SucI)
haftmann@25919
  1792
done
haftmann@25919
  1793
haftmann@25919
  1794
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1795
haftmann@25919
  1796
lemma nat_intermed_int_val:
haftmann@25919
  1797
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1798
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1799
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1800
       in int_val_lemma)
haftmann@25919
  1801
apply simp
haftmann@25919
  1802
apply (erule exE)
haftmann@25919
  1803
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1804
done
haftmann@25919
  1805
haftmann@25919
  1806
haftmann@25919
  1807
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1808
haftmann@25919
  1809
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1810
by arith
haftmann@25919
  1811
haftmann@25919
  1812
lemma abs_zmult_eq_1: "(\<bar>m * n\<bar> = 1) ==> \<bar>m\<bar> = (1::int)"
haftmann@25919
  1813
apply (cases "\<bar>n\<bar>=1") 
haftmann@25919
  1814
apply (simp add: abs_mult) 
haftmann@25919
  1815
apply (rule ccontr) 
haftmann@25919
  1816
apply (auto simp add: linorder_neq_iff abs_mult) 
haftmann@25919
  1817
apply (subgoal_tac "2 \<le> \<bar>m\<bar> & 2 \<le> \<bar>n\<bar>")
haftmann@25919
  1818
 prefer 2 apply arith 
haftmann@25919
  1819
apply (subgoal_tac "2*2 \<le> \<bar>m\<bar> * \<bar>n\<bar>", simp) 
haftmann@25919
  1820
apply (rule mult_mono, auto) 
haftmann@25919
  1821
done
haftmann@25919
  1822
haftmann@25919
  1823
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1824
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1825
haftmann@25919
  1826
lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)"
haftmann@25919
  1827
apply (auto dest: pos_zmult_eq_1_iff_lemma) 
haftmann@25919
  1828
apply (simp add: mult_commute [of m]) 
haftmann@25919
  1829
apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1830
done
haftmann@25919
  1831
haftmann@25919
  1832
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1833
apply (rule iffI) 
haftmann@25919
  1834
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919
  1835
 apply (simp add: mult_commute [of m]) 
haftmann@25919
  1836
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1837
done
haftmann@25919
  1838
haftmann@25919
  1839
(* Could be simplified but Presburger only becomes available too late *)
haftmann@25919
  1840
lemma infinite_UNIV_int: "~finite(UNIV::int set)"
haftmann@25919
  1841
proof
haftmann@25919
  1842
  assume "finite(UNIV::int set)"
haftmann@25919
  1843
  moreover have "~(EX i::int. 2*i = 1)"
haftmann@25919
  1844
    by (auto simp: pos_zmult_eq_1_iff)
haftmann@25919
  1845
  ultimately show False using finite_UNIV_inj_surj[of "%n::int. n+n"]
haftmann@25919
  1846
    by (simp add:inj_on_def surj_def) (blast intro:sym)
haftmann@25919
  1847
qed
haftmann@25919
  1848
haftmann@25919
  1849
haftmann@25961
  1850
subsection{*Integer Powers*} 
haftmann@25961
  1851
haftmann@25961
  1852
instantiation int :: recpower
haftmann@25961
  1853
begin
haftmann@25961
  1854
haftmann@25961
  1855
primrec power_int where
haftmann@25961
  1856
  "p ^ 0 = (1\<Colon>int)"
haftmann@25961
  1857
  | "p ^ (Suc n) = (p\<Colon>int) * (p ^ n)"
haftmann@25961
  1858
haftmann@25961
  1859
instance proof
haftmann@25961
  1860
  fix z :: int
haftmann@25961
  1861
  fix n :: nat
haftmann@25961
  1862
  show "z ^ 0 = 1" by simp
haftmann@25961
  1863
  show "z ^ Suc n = z * (z ^ n)" by simp
haftmann@25961
  1864
qed
haftmann@25961
  1865
haftmann@25961
  1866
end
haftmann@25961
  1867
haftmann@25961
  1868
lemma zpower_zadd_distrib: "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
haftmann@25961
  1869
  by (rule Power.power_add)
haftmann@25961
  1870
haftmann@25961
  1871
lemma zpower_zpower: "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@25961
  1872
  by (rule Power.power_mult [symmetric])
haftmann@25961
  1873
haftmann@25961
  1874
lemma zero_less_zpower_abs_iff [simp]:
haftmann@25961
  1875
  "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
haftmann@25961
  1876
  by (induct n) (auto simp add: zero_less_mult_iff)
haftmann@25961
  1877
haftmann@25961
  1878
