src/HOL/Int.thy
author haftmann
Fri Jan 25 14:53:52 2008 +0100 (2008-01-25)
changeset 25961 ec39d7e40554
parent 25928 042e877d9841
child 26072 f65a7fa2da6c
permissions -rw-r--r--
moved definition of power on ints to theory Int
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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 Wellfounded_Relations Datatype Nat
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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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  ("int_arith1.ML")
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begin
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subsection {* The equivalence relation underlying the integers *}
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definition
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  intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set"
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where
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  "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"
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typedef (Integ)
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  int = "UNIV//intrel"
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  by (auto simp add: quotient_def)
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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 func del]: "0 = Abs_Integ (intrel `` {(0, 0)})"
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definition
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  One_int_def [code func del]: "1 = Abs_Integ (intrel `` {(1, 0)})"
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definition
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  add_int_def [code func 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 func 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 func del]:  "z - w = z + (-w \<Colon> int)"
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definition
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  mult_int_def [code func 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 func 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 func 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 ring_simps)
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  show "i * j = j * i"
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    by (cases i, cases j) (simp add: mult ring_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 ring_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 "(i < j) = (i \<le> j \<and> i \<noteq> j)"
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    by (simp add: less_int_def)
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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 \<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 \<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 func 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 compare_rls 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"
haftmann@25919
   319
  by (cases w, cases z, simp add: compare_rls of_int OrderedGroup.compare_rls add)
haftmann@25919
   320
haftmann@25919
   321
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
haftmann@25919
   322
  by (cases z, simp add: compare_rls of_int minus)
haftmann@25919
   323
haftmann@25919
   324
lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
haftmann@25919
   325
  by (simp add: OrderedGroup.diff_minus diff_minus)
haftmann@25919
   326
haftmann@25919
   327
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
haftmann@25919
   328
apply (cases w, cases z)
haftmann@25919
   329
apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
haftmann@25919
   330
                 mult add_ac of_nat_mult)
haftmann@25919
   331
done
haftmann@25919
   332
haftmann@25919
   333
text{*Collapse nested embeddings*}
haftmann@25919
   334
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
haftmann@25919
   335
  by (induct n) auto
haftmann@25919
   336
haftmann@25919
   337
end
haftmann@25919
   338
haftmann@25919
   339
context ordered_idom
haftmann@25919
   340
begin
haftmann@25919
   341
haftmann@25919
   342
lemma of_int_le_iff [simp]:
haftmann@25919
   343
  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
haftmann@25919
   344
  by (cases w, cases z, simp add: of_int le minus compare_rls of_nat_add [symmetric] del: of_nat_add)
haftmann@25919
   345
haftmann@25919
   346
text{*Special cases where either operand is zero*}
haftmann@25919
   347
lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
haftmann@25919
   348
lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
haftmann@25919
   349
haftmann@25919
   350
lemma of_int_less_iff [simp]:
haftmann@25919
   351
  "of_int w < of_int z \<longleftrightarrow> w < z"
haftmann@25919
   352
  by (simp add: not_le [symmetric] linorder_not_le [symmetric])
haftmann@25919
   353
haftmann@25919
   354
text{*Special cases where either operand is zero*}
haftmann@25919
   355
lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
haftmann@25919
   356
lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
haftmann@25919
   357
haftmann@25919
   358
end
haftmann@25919
   359
haftmann@25919
   360
text{*Class for unital rings with characteristic zero.
haftmann@25919
   361
 Includes non-ordered rings like the complex numbers.*}
haftmann@25919
   362
class ring_char_0 = ring_1 + semiring_char_0
haftmann@25919
   363
begin
haftmann@25919
   364
haftmann@25919
   365
lemma of_int_eq_iff [simp]:
haftmann@25919
   366
   "of_int w = of_int z \<longleftrightarrow> w = z"
haftmann@25919
   367
apply (cases w, cases z, simp add: of_int)
haftmann@25919
   368
apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
haftmann@25919
   369
apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
haftmann@25919
   370
done
haftmann@25919
   371
haftmann@25919
   372
text{*Special cases where either operand is zero*}
haftmann@25919
   373
lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
haftmann@25919
   374
lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
haftmann@25919
   375
haftmann@25919
   376
end
haftmann@25919
   377
haftmann@25919
   378
text{*Every @{text ordered_idom} has characteristic zero.*}
haftmann@25919
   379
subclass (in ordered_idom) ring_char_0 by intro_locales
haftmann@25919
   380
haftmann@25919
   381
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   382
proof
haftmann@25919
   383
  fix z show "of_int z = id z"
haftmann@25919
   384
    by (cases z) (simp add: of_int add minus int_def diff_minus)
haftmann@25919
   385
qed
haftmann@25919
   386
haftmann@25919
   387
haftmann@25919
   388
subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
haftmann@25919
   389
haftmann@25919
   390
definition
haftmann@25919
   391
  nat :: "int \<Rightarrow> nat"
haftmann@25919
   392
where
haftmann@25919
   393
  [code func del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
haftmann@25919
   394
haftmann@25919
   395
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
haftmann@25919
   396
proof -
haftmann@25919
   397
  have "(\<lambda>(x,y). {x-y}) respects intrel"
haftmann@25919
   398
    by (simp add: congruent_def) arith
haftmann@25919
   399
  thus ?thesis
haftmann@25919
   400
    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
haftmann@25919
   401
qed
haftmann@25919
   402
haftmann@25919
   403
lemma nat_int [simp]: "nat (of_nat n) = n"
haftmann@25919
   404
by (simp add: nat int_def)
haftmann@25919
   405
haftmann@25919
   406
lemma nat_zero [simp]: "nat 0 = 0"
haftmann@25919
   407
by (simp add: Zero_int_def nat)
haftmann@25919
   408
haftmann@25919
   409
lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
haftmann@25919
   410
by (cases z, simp add: nat le int_def Zero_int_def)
haftmann@25919
   411
haftmann@25919
   412
corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
haftmann@25919
   413
by simp
haftmann@25919
   414
haftmann@25919
   415
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
haftmann@25919
   416
by (cases z, simp add: nat le Zero_int_def)
haftmann@25919
   417
haftmann@25919
   418
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
haftmann@25919
   419
apply (cases w, cases z) 
haftmann@25919
   420
apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
haftmann@25919
   421
done
haftmann@25919
   422
haftmann@25919
   423
text{*An alternative condition is @{term "0 \<le> w"} *}
haftmann@25919
   424
corollary nat_mono_iff: "0 < z ==> (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
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
haftmann@25919
   428
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919
   429
haftmann@25919
   430
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
haftmann@25919
   431
apply (cases w, cases z) 
haftmann@25919
   432
apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
haftmann@25919
   433
done
haftmann@25919
   434
haftmann@25919
   435
lemma nonneg_eq_int:
haftmann@25919
   436
  fixes z :: int
haftmann@25919
   437
  assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
haftmann@25919
   438
  shows P
haftmann@25919
   439
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   440
haftmann@25919
   441
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
haftmann@25919
   442
by (cases w, simp add: nat le int_def Zero_int_def, arith)
haftmann@25919
   443
haftmann@25919
   444
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
haftmann@25919
   445
by (simp only: eq_commute [of m] nat_eq_iff)
haftmann@25919
   446
haftmann@25919
   447
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
haftmann@25919
   448
apply (cases w)
haftmann@25919
   449
apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
haftmann@25919
   450
done
haftmann@25919
   451
haftmann@25919
   452
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   453
by (auto simp add: nat_eq_iff2)
haftmann@25919
   454
haftmann@25919
   455
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   456
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   457
haftmann@25919
   458
lemma nat_add_distrib:
haftmann@25919
   459
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
haftmann@25919
   460
by (cases z, cases z', simp add: nat add le Zero_int_def)
haftmann@25919
   461
haftmann@25919
   462
lemma nat_diff_distrib:
haftmann@25919
   463
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
haftmann@25919
   464
by (cases z, cases z', 
haftmann@25919
   465
    simp add: nat add minus diff_minus le Zero_int_def)
haftmann@25919
   466
haftmann@25919
   467
lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
haftmann@25919
   468
by (simp add: int_def minus nat Zero_int_def) 
haftmann@25919
   469
haftmann@25919
   470
lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
haftmann@25919
   471
by (cases z, simp add: nat less int_def, arith)
haftmann@25919
   472
haftmann@25919
   473
context ring_1
haftmann@25919
   474
begin
haftmann@25919
   475
haftmann@25919
   476
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
haftmann@25919
   477
  by (cases z rule: eq_Abs_Integ)
haftmann@25919
   478
   (simp add: nat le of_int Zero_int_def of_nat_diff)
haftmann@25919
   479
haftmann@25919
   480
end
haftmann@25919
   481
haftmann@25919
   482
haftmann@25919
   483
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   484
haftmann@25919
   485
lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   486
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   487
haftmann@25919
   488
lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
haftmann@25919
   489
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   490
haftmann@25919
   491
lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
haftmann@25919
   492
by (simp add: minus_le_iff)
haftmann@25919
   493
haftmann@25919
   494
lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
haftmann@25919
   495
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   496
haftmann@25919
   497
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
haftmann@25919
   498
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   499
haftmann@25919
   500
lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   501
by (simp add: int_def le minus Zero_int_def)
haftmann@25919
   502
haftmann@25919
   503
lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
haftmann@25919
   504
by (simp add: linorder_not_less)
haftmann@25919
   505
haftmann@25919
   506
lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   507
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   508
haftmann@25919
   509
lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
haftmann@25919
   510
proof -
haftmann@25919
   511
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   512
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   513
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   514
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   515
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
haftmann@25919
   516
    by (simp only: group_simps)
haftmann@25919
   517
  finally show ?thesis .