lemma zero_le_zpower_abs [simp]: "(0::int) \<le> abs x ^ n"
haftmann@25961
  1879
  by (induct n) (auto simp add: zero_le_mult_iff)
haftmann@25961
  1880
haftmann@25961
  1881
lemma of_int_power:
haftmann@25961
  1882
  "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})"
haftmann@25961
  1883
  by (induct n) (simp_all add: power_Suc)
haftmann@25961
  1884
haftmann@25961
  1885
lemma int_power: "int (m^n) = (int m) ^ n"
haftmann@25961
  1886
  by (rule of_nat_power)
haftmann@25961
  1887
haftmann@25961
  1888
lemmas zpower_int = int_power [symmetric]
haftmann@25961
  1889
haftmann@25919
  1890
subsection {* Configuration of the code generator *}
haftmann@25919
  1891
haftmann@26507
  1892
code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"
haftmann@26507
  1893
haftmann@28562
  1894
lemmas pred_succ_numeral_code [code] =
haftmann@26507
  1895
  pred_bin_simps succ_bin_simps
haftmann@26507
  1896
haftmann@28562
  1897
lemmas plus_numeral_code [code] =
haftmann@26507
  1898
  add_bin_simps
haftmann@26507
  1899
  arith_extra_simps(1) [where 'a = int]
haftmann@26507
  1900
haftmann@28562
  1901
lemmas minus_numeral_code [code] =
haftmann@26507
  1902
  minus_bin_simps
haftmann@26507
  1903
  arith_extra_simps(2) [where 'a = int]
haftmann@26507
  1904
  arith_extra_simps(5) [where 'a = int]
haftmann@26507
  1905
haftmann@28562
  1906
lemmas times_numeral_code [code] =
haftmann@26507
  1907
  mult_bin_simps
haftmann@26507
  1908
  arith_extra_simps(4) [where 'a = int]
haftmann@26507
  1909
haftmann@26507
  1910
instantiation int :: eq
haftmann@26507
  1911
begin
haftmann@26507
  1912
haftmann@28562
  1913
definition [code del]: "eq_class.eq k l \<longleftrightarrow> k - l = (0\<Colon>int)"
haftmann@26507
  1914
haftmann@26507
  1915
instance by default (simp add: eq_int_def)
haftmann@26507
  1916
haftmann@26507
  1917
end
haftmann@26507
  1918
haftmann@28562
  1919
lemma eq_number_of_int_code [code]:
haftmann@26732
  1920
  "eq_class.eq (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> eq_class.eq k l"
haftmann@26507
  1921
  unfolding eq_int_def number_of_is_id ..
haftmann@26507
  1922
haftmann@28562
  1923
lemma eq_int_code [code]:
haftmann@26732
  1924
  "eq_class.eq Int.Pls Int.Pls \<longleftrightarrow> True"
haftmann@26732
  1925
  "eq_class.eq Int.Pls Int.Min \<longleftrightarrow> False"
haftmann@26732
  1926
  "eq_class.eq Int.Pls (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq Int.Pls k2"
haftmann@26732
  1927
  "eq_class.eq Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@26732
  1928
  "eq_class.eq Int.Min Int.Pls \<longleftrightarrow> False"
haftmann@26732
  1929
  "eq_class.eq Int.Min Int.Min \<longleftrightarrow> True"
haftmann@26732
  1930
  "eq_class.eq Int.Min (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@26732
  1931
  "eq_class.eq Int.Min (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq Int.Min k2"
huffman@28958
  1932
  "eq_class.eq (Int.Bit0 k1) Int.Pls \<longleftrightarrow> eq_class.eq k1 Int.Pls"
haftmann@26732
  1933
  "eq_class.eq (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False"
haftmann@26732
  1934
  "eq_class.eq (Int.Bit0 k1) Int.Min \<longleftrightarrow> False"
huffman@28958
  1935
  "eq_class.eq (Int.Bit1 k1) Int.Min \<longleftrightarrow> eq_class.eq k1 Int.Min"
haftmann@26732
  1936
  "eq_class.eq (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq k1 k2"
haftmann@26732
  1937
  "eq_class.eq (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@26732
  1938
  "eq_class.eq (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@26732
  1939
  "eq_class.eq (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq k1 k2"
huffman@28958
  1940
  unfolding eq_equals by simp_all
haftmann@25919
  1941
haftmann@28351
  1942
lemma eq_int_refl [code nbe]:
haftmann@28351
  1943
  "eq_class.eq (k::int) k \<longleftrightarrow> True"
haftmann@28351
  1944
  by (rule HOL.eq_refl)
haftmann@28351
  1945
haftmann@28562
  1946
lemma less_eq_number_of_int_code [code]:
haftmann@26507
  1947
  "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
haftmann@26507
  1948
  unfolding number_of_is_id ..