haftmann@25919
   518
qed
haftmann@25919
   519
haftmann@25919
   520
lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
haftmann@25919
   521
by simp
haftmann@25919
   522
haftmann@25919
   523
lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
haftmann@25919
   524
by simp
haftmann@25919
   525
haftmann@25919
   526
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   527
      It is proved here because attribute @{text arith_split} is not available
haftmann@25919
   528
      in theory @{text Ring_and_Field}.
haftmann@25919
   529
      But is it really better than just rewriting with @{text abs_if}?*}
haftmann@25919
   530
lemma abs_split [arith_split,noatp]:
haftmann@25919
   531
     "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   532
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   533
haftmann@25919
   534
lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
haftmann@25919
   535
apply (cases x)
haftmann@25919
   536
apply (auto simp add: le minus Zero_int_def int_def order_less_le)
haftmann@25919
   537
apply (rule_tac x="y - Suc x" in exI, arith)
haftmann@25919
   538
done
haftmann@25919
   539
haftmann@25919
   540
haftmann@25919
   541
subsection {* Cases and induction *}
haftmann@25919
   542
haftmann@25919
   543
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   544
whether an integer is negative or not.*}
haftmann@25919
   545
haftmann@25919
   546
theorem int_cases [cases type: int, case_names nonneg neg]:
haftmann@25919
   547
  "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"
haftmann@25919
   548
apply (cases "z < 0", blast dest!: negD)
haftmann@25919
   549
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   550
apply auto
haftmann@25919
   551
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   552
done
haftmann@25919
   553
haftmann@25919
   554
theorem int_induct [induct type: int, case_names nonneg neg]:
haftmann@25919
   555
     "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"
haftmann@25919
   556
  by (cases z rule: int_cases) auto
haftmann@25919
   557
haftmann@25919
   558
text{*Contributed by Brian Huffman*}
haftmann@25919
   559
theorem int_diff_cases:
haftmann@25919
   560
  obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
haftmann@25919
   561
apply (cases z rule: eq_Abs_Integ)
haftmann@25919
   562
apply (rule_tac m=x and n=y in diff)
haftmann@25919
   563
apply (simp add: int_def diff_def minus add)
haftmann@25919
   564
done
haftmann@25919
   565
haftmann@25919
   566
haftmann@25919
   567
subsection {* Binary representation *}
haftmann@25919
   568
haftmann@25919
   569
text {*
haftmann@25919
   570
  This formalization defines binary arithmetic in terms of the integers
haftmann@25919
   571
  rather than using a datatype. This avoids multiple representations (leading
haftmann@25919
   572
  zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
haftmann@25919
   573
  int_of_binary}, for the numerical interpretation.
haftmann@25919
   574
haftmann@25919
   575
  The representation expects that @{text "(m mod 2)"} is 0 or 1,
haftmann@25919
   576
  even if m is negative;
haftmann@25919
   577
  For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
haftmann@25919
   578
  @{text "-5 = (-3)*2 + 1"}.
haftmann@25919
   579
  
haftmann@25919
   580
  This two's complement binary representation derives from the paper 
haftmann@25919
   581
  "An Efficient Representation of Arithmetic for Term Rewriting" by
haftmann@25919
   582
  Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
haftmann@25919
   583
  Springer LNCS 488 (240-251), 1991.
haftmann@25919
   584
*}
haftmann@25919
   585
haftmann@25919
   586
datatype bit = B0 | B1
haftmann@25919
   587
haftmann@25919
   588
text{*
haftmann@25919
   589
  Type @{typ bit} avoids the use of type @{typ bool}, which would make
haftmann@25919
   590
  all of the rewrite rules higher-order.
haftmann@25919
   591
*}
haftmann@25919
   592
haftmann@25919
   593
definition
haftmann@25919
   594
  Pls :: int where
haftmann@25919
   595
  [code func del]: "Pls = 0"
haftmann@25919
   596
haftmann@25919
   597
definition
haftmann@25919
   598
  Min :: int where
haftmann@25919
   599
  [code func del]: "Min = - 1"
haftmann@25919
   600
haftmann@25919
   601
definition
haftmann@25919
   602
  Bit :: "int \<Rightarrow> bit \<Rightarrow> int" (infixl "BIT" 90) where
haftmann@25919
   603
  [code func del]: "k BIT b = (case b of B0 \<Rightarrow> 0 | B1 \<Rightarrow> 1) + k + k"
haftmann@25919
   604
haftmann@25919
   605
class number = type + -- {* for numeric types: nat, int, real, \dots *}
haftmann@25919
   606
  fixes number_of :: "int \<Rightarrow> 'a"
haftmann@25919
   607
haftmann@25919
   608
use "Tools/numeral.ML"
haftmann@25919
   609
haftmann@25919
   610
syntax
haftmann@25919
   611
  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
haftmann@25919
   612
haftmann@25919
   613
use "Tools/numeral_syntax.ML"
haftmann@25919
   614
setup NumeralSyntax.setup
haftmann@25919
   615
haftmann@25919
   616
abbreviation
haftmann@25919
   617
  "Numeral0 \<equiv> number_of Pls"
haftmann@25919
   618
haftmann@25919
   619
abbreviation
haftmann@25919
   620
  "Numeral1 \<equiv> number_of (Pls BIT B1)"
haftmann@25919
   621
haftmann@25919
   622
lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
haftmann@25919
   623
  -- {* Unfold all @{text let}s involving constants *}
haftmann@25919
   624
  unfolding Let_def ..