haftmann@26507
  1949
haftmann@28562
  1950
lemma less_eq_int_code [code]:
haftmann@26507
  1951
  "Int.Pls \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507
  1952
  "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
haftmann@26507
  1953
  "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  1954
  "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  1955
  "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507
  1956
  "Int.Min \<le> Int.Min \<longleftrightarrow> True"
haftmann@26507
  1957
  "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  1958
  "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"
haftmann@26507
  1959
  "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
haftmann@26507
  1960
  "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  1961
  "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  1962
  "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  1963
  "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  1964
  "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  1965
  "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  1966
  "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
huffman@28958
  1967
  by simp_all
haftmann@26507
  1968
haftmann@28562
  1969
lemma less_number_of_int_code [code]:
haftmann@26507
  1970
  "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
haftmann@26507
  1971
  unfolding number_of_is_id ..
haftmann@26507
  1972
haftmann@28562
  1973
lemma less_int_code [code]:
haftmann@26507
  1974
  "Int.Pls < Int.Pls \<longleftrightarrow> False"
haftmann@26507
  1975
  "Int.Pls < Int.Min \<longleftrightarrow> False"
haftmann@26507
  1976
  "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"
haftmann@26507
  1977
  "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  1978
  "Int.Min < Int.Pls \<longleftrightarrow> True"
haftmann@26507
  1979
  "Int.Min < Int.Min \<longleftrightarrow> False"
haftmann@26507
  1980
  "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  1981
  "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  1982
  "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  1983
  "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  1984
  "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  1985
  "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"
haftmann@26507
  1986
  "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  1987
  "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  1988
  "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  1989
  "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"
huffman@28958
  1990
  by simp_all
haftmann@25919
  1991
haftmann@25919
  1992
definition
haftmann@25919
  1993
  nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
haftmann@25919
  1994
  "nat_aux i n = nat i + n"
haftmann@25919
  1995
haftmann@25919
  1996
lemma [code]:
haftmann@25919
  1997
  "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
haftmann@25919
  1998
  by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
haftmann@25919
  1999
    dest: zless_imp_add1_zle)
haftmann@25919
  2000
haftmann@25919
  2001
lemma [code]: "nat i = nat_aux i 0"
haftmann@25919
  2002
  by (simp add: nat_aux_def)
haftmann@25919
  2003
haftmann@28514
  2004
hide (open) const nat_aux
haftmann@25928
  2005
haftmann@28562
  2006
lemma zero_is_num_zero [code, code inline, symmetric, code post]:
haftmann@25919
  2007
  "(0\<Colon>int) = Numeral0" 
haftmann@25919
  2008
  by simp
haftmann@25919
  2009
haftmann@28562
  2010
lemma one_is_num_one [code, code inline, symmetric, code post]:
haftmann@25919
  2011
  "(1\<Colon>int) = Numeral1" 
haftmann@25961
  2012
  by simp
haftmann@25919
  2013
haftmann@25919
  2014
code_modulename SML
haftmann@25928
  2015
  Int Integer
haftmann@25919
  2016
haftmann@25919
  2017
code_modulename OCaml
haftmann@25928
  2018
  Int Integer
haftmann@25919
  2019
haftmann@25919
  2020
code_modulename Haskell
haftmann@25928
  2021
  Int Integer
haftmann@25919
  2022
haftmann@25919
  2023
types_code
haftmann@25919
  2024
  "int" ("int")
haftmann@25919
  2025
attach (term_of) {*
haftmann@25919
  2026
val term_of_int = HOLogic.mk_number HOLogic.intT;
haftmann@25919
  2027
*}
haftmann@25919
  2028
attach (test) {*
haftmann@25919
  2029
fun gen_int i =
haftmann@25919
  2030
  let val j = one_of [~1, 1] * random_range 0 i
haftmann@25919
  2031
  in (j, fn () => term_of_int j) end;
haftmann@25919
  2032
*}
haftmann@25919
  2033
haftmann@25919
  2034
setup {*
haftmann@25919
  2035
let
haftmann@25919
  2036
haftmann@25919
  2037
fun strip_number_of (@{term "Int.number_of :: int => int"} $ t) = t
haftmann@25919
  2038
  | strip_number_of t = t;
haftmann@25919
  2039
haftmann@28537
  2040
fun numeral_codegen thy defs dep module b t gr =
haftmann@25919
  2041
  let val i = HOLogic.dest_numeral (strip_number_of t)
haftmann@25919
  2042
  in
haftmann@28537