haftmann@25919
   625
haftmann@25919
   626
definition
haftmann@25919
   627
  succ :: "int \<Rightarrow> int" where
haftmann@25919
   628
  [code func del]: "succ k = k + 1"
haftmann@25919
   629
haftmann@25919
   630
definition
haftmann@25919
   631
  pred :: "int \<Rightarrow> int" where
haftmann@25919
   632
  [code func del]: "pred k = k - 1"
haftmann@25919
   633
haftmann@25919
   634
lemmas
haftmann@25919
   635
  max_number_of [simp] = max_def
haftmann@25919
   636
    [of "number_of u" "number_of v", standard, simp]
haftmann@25919
   637
and
haftmann@25919
   638
  min_number_of [simp] = min_def 
haftmann@25919
   639
    [of "number_of u" "number_of v", standard, simp]
haftmann@25919
   640
  -- {* unfolding @{text minx} and @{text max} on numerals *}
haftmann@25919
   641
haftmann@25919
   642
lemmas numeral_simps = 
haftmann@25919
   643
  succ_def pred_def Pls_def Min_def Bit_def
haftmann@25919
   644
haftmann@25919
   645
text {* Removal of leading zeroes *}
haftmann@25919
   646
haftmann@25919
   647
lemma Pls_0_eq [simp, code post]:
haftmann@25919
   648
  "Pls BIT B0 = Pls"
haftmann@25919
   649
  unfolding numeral_simps by simp
haftmann@25919
   650
haftmann@25919
   651
lemma Min_1_eq [simp, code post]:
haftmann@25919
   652
  "Min BIT B1 = Min"
haftmann@25919
   653
  unfolding numeral_simps by simp
haftmann@25919
   654
haftmann@25919
   655
haftmann@25919
   656
subsection {* The Functions @{term succ}, @{term pred} and @{term uminus} *}
haftmann@25919
   657
haftmann@25919
   658
lemma succ_Pls [simp]:
haftmann@25919
   659
  "succ Pls = Pls BIT B1"
haftmann@25919
   660
  unfolding numeral_simps by simp
haftmann@25919
   661
haftmann@25919
   662
lemma succ_Min [simp]:
haftmann@25919
   663
  "succ Min = Pls"
haftmann@25919
   664
  unfolding numeral_simps by simp
haftmann@25919
   665
haftmann@25919
   666
lemma succ_1 [simp]:
haftmann@25919
   667
  "succ (k BIT B1) = succ k BIT B0"
haftmann@25919
   668
  unfolding numeral_simps by simp
haftmann@25919
   669
haftmann@25919
   670
lemma succ_0 [simp]:
haftmann@25919
   671
  "succ (k BIT B0) = k BIT B1"
haftmann@25919
   672
  unfolding numeral_simps by simp
haftmann@25919
   673
haftmann@25919
   674
lemma pred_Pls [simp]:
haftmann@25919
   675
  "pred Pls = Min"
haftmann@25919
   676
  unfolding numeral_simps by simp
haftmann@25919
   677
haftmann@25919
   678
lemma pred_Min [simp]:
haftmann@25919
   679
  "pred Min = Min BIT B0"
haftmann@25919
   680
  unfolding numeral_simps by simp
haftmann@25919
   681
haftmann@25919
   682
lemma pred_1 [simp]:
haftmann@25919
   683
  "pred (k BIT B1) = k BIT B0"
haftmann@25919
   684
  unfolding numeral_simps by simp
haftmann@25919
   685
haftmann@25919
   686
lemma pred_0 [simp]:
haftmann@25919
   687
  "pred (k BIT B0) = pred k BIT B1"
haftmann@25919
   688
  unfolding numeral_simps by simp 
haftmann@25919
   689
haftmann@25919
   690
lemma minus_Pls [simp]:
haftmann@25919
   691
  "- Pls = Pls"
haftmann@25919
   692
  unfolding numeral_simps by simp 
haftmann@25919
   693
haftmann@25919
   694
lemma minus_Min [simp]:
haftmann@25919
   695
  "- Min = Pls BIT B1"
haftmann@25919
   696
  unfolding numeral_simps by simp 
haftmann@25919
   697
haftmann@25919
   698
lemma minus_1 [simp]:
haftmann@25919
   699
  "- (k BIT B1) = pred (- k) BIT B1"
haftmann@25919
   700
  unfolding numeral_simps by simp 
haftmann@25919
   701
haftmann@25919
   702
lemma minus_0 [simp]:
haftmann@25919
   703
  "- (k BIT B0) = (- k) BIT B0"
haftmann@25919
   704
  unfolding numeral_simps by simp 
haftmann@25919
   705
haftmann@25919
   706
haftmann@25919
   707
subsection {*
haftmann@25919
   708
  Binary Addition and Multiplication: @{term "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
haftmann@25919
   709
    and @{term "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"}
haftmann@25919
   710
*}
haftmann@25919
   711
haftmann@25919
   712
lemma add_Pls [simp]:
haftmann@25919
   713
  "Pls + k = k"
haftmann@25919
   714
  unfolding numeral_simps by simp 
haftmann@25919
   715
haftmann@25919
   716
lemma add_Min [simp]:
haftmann@25919
   717
  "Min + k = pred k"
haftmann@25919
   718
  unfolding numeral_simps by simp
haftmann@25919
   719
haftmann@25919
   720
lemma add_BIT_11 [simp]:
haftmann@25919
   721
  "(k BIT B1) + (l BIT B1) = (k + succ l) BIT B0"
haftmann@25919
   722
  unfolding numeral_simps by simp
haftmann@25919
   723
haftmann@25919
   724
lemma add_BIT_10 [simp]:
haftmann@25919
   725
  "(k BIT B1) + (l BIT B0) = (k + l) BIT B1"
haftmann@25919
   726
  unfolding numeral_simps by simp
haftmann@25919
   727
haftmann@25919
   728
lemma add_BIT_0 [simp]:
haftmann@25919
   729
  "(k BIT B0) + (l BIT b) = (k + l) BIT b"
haftmann@25919
   730
  unfolding numeral_simps by simp 
haftmann@25919
   731
haftmann@25919
   732
lemma add_Pls_right [simp]:
haftmann@25919
   733
  "k + Pls = k"
haftmann@25919
   734
  unfolding numeral_simps by simp 
haftmann@25919
   735
haftmann@25919
   736
lemma add_Min_right [simp]:
haftmann@25919
   737
  "k + Min = pred k"
haftmann@25919
   738
  unfolding numeral_simps by simp 
haftmann@25919
   739
haftmann@25919
   740
lemma mult_Pls [simp]:
haftmann@25919
   741
  "Pls * w = Pls"
haftmann@25919
   742
  unfolding numeral_simps by simp 
haftmann@25919
   743
haftmann@25919
   744
lemma mult_Min [simp]:
haftmann@25919
   745
  "Min * k = - k"
haftmann@25919
   746
  unfolding numeral_simps by simp
haftmann@25919
   747
haftmann@25919
   748
lemma mult_num1 [simp]:
haftmann@25919
   749
  "(k BIT B1) * l = ((k * l) BIT B0) + l"
haftmann@25919
   750
  unfolding numeral_simps int_distrib by simp 
haftmann@25919
   751
haftmann@25919
   752
lemma mult_num0 [simp]:
haftmann@25919
   753
  "(k BIT B0) * l = (k * l) BIT B0"
haftmann@25919
   754
  unfolding numeral_simps int_distrib by simp 
haftmann@25919
   755
haftmann@25919
   756
haftmann@25919
   757
subsection {* Converting Numerals to Rings: @{term number_of} *}
haftmann@25919
   758
haftmann@25919
   759
class number_ring = number + comm_ring_1 +
haftmann@25919
   760
  assumes number_of_eq: "number_of k = of_int k"
haftmann@25919
   761
haftmann@25919
   762
text {* self-embedding of the integers *}
haftmann@25919
   763
haftmann@25919
   764
instantiation int :: number_ring
haftmann@25919
   765
begin
haftmann@25919
   766
haftmann@25919
   767
definition
haftmann@25919
   768
  int_number_of_def [code func del]: "number_of w = (of_int w \<Colon> int)"
haftmann@25919
   769
haftmann@25919
   770
instance
haftmann@25919
   771
  by intro_classes (simp only: int_number_of_def)
haftmann@25919
   772
haftmann@25919
   773
end
haftmann@25919
   774
haftmann@25919
   775
lemma number_of_is_id:
haftmann@25919
   776
  "number_of (k::int) = k"
haftmann@25919
   777
  unfolding int_number_of_def by simp
haftmann@25919
   778
haftmann@25919
   779
lemma number_of_succ:
haftmann@25919
   780
  "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
haftmann@25919
   781
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   782
haftmann@25919
   783
lemma number_of_pred:
haftmann@25919
   784
  "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
haftmann@25919
   785
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   786
haftmann@25919
   787
lemma number_of_minus:
haftmann@25919
   788
  "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
haftmann@25919
   789
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   790
haftmann@25919
   791
lemma number_of_add:
haftmann@25919
   792
  "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
haftmann@25919
   793
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   794
haftmann@25919
   795
lemma number_of_mult:
haftmann@25919
   796
  "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
haftmann@25919
   797
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   798
haftmann@25919
   799
text {*
haftmann@25919
   800
  The correctness of shifting.