  2043
    SOME (Codegen.str (string_of_int i),
haftmann@28537
  2044
      snd (Codegen.invoke_tycodegen thy defs dep module false HOLogic.intT gr))
haftmann@25919
  2045
  end handle TERM _ => NONE;
haftmann@25919
  2046
haftmann@25919
  2047
in
haftmann@25919
  2048
haftmann@25919
  2049
Codegen.add_codegen "numeral_codegen" numeral_codegen
haftmann@25919
  2050
haftmann@25919
  2051
end
haftmann@25919
  2052
*}
haftmann@25919
  2053
haftmann@25919
  2054
consts_code
haftmann@25919
  2055
  "number_of :: int \<Rightarrow> int"    ("(_)")
haftmann@25919
  2056
  "0 :: int"                   ("0")
haftmann@25919
  2057
  "1 :: int"                   ("1")
haftmann@25919
  2058
  "uminus :: int => int"       ("~")
haftmann@25919
  2059
  "op + :: int => int => int"  ("(_ +/ _)")
haftmann@25919
  2060
  "op * :: int => int => int"  ("(_ */ _)")
haftmann@25919
  2061
  "op \<le> :: int => int => bool" ("(_ <=/ _)")
haftmann@25919
  2062
  "op < :: int => int => bool" ("(_ </ _)")
haftmann@25919
  2063
haftmann@25919
  2064
quickcheck_params [default_type = int]
haftmann@25919
  2065
huffman@26086
  2066
hide (open) const Pls Min Bit0 Bit1 succ pred
haftmann@25919
  2067
haftmann@25919
  2068
haftmann@25919
  2069
subsection {* Legacy theorems *}
haftmann@25919
  2070
haftmann@25919
  2071
lemmas zminus_zminus = minus_minus [of "z::int", standard]
haftmann@25919
  2072
lemmas zminus_0 = minus_zero [where 'a=int]
haftmann@25919
  2073
lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
haftmann@25919
  2074
lemmas zadd_commute = add_commute [of "z::int" "w", standard]
haftmann@25919
  2075
lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919
  2076
lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]
haftmann@25919
  2077
lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
haftmann@25919
  2078
lemmas zmult_ac = OrderedGroup.mult_ac
haftmann@25919
  2079
lemmas zadd_0 = OrderedGroup.add_0_left [of "z::int", standard]
haftmann@25919
  2080
lemmas zadd_0_right = OrderedGroup.add_0_left [of "z::int", standard]
haftmann@25919
  2081
lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]
haftmann@25919
  2082
lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]
haftmann@25919
  2083
lemmas zmult_commute = mult_commute [of "z::int" "w", standard]
haftmann@25919
  2084
lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919
  2085
lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919
  2086
lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919
  2087
lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919
  2088
lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919
  2089
haftmann@25919
  2090
lemmas zmult_1 = mult_1_left [of "z::int", standard]
haftmann@25919
  2091
lemmas zmult_1_right = mult_1_right [of "z::int", standard]
haftmann@25919
  2092
haftmann@25919
  2093
lemmas zle_refl = order_refl [of "w::int", standard]
haftmann@25919
  2094
lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]
haftmann@25919
  2095
lemmas zle_anti_sym = order_antisym [of "z::int" "w", standard]
haftmann@25919
  2096
lemmas zle_linear = linorder_linear [of "z::int" "w", standard]
haftmann@25919
  2097
lemmas zless_linear = linorder_less_linear [where 'a = int]
haftmann@25919
  2098
haftmann@25919
  2099
lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]
haftmann@25919
  2100
lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]
haftmann@25919
  2101
lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]
haftmann@25919
  2102
haftmann@25919
  2103
lemmas int_0_less_1 = zero_less_one [where 'a=int]
haftmann@25919
  2104
lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
haftmann@25919
  2105
haftmann@25919
  2106
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  2107
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  2108
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  2109
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
haftmann@25919
  2110
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  2111
lemmas zless_int = of_nat_less_iff [where 'a=int]
haftmann@25919
  2112
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
haftmann@25919
  2113
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  2114
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
haftmann@25919
  2115
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  2116
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  2117
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  2118
lemmas int_Suc = of_nat_Suc [where 'a=int]
haftmann@25919
  2119
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
haftmann@25919
  2120
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  2121
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
haftmann@25919
  2122
lemmas zless_le = less_int_def
haftmann@25919
  2123
lemmas int_eq_of_nat = TrueI
haftmann@25919
  2124
haftmann@25919
  2125
end