haftmann@25919
   801
  But it doesn't seem to give a measurable speed-up.
haftmann@25919
   802
*}
haftmann@25919
   803
haftmann@25919
   804
lemma double_number_of_BIT:
haftmann@25919
   805
  "(1 + 1) * number_of w = (number_of (w BIT B0) ::'a::number_ring)"
haftmann@25919
   806
  unfolding number_of_eq numeral_simps left_distrib by simp
haftmann@25919
   807
haftmann@25919
   808
text {*
haftmann@25919
   809
  Converting numerals 0 and 1 to their abstract versions.
haftmann@25919
   810
*}
haftmann@25919
   811
haftmann@25919
   812
lemma numeral_0_eq_0 [simp]:
haftmann@25919
   813
  "Numeral0 = (0::'a::number_ring)"
haftmann@25919
   814
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   815
haftmann@25919
   816
lemma numeral_1_eq_1 [simp]:
haftmann@25919
   817
  "Numeral1 = (1::'a::number_ring)"
haftmann@25919
   818
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   819
haftmann@25919
   820
text {*
haftmann@25919
   821
  Special-case simplification for small constants.
haftmann@25919
   822
*}
haftmann@25919
   823
haftmann@25919
   824
text{*
haftmann@25919
   825
  Unary minus for the abstract constant 1. Cannot be inserted
haftmann@25919
   826
  as a simprule until later: it is @{text number_of_Min} re-oriented!
haftmann@25919
   827
*}
haftmann@25919
   828
haftmann@25919
   829
lemma numeral_m1_eq_minus_1:
haftmann@25919
   830
  "(-1::'a::number_ring) = - 1"
haftmann@25919
   831
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   832
haftmann@25919
   833
lemma mult_minus1 [simp]:
haftmann@25919
   834
  "-1 * z = -(z::'a::number_ring)"
haftmann@25919
   835
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   836
haftmann@25919
   837
lemma mult_minus1_right [simp]:
haftmann@25919
   838
  "z * -1 = -(z::'a::number_ring)"
haftmann@25919
   839
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   840
haftmann@25919
   841
(*Negation of a coefficient*)
haftmann@25919
   842
lemma minus_number_of_mult [simp]:
haftmann@25919
   843
   "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
haftmann@25919
   844
   unfolding number_of_eq by simp
haftmann@25919
   845
haftmann@25919
   846
text {* Subtraction *}
haftmann@25919
   847
haftmann@25919
   848
lemma diff_number_of_eq:
haftmann@25919
   849
  "number_of v - number_of w =
haftmann@25919
   850
    (number_of (v + uminus w)::'a::number_ring)"
haftmann@25919
   851
  unfolding number_of_eq by simp
haftmann@25919
   852
haftmann@25919
   853
lemma number_of_Pls:
haftmann@25919
   854
  "number_of Pls = (0::'a::number_ring)"
haftmann@25919
   855
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   856
haftmann@25919
   857
lemma number_of_Min:
haftmann@25919
   858
  "number_of Min = (- 1::'a::number_ring)"
haftmann@25919
   859
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   860
haftmann@25919
   861
lemma number_of_BIT:
haftmann@25919
   862
  "number_of(w BIT x) = (case x of B0 => 0 | B1 => (1::'a::number_ring))
haftmann@25919
   863
    + (number_of w) + (number_of w)"
haftmann@25919
   864
  unfolding number_of_eq numeral_simps by (simp split: bit.split)
haftmann@25919
   865
haftmann@25919
   866
haftmann@25919
   867
subsection {* Equality of Binary Numbers *}
haftmann@25919
   868
haftmann@25919
   869
text {* First version by Norbert Voelker *}
haftmann@25919
   870
haftmann@25919
   871
definition
haftmann@25919
   872
  neg  :: "'a\<Colon>ordered_idom \<Rightarrow> bool"
haftmann@25919
   873
where
haftmann@25919
   874
  "neg Z \<longleftrightarrow> Z < 0"
haftmann@25919
   875
haftmann@25919
   876
definition (*for simplifying equalities*)
haftmann@25919
   877
  iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
haftmann@25919
   878
where
haftmann@25919
   879
  "iszero z \<longleftrightarrow> z = 0"
haftmann@25919
   880
haftmann@25919
   881
lemma not_neg_int [simp]: "~ neg (of_nat n)"
haftmann@25919
   882
by (simp add: neg_def)
haftmann@25919
   883
haftmann@25919
   884
lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
haftmann@25919
   885
by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
haftmann@25919
   886
haftmann@25919
   887
lemmas neg_eq_less_0 = neg_def
haftmann@25919
   888
haftmann@25919
   889
lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
haftmann@25919
   890
by (simp add: neg_def linorder_not_less)
haftmann@25919
   891
haftmann@25919
   892
text{*To simplify inequalities when Numeral1 can get simplified to 1*}
haftmann@25919
   893
haftmann@25919
   894
lemma not_neg_0: "~ neg 0"
haftmann@25919
   895
by (simp add: One_int_def neg_def)
haftmann@25919
   896
haftmann@25919
   897
lemma not_neg_1: "~ neg 1"
haftmann@25919
   898
by (simp add: neg_def linorder_not_less zero_le_one)
haftmann@25919
   899
haftmann@25919
   900
lemma iszero_0: "iszero 0"
haftmann@25919
   901
by (simp add: iszero_def)
haftmann@25919
   902
haftmann@25919
   903
lemma not_iszero_1: "~ iszero 1"
haftmann@25919
   904
by (simp add: iszero_def eq_commute)
haftmann@25919
   905
haftmann@25919
   906
lemma neg_nat: "neg z ==> nat z = 0"
haftmann@25919
   907
by (simp add: neg_def order_less_imp_le) 
haftmann@25919
   908
haftmann@25919
   909
lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
haftmann@25919
   910
by (simp add: linorder_not_less neg_def)
haftmann@25919
   911
haftmann@25919
   912
lemma eq_number_of_eq:
haftmann@25919
   913
  "((number_of x::'a::number_ring) = number_of y) =
haftmann@25919
   914
   iszero (number_of (x + uminus y) :: 'a)"
haftmann@25919
   915
  unfolding iszero_def number_of_add number_of_minus
haftmann@25919
   916
  by (simp add: compare_rls)
haftmann@25919
   917
haftmann@25919
   918
lemma iszero_number_of_Pls:
haftmann@25919
   919
  "iszero ((number_of Pls)::'a::number_ring)"
haftmann@25919
   920
  unfolding iszero_def numeral_0_eq_0 ..
haftmann@25919
   921
haftmann@25919
   922
lemma nonzero_number_of_Min:
haftmann@25919
   923
  "~ iszero ((number_of Min)::'a::number_ring)"
haftmann@25919
   924
  unfolding iszero_def numeral_m1_eq_minus_1 by simp
haftmann@25919
   925
haftmann@25919
   926
haftmann@25919
   927
subsection {* Comparisons, for Ordered Rings *}
haftmann@25919
   928
haftmann@25919
   929
lemmas double_eq_0_iff = double_zero
haftmann@25919
   930
haftmann@25919
   931
lemma le_imp_0_less: 
haftmann@25919
   932
  assumes le: "0 \<le> z"
haftmann@25919
   933
  shows "(0::int) < 1 + z"
haftmann@25919
   934
proof -
haftmann@25919
   935
  have "0 \<le> z" by fact
haftmann@25919
   936
  also have "... < z + 1" by (rule less_add_one) 
haftmann@25919
   937
  also have "... = 1 + z" by (simp add: add_ac)
haftmann@25919
   938
  finally show "0 < 1 + z" .
haftmann@25919
   939
qed
haftmann@25919
   940
haftmann@25919
   941
lemma odd_nonzero:
haftmann@25919
   942
  "1 + z + z \<noteq> (0::int)";
haftmann@25919
   943
proof (cases z rule: int_cases)
haftmann@25919
   944
  case (nonneg n)
haftmann@25919
   945
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
   946
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919
   947
    by (auto simp add: add_assoc) 
haftmann@25919
   948
next
haftmann@25919
   949
  case (neg n)
haftmann@25919
   950
  show ?thesis
haftmann@25919
   951
  proof
haftmann@25919
   952
    assume eq: "1 + z + z = 0"
haftmann@25919
   953
    have "(0::int) < 1 + (of_nat n + of_nat n)"
haftmann@25919
   954
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
   955
    also have "... = - (1 + z + z)" 
haftmann@25919
   956
      by (simp add: neg add_assoc [symmetric]) 
haftmann@25919
   957
    also have "... = 0" by (simp add: eq) 
haftmann@25919
   958
    finally have "0<0" ..
haftmann@25919
   959
    thus False by blast
haftmann@25919
   960
  qed
haftmann@25919
   961
qed
haftmann@25919
   962
haftmann@25919
   963
lemma iszero_number_of_BIT:
haftmann@25919
   964
  "iszero (number_of (w BIT x)::'a) = 
haftmann@25919
   965
   (x = B0 \<and> iszero (number_of w::'a::{ring_char_0,number_ring}))"
haftmann@25919
   966
proof -
haftmann@25919
   967
  have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
haftmann@25919
   968
  proof -
haftmann@25919
   969
    assume eq: "of_int w + of_int w = (0::'a)"
haftmann@25919
   970
    then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
haftmann@25919
   971
    then have "w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   972
    then show "w = 0" by (simp only: double_eq_0_iff)
haftmann@25919
   973
  qed
haftmann@25919
   974
  moreover have "1 + of_int w + of_int w \<noteq> (0::'a)"
haftmann@25919
   975
  proof
haftmann@25919
   976
    assume eq: "1 + of_int w + of_int w = (0::'a)"
haftmann@25919
   977
    hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp 
haftmann@25919
   978
    hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
   979
    with odd_nonzero show False by blast
haftmann@25919
   980
  qed
haftmann@25919
   981
  ultimately show ?thesis
haftmann@25919
   982
    by (auto simp add: iszero_def number_of_eq numeral_simps 
haftmann@25919
   983
     split: bit.split)
haftmann@25919
   984
qed
haftmann@25919
   985
haftmann@25919
   986
lemma iszero_number_of_0:
haftmann@25919
   987
  "iszero (number_of (w BIT B0) :: 'a::{ring_char_0,number_ring}) = 
haftmann@25919
   988
  iszero (number_of w :: 'a)"
haftmann@25919
   989
  by (simp only: iszero_number_of_BIT simp_thms)
haftmann@25919
   990
haftmann@25919
   991
lemma iszero_number_of_1:
haftmann@25919
   992
  "~ iszero (number_of (w BIT B1)::'a::{ring_char_0,number_ring})"
haftmann@25919
   993
  by (simp add: iszero_number_of_BIT) 
haftmann@25919
   994
haftmann@25919
   995
haftmann@25919
   996
subsection {* The Less-Than Relation *}
haftmann@25919
   997
haftmann@25919
   998
lemma less_number_of_eq_neg:
haftmann@25919
   999
  "((number_of x::'a::{ordered_idom,number_ring}) < number_of y)
haftmann@25919
  1000
  = neg (number_of (x + uminus y) :: 'a)"
haftmann@25919
  1001
apply (subst less_iff_diff_less_0) 
haftmann@25919
  1002
apply (simp add: neg_def diff_minus number_of_add number_of_minus)
haftmann@25919
  1003
done
haftmann@25919
  1004
haftmann@25919
  1005
text {*
haftmann@25919
  1006
  If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
haftmann@25919
  1007
  @{term Numeral0} IS @{term "number_of Pls"}
haftmann@25919
  1008
*}
haftmann@25919
  1009
haftmann@25919
  1010
lemma not_neg_number_of_Pls:
haftmann@25919
  1011
  "~ neg (number_of Pls ::'a::{ordered_idom,number_ring})"
haftmann@25919
  1012
  by (simp add: neg_def numeral_0_eq_0)
haftmann@25919
  1013
haftmann@25919
  1014
lemma neg_number_of_Min:
haftmann@25919
  1015
  "neg (number_of Min ::'a::{ordered_idom,number_ring})"
haftmann@25919
  1016
  by (simp add: neg_def zero_less_one numeral_m1_eq_minus_1)
haftmann@25919
  1017
haftmann@25919
  1018
lemma double_less_0_iff:
haftmann@25919
  1019
  "(a + a < 0) = (a < (0::'a::ordered_idom))"
haftmann@25919
  1020
proof -
haftmann@25919
  1021
  have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
haftmann@25919
  1022
  also have "... = (a < 0)"
haftmann@25919
  1023
    by (simp add: mult_less_0_iff zero_less_two 
haftmann@25919
  1024
                  order_less_not_sym [OF zero_less_two]) 
haftmann@25919
  1025
  finally show ?thesis .
haftmann@25919
  1026
qed
haftmann@25919
  1027
haftmann@25919
  1028
lemma odd_less_0:
haftmann@25919
  1029
  "(1 + z + z < 0) = (z < (0::int))";
haftmann@25919
  1030
proof (cases z rule: int_cases)
haftmann@25919
  1031
  case (nonneg n)
haftmann@25919
  1032
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
haftmann@25919
  1033
                             le_imp_0_less [THEN order_less_imp_le])  
haftmann@25919
  1034
next
haftmann@25919
  1035
  case (neg n)
haftmann@25919
  1036
  thus ?thesis by (simp del: of_nat_Suc of_nat_add
haftmann@25919
  1037
    add: compare_rls of_nat_1 [symmetric] of_nat_add [symmetric])
haftmann@25919
  1038
qed
haftmann@25919
  1039
haftmann@25919
  1040
lemma neg_number_of_BIT:
haftmann@25919
  1041
  "neg (number_of (w BIT x)::'a) = 
haftmann@25919
  1042
  neg (number_of w :: 'a::{ordered_idom,number_ring})"
haftmann@25919
  1043
proof -
haftmann@25919
  1044
  have "((1::'a) + of_int w + of_int w < 0) = (of_int (1 + w + w) < (of_int 0 :: 'a))"
haftmann@25919
  1045
    by simp
haftmann@25919
  1046
  also have "... = (w < 0)" by (simp only: of_int_less_iff odd_less_0)
haftmann@25919
  1047
  finally show ?thesis
haftmann@25919
  1048
  by ( simp add: neg_def number_of_eq numeral_simps double_less_0_iff
haftmann@25919
  1049
    split: bit.split)
haftmann@25919
  1050
qed
haftmann@25919
  1051
haftmann@25919
  1052
haftmann@25919
  1053
text {* Less-Than or Equals *}
haftmann@25919
  1054
haftmann@25919
  1055
text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
haftmann@25919
  1056
haftmann@25919
  1057
lemmas le_number_of_eq_not_less =
haftmann@25919
  1058
  linorder_not_less [of "number_of w" "number_of v", symmetric, 
haftmann@25919
  1059
  standard]
haftmann@25919
  1060
haftmann@25919
  1061
lemma le_number_of_eq:
haftmann@25919
  1062
    "((number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y)
haftmann@25919
  1063
     = (~ (neg (number_of (y + uminus x) :: 'a)))"
haftmann@25919
  1064
by (simp add: le_number_of_eq_not_less less_number_of_eq_neg)
haftmann@25919
  1065
haftmann@25919
  1066
haftmann@25919
  1067
text {* Absolute value (@{term abs}) *}
haftmann@25919
  1068
haftmann@25919
  1069
lemma abs_number_of:
haftmann@25919
  1070
  "abs(number_of x::'a::{ordered_idom,number_ring}) =
haftmann@25919
  1071
   (if number_of x < (0::'a) then -number_of x else number_of x)"
haftmann@25919
  1072
  by (simp add: abs_if)
haftmann@25919
  1073
haftmann@25919
  1074
haftmann@25919
  1075
text {* Re-orientation of the equation nnn=x *}
haftmann@25919
  1076
haftmann@25919
  1077
lemma number_of_reorient:
haftmann@25919
  1078
  "(number_of w = x) = (x = number_of w)"
haftmann@25919
  1079
  by auto
haftmann@25919
  1080
haftmann@25919
  1081
haftmann@25919
  1082
subsection {* Simplification of arithmetic operations on integer constants. *}
haftmann@25919
  1083
haftmann@25919
  1084
lemmas arith_extra_simps [standard, simp] =
haftmann@25919
  1085
  number_of_add [symmetric]
haftmann@25919
  1086
  number_of_minus [symmetric] numeral_m1_eq_minus_1 [symmetric]
haftmann@25919
  1087
  number_of_mult [symmetric]
haftmann@25919
  1088
  diff_number_of_eq abs_number_of 
haftmann@25919
  1089
haftmann@25919
  1090
text {*
haftmann@25919
  1091
  For making a minimal simpset, one must include these default simprules.
haftmann@25919
  1092
  Also include @{text simp_thms}.
haftmann@25919
  1093
*}
haftmann@25919
  1094
haftmann@25919
  1095
lemmas arith_simps = 
haftmann@25919
  1096
  bit.distinct
haftmann@25919
  1097
  Pls_0_eq Min_1_eq
haftmann@25919
  1098
  pred_Pls pred_Min pred_1 pred_0
haftmann@25919
  1099
  succ_Pls succ_Min succ_1 succ_0
haftmann@25919
  1100
  add_Pls add_Min add_BIT_0 add_BIT_10 add_BIT_11
haftmann@25919
  1101
  minus_Pls minus_Min minus_1 minus_0
haftmann@25919
  1102
  mult_Pls mult_Min mult_num1 mult_num0 
haftmann@25919
  1103
  add_Pls_right add_Min_right
haftmann@25919
  1104
  abs_zero abs_one arith_extra_simps
haftmann@25919
  1105
haftmann@25919
  1106
text {* Simplification of relational operations *}
haftmann@25919
  1107
haftmann@25919
  1108
lemmas rel_simps [simp] = 
haftmann@25919
  1109
  eq_number_of_eq iszero_0 nonzero_number_of_Min
haftmann@25919
  1110
  iszero_number_of_0 iszero_number_of_1
haftmann@25919
  1111
  less_number_of_eq_neg
haftmann@25919
  1112
  not_neg_number_of_Pls not_neg_0 not_neg_1 not_iszero_1
haftmann@25919
  1113
  neg_number_of_Min neg_number_of_BIT
haftmann@25919
  1114
  le_number_of_eq
haftmann@25919
  1115
(* iszero_number_of_Pls would never be used
haftmann@25919
  1116
   because its lhs simplifies to "iszero 0" *)
haftmann@25919
  1117
haftmann@25919
  1118
haftmann@25919
  1119
subsection {* Simplification of arithmetic when nested to the right. *}
haftmann@25919
  1120
haftmann@25919
  1121
lemma add_number_of_left [simp]:
haftmann@25919
  1122
  "number_of v + (number_of w + z) =
haftmann@25919
  1123
   (number_of(v + w) + z::'a::number_ring)"
haftmann@25919
  1124
  by (simp add: add_assoc [symmetric])
haftmann@25919
  1125
haftmann@25919
  1126
lemma mult_number_of_left [simp]:
haftmann@25919
  1127
  "number_of v * (number_of w * z) =
haftmann@25919
  1128
   (number_of(v * w) * z::'a::number_ring)"
haftmann@25919
  1129
  by (simp add: mult_assoc [symmetric])
haftmann@25919
  1130
haftmann@25919
  1131
lemma add_number_of_diff1:
haftmann@25919
  1132
  "number_of v + (number_of w - c) = 
haftmann@25919
  1133
  number_of(v + w) - (c::'a::number_ring)"
haftmann@25919
  1134
  by (simp add: diff_minus add_number_of_left)
haftmann@25919
  1135
haftmann@25919
  1136
lemma add_number_of_diff2 [simp]:
haftmann@25919
  1137
  "number_of v + (c - number_of w) =
haftmann@25919
  1138
   number_of (v + uminus w) + (c::'a::number_ring)"
haftmann@25919
  1139
apply (subst diff_number_of_eq [symmetric])
haftmann@25919
  1140
apply (simp only: compare_rls)
haftmann@25919
  1141
done
haftmann@25919
  1142
haftmann@25919
  1143
haftmann@25919
  1144
subsection {* The Set of Integers *}
haftmann@25919
  1145
haftmann@25919
  1146
context ring_1
haftmann@25919
  1147
begin
haftmann@25919
  1148
haftmann@25919
  1149
definition
haftmann@25919
  1150
  Ints  :: "'a set"
haftmann@25919
  1151
where
haftmann@25919
  1152
  "Ints = range of_int"
haftmann@25919
  1153
haftmann@25919
  1154
end
haftmann@25919
  1155
haftmann@25919
  1156
notation (xsymbols)
haftmann@25919
  1157
  Ints  ("\<int>")
haftmann@25919
  1158
haftmann@25919
  1159
context ring_1
haftmann@25919
  1160
begin
haftmann@25919
  1161
haftmann@25919
  1162
lemma Ints_0 [simp]: "0 \<in> \<int>"
haftmann@25919
  1163
apply (simp add: Ints_def)
haftmann@25919
  1164
apply (rule range_eqI)
haftmann@25919
  1165
apply (rule of_int_0 [symmetric])
haftmann@25919
  1166
done
haftmann@25919
  1167
haftmann@25919
  1168
lemma Ints_1 [simp]: "1 \<in> \<int>"
haftmann@25919
  1169
apply (simp add: Ints_def)
haftmann@25919
  1170
apply (rule range_eqI)
haftmann@25919
  1171
apply (rule of_int_1 [symmetric])
haftmann@25919
  1172
done
haftmann@25919
  1173
haftmann@25919
  1174
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
  1175
apply (auto simp add: Ints_def)
haftmann@25919
  1176
apply (rule range_eqI)
haftmann@25919
  1177
apply (rule of_int_add [symmetric])
haftmann@25919
  1178
done
haftmann@25919
  1179
haftmann@25919
  1180
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
  1181
apply (auto simp add: Ints_def)
haftmann@25919
  1182
apply (rule range_eqI)
haftmann@25919
  1183
apply (rule of_int_minus [symmetric])
haftmann@25919
  1184
done
haftmann@25919
  1185
haftmann@25919
  1186
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
  1187
apply (auto simp add: Ints_def)
haftmann@25919
  1188
apply (rule range_eqI)
haftmann@25919
  1189
apply (rule of_int_mult [symmetric])
haftmann@25919
  1190
done
haftmann@25919
  1191
haftmann@25919
  1192
lemma Ints_cases [cases set: Ints]:
haftmann@25919
  1193
  assumes "q \<in> \<int>"
haftmann@25919
  1194
  obtains (of_int) z where "q = of_int z"
haftmann@25919
  1195
  unfolding Ints_def
haftmann@25919
  1196
proof -
haftmann@25919
  1197
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
  1198
  then obtain z where "q = of_int z" ..
haftmann@25919
  1199
  then show thesis ..
haftmann@25919
  1200
qed
haftmann@25919
  1201
haftmann@25919
  1202
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
  1203
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
  1204
  by (rule Ints_cases) auto
haftmann@25919
  1205
haftmann@25919
  1206
end
haftmann@25919
  1207
haftmann@25919
  1208
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a-b \<in> \<int>"
haftmann@25919
  1209
apply (auto simp add: Ints_def)
haftmann@25919
  1210
apply (rule range_eqI)
haftmann@25919
  1211
apply (rule of_int_diff [symmetric])
haftmann@25919
  1212
done
haftmann@25919
  1213
haftmann@25919
  1214
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
  1215
haftmann@25919
  1216
lemma Ints_double_eq_0_iff:
haftmann@25919
  1217
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1218
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
  1219
proof -
haftmann@25919
  1220
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1221
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1222
  show ?thesis
haftmann@25919
  1223
  proof
haftmann@25919
  1224
    assume "a = 0"
haftmann@25919
  1225
    thus "a + a = 0" by simp
haftmann@25919
  1226
  next
haftmann@25919
  1227
    assume eq: "a + a = 0"
haftmann@25919
  1228
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1229
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1230
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
  1231
    thus "a = 0" by (simp add: a)
haftmann@25919
  1232
  qed
haftmann@25919
  1233
qed
haftmann@25919
  1234
haftmann@25919
  1235
lemma Ints_odd_nonzero:
haftmann@25919
  1236
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1237
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
  1238
proof -
haftmann@25919
  1239
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1240
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1241
  show ?thesis
haftmann@25919
  1242
  proof
haftmann@25919
  1243
    assume eq: "1 + a + a = 0"
haftmann@25919
  1244
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1245
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1246
    with odd_nonzero show False by blast
haftmann@25919
  1247
  qed
haftmann@25919
  1248
qed 
haftmann@25919
  1249
haftmann@25919
  1250
lemma Ints_number_of:
haftmann@25919
  1251
  "(number_of w :: 'a::number_ring) \<in> Ints"
haftmann@25919
  1252
  unfolding number_of_eq Ints_def by simp
haftmann@25919
  1253
haftmann@25919
  1254
lemma Ints_odd_less_0: 
haftmann@25919
  1255
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1256
  shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))";
haftmann@25919
  1257
proof -
haftmann@25919
  1258
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1259
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1260
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
  1261
    by (simp add: a)
haftmann@25919
  1262
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
haftmann@25919
  1263
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
  1264
  finally show ?thesis .
haftmann@25919
  1265
qed
haftmann@25919
  1266
haftmann@25919
  1267
haftmann@25919
  1268
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
  1269
haftmann@25919
  1270
text {*By Jeremy Avigad*}
haftmann@25919
  1271
haftmann@25919
  1272
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
  1273
  apply (cases "finite A")
haftmann@25919
  1274
  apply (erule finite_induct, auto)
haftmann@25919
  1275
  done
haftmann@25919
  1276
haftmann@25919
  1277
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
  1278
  apply (cases "finite A")
haftmann@25919
  1279
  apply (erule finite_induct, auto)
haftmann@25919
  1280
  done
haftmann@25919
  1281
haftmann@25919
  1282
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
  1283
  apply (cases "finite A")
haftmann@25919
  1284
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
  1285
  done
haftmann@25919
  1286
haftmann@25919
  1287
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
  1288
  apply (cases "finite A")
haftmann@25919
  1289
  apply (erule finite_induct, auto)
haftmann@25919
  1290
  done
haftmann@25919
  1291
haftmann@25919
  1292
lemma setprod_nonzero_nat:
haftmann@25919
  1293
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
haftmann@25919
  1294
  by (rule setprod_nonzero, auto)
haftmann@25919
  1295
haftmann@25919
  1296
lemma setprod_zero_eq_nat:
haftmann@25919
  1297
    "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
haftmann@25919
  1298
  by (rule setprod_zero_eq, auto)
haftmann@25919
  1299
haftmann@25919
  1300
lemma setprod_nonzero_int:
haftmann@25919
  1301
    "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
haftmann@25919
  1302
  by (rule setprod_nonzero, auto)
haftmann@25919
  1303
haftmann@25919
  1304
lemma setprod_zero_eq_int:
haftmann@25919
  1305
    "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
haftmann@25919
  1306
  by (rule setprod_zero_eq, auto)
haftmann@25919
  1307
haftmann@25919
  1308
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
  1309
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
  1310
haftmann@25919
  1311
haftmann@25919
  1312
subsection{*Inequality Reasoning for the Arithmetic Simproc*}
haftmann@25919
  1313
haftmann@25919
  1314
lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
haftmann@25919
  1315
by simp 
haftmann@25919
  1316
haftmann@25919
  1317
lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
haftmann@25919
  1318
by simp
haftmann@25919
  1319
haftmann@25919
  1320
lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
haftmann@25919
  1321
by simp 
haftmann@25919
  1322
haftmann@25919
  1323
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
haftmann@25919
  1324
by simp
haftmann@25919
  1325
haftmann@25919
  1326
lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
haftmann@25919
  1327
by simp
haftmann@25919
  1328
haftmann@25919
  1329
lemma inverse_numeral_1:
haftmann@25919
  1330
  "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
haftmann@25919
  1331
by simp
haftmann@25919
  1332
haftmann@25919
  1333
text{*Theorem lists for the cancellation simprocs. The use of binary numerals
haftmann@25919
  1334
for 0 and 1 reduces the number of special cases.*}
haftmann@25919
  1335
haftmann@25919
  1336
lemmas add_0s = add_numeral_0 add_numeral_0_right
haftmann@25919
  1337
lemmas mult_1s = mult_numeral_1 mult_numeral_1_right 
haftmann@25919
  1338
                 mult_minus1 mult_minus1_right
haftmann@25919
  1339
haftmann@25919
  1340
haftmann@25919
  1341
subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
haftmann@25919
  1342
haftmann@25919
  1343
text{*Arithmetic computations are defined for binary literals, which leaves 0
haftmann@25919
  1344
and 1 as special cases. Addition already has rules for 0, but not 1.
haftmann@25919
  1345
Multiplication and unary minus already have rules for both 0 and 1.*}
haftmann@25919
  1346
haftmann@25919
  1347
haftmann@25919
  1348
lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
haftmann@25919
  1349
by simp
haftmann@25919
  1350
haftmann@25919
  1351
haftmann@25919
  1352
lemmas add_number_of_eq = number_of_add [symmetric]
haftmann@25919
  1353
haftmann@25919
  1354
text{*Allow 1 on either or both sides*}
haftmann@25919
  1355
lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
haftmann@25919
  1356
by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric] add_number_of_eq)
haftmann@25919
  1357
haftmann@25919
  1358
lemmas add_special =
haftmann@25919
  1359
    one_add_one_is_two
haftmann@25919
  1360
    binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1361
    binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1362
haftmann@25919
  1363
text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
haftmann@25919
  1364
lemmas diff_special =
haftmann@25919
  1365
    binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1366
    binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1367
haftmann@25919
  1368
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1369
lemmas eq_special =
haftmann@25919
  1370
    binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
haftmann@25919
  1371
    binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1372
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
haftmann@25919
  1373
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1374
haftmann@25919
  1375
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1376
lemmas less_special =
haftmann@25919
  1377
  binop_eq [of "op <", OF less_number_of_eq_neg numeral_0_eq_0 refl, standard]
haftmann@25919
  1378
  binop_eq [of "op <", OF less_number_of_eq_neg numeral_1_eq_1 refl, standard]
haftmann@25919
  1379
  binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_0_eq_0, standard]
haftmann@25919
  1380
  binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_1_eq_1, standard]
haftmann@25919
  1381
haftmann@25919
  1382
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1383
lemmas le_special =
haftmann@25919
  1384
    binop_eq [of "op \<le>", OF le_number_of_eq numeral_0_eq_0 refl, standard]
haftmann@25919
  1385
    binop_eq [of "op \<le>", OF le_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1386
    binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_0_eq_0, standard]
haftmann@25919
  1387
    binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1388
haftmann@25919
  1389
lemmas arith_special[simp] = 
haftmann@25919
  1390
       add_special diff_special eq_special less_special le_special
haftmann@25919
  1391
haftmann@25919
  1392
haftmann@25919
  1393
lemma min_max_01: "min (0::int) 1 = 0 & min (1::int) 0 = 0 &
haftmann@25919
  1394
                   max (0::int) 1 = 1 & max (1::int) 0 = 1"
haftmann@25919
  1395
by(simp add:min_def max_def)
haftmann@25919
  1396
haftmann@25919
  1397
lemmas min_max_special[simp] =
haftmann@25919
  1398
 min_max_01
haftmann@25919
  1399
 max_def[of "0::int" "number_of v", standard, simp]
haftmann@25919
  1400
 min_def[of "0::int" "number_of v", standard, simp]
haftmann@25919
  1401
 max_def[of "number_of u" "0::int", standard, simp]
haftmann@25919
  1402
 min_def[of "number_of u" "0::int", standard, simp]
haftmann@25919
  1403
 max_def[of "1::int" "number_of v", standard, simp]
haftmann@25919
  1404
 min_def[of "1::int" "number_of v", standard, simp]
haftmann@25919
  1405
 max_def[of "number_of u" "1::int", standard, simp]
haftmann@25919
  1406
 min_def[of "number_of u" "1::int", standard, simp]
haftmann@25919
  1407
haftmann@25919
  1408
text {* Legacy theorems *}
haftmann@25919
  1409
haftmann@25919
  1410
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
  1411
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@25919
  1412
haftmann@25919
  1413
use "~~/src/Provers/Arith/assoc_fold.ML"
haftmann@25919
  1414
use "int_arith1.ML"
haftmann@25919
  1415
declaration {* K int_arith_setup *}
haftmann@25919
  1416
haftmann@25919
  1417
haftmann@25919
  1418
subsection{*Lemmas About Small Numerals*}
haftmann@25919
  1419
haftmann@25919
  1420
lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
haftmann@25919
  1421
proof -
haftmann@25919
  1422
  have "(of_int -1 :: 'a) = of_int (- 1)" by simp
haftmann@25919
  1423
  also have "... = - of_int 1" by (simp only: of_int_minus)
haftmann@25919
  1424
  also have "... = -1" by simp
haftmann@25919
  1425
  finally show ?thesis .
haftmann@25919
  1426
qed
haftmann@25919
  1427
haftmann@25919
  1428
lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
haftmann@25919
  1429
by (simp add: abs_if)
haftmann@25919
  1430
haftmann@25919
  1431
lemma abs_power_minus_one [simp]:
haftmann@25919
  1432
     "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
haftmann@25919
  1433
by (simp add: power_abs)
haftmann@25919
  1434
haftmann@25919
  1435
lemma of_int_number_of_eq:
haftmann@25919
  1436
     "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
haftmann@25919
  1437
by (simp add: number_of_eq) 
haftmann@25919
  1438
haftmann@25919
  1439
text{*Lemmas for specialist use, NOT as default simprules*}
haftmann@25919
  1440
lemma mult_2: "2 * z = (z+z::'a::number_ring)"
haftmann@25919
  1441
proof -
haftmann@25919
  1442
  have "2*z = (1 + 1)*z" by simp
haftmann@25919
  1443
  also have "... = z+z" by (simp add: left_distrib)
haftmann@25919
  1444
  finally show ?thesis .
haftmann@25919
  1445
qed
haftmann@25919
  1446
haftmann@25919
  1447
lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
haftmann@25919
  1448
by (subst mult_commute, rule mult_2)
haftmann@25919
  1449
haftmann@25919
  1450
haftmann@25919
  1451
subsection{*More Inequality Reasoning*}
haftmann@25919
  1452
haftmann@25919
  1453
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
  1454
by arith
haftmann@25919
  1455
haftmann@25919
  1456
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
  1457
by arith
haftmann@25919
  1458
haftmann@25919
  1459
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
  1460
by arith
haftmann@25919
  1461
haftmann@25919
  1462
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
  1463
by arith
haftmann@25919
  1464
haftmann@25919
  1465
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
  1466
by arith
haftmann@25919
  1467
haftmann@25919
  1468
haftmann@25919
  1469
subsection{*The Functions @{term nat} and @{term int}*}
haftmann@25919
  1470
haftmann@25919
  1471
text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
haftmann@25919
  1472
  @{term "w + - z"}*}
haftmann@25919
  1473
declare Zero_int_def [symmetric, simp]
haftmann@25919
  1474
declare One_int_def [symmetric, simp]
haftmann@25919
  1475
haftmann@25919
  1476
lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
haftmann@25919
  1477
haftmann@25919
  1478
lemma nat_0: "nat 0 = 0"
haftmann@25919
  1479
by (simp add: nat_eq_iff)
haftmann@25919
  1480
haftmann@25919
  1481
lemma nat_1: "nat 1 = Suc 0"
haftmann@25919
  1482
by (subst nat_eq_iff, simp)
haftmann@25919
  1483
haftmann@25919
  1484
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@25919
  1485
by (subst nat_eq_iff, simp)
haftmann@25919
  1486
haftmann@25919
  1487
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
  1488
apply (insert zless_nat_conj [of 1 z])
haftmann@25919
  1489
apply (auto simp add: nat_1)
haftmann@25919
  1490
done
haftmann@25919
  1491
haftmann@25919
  1492
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
  1493
      z is an integer literal.*}
haftmann@25919
  1494
lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
haftmann@25919
  1495
haftmann@25919
  1496
lemma split_nat [arith_split]:
haftmann@25919
  1497
  "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
  1498
  (is "?P = (?L & ?R)")
haftmann@25919
  1499
proof (cases "i < 0")
haftmann@25919
  1500
  case True thus ?thesis by auto
haftmann@25919
  1501
next
haftmann@25919
  1502
  case False
haftmann@25919
  1503
  have "?P = ?L"
haftmann@25919
  1504
  proof
haftmann@25919
  1505
    assume ?P thus ?L using False by clarsimp
haftmann@25919
  1506
  next
haftmann@25919
  1507
    assume ?L thus ?P using False by simp
haftmann@25919
  1508
  qed
haftmann@25919
  1509
  with False show ?thesis by simp
haftmann@25919
  1510
qed
haftmann@25919
  1511
haftmann@25919
  1512
context ring_1
haftmann@25919
  1513
begin
haftmann@25919
  1514
haftmann@25919
  1515
lemma of_int_of_nat:
haftmann@25919
  1516
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1517
proof (cases "k < 0")
haftmann@25919
  1518
  case True then have "0 \<le> - k" by simp
haftmann@25919
  1519
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1520
  with True show ?thesis by simp
haftmann@25919
  1521
next
haftmann@25919
  1522
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
  1523
qed
haftmann@25919
  1524
haftmann@25919
  1525
end
haftmann@25919
  1526
haftmann@25919
  1527
lemma nat_mult_distrib:
haftmann@25919
  1528
  fixes z z' :: int
haftmann@25919
  1529
  assumes "0 \<le> z"
haftmann@25919
  1530
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1531
proof (cases "0 \<le> z'")
haftmann@25919
  1532
  case False with assms have "z * z' \<le> 0"
haftmann@25919
  1533
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1534
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1535
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1536
  ultimately show ?thesis by simp
haftmann@25919
  1537
next
haftmann@25919
  1538
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1539
  show ?thesis
haftmann@25919
  1540
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1541
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1542
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1543
qed
haftmann@25919
  1544
haftmann@25919
  1545
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
  1546
apply (rule trans)
haftmann@25919
  1547
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
  1548
done
haftmann@25919
  1549
haftmann@25919
  1550
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
  1551
apply (cases "z=0 | w=0")
haftmann@25919
  1552
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
  1553
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1554
done
haftmann@25919
  1555
haftmann@25919
  1556
haftmann@25919
  1557
subsection "Induction principles for int"
haftmann@25919
  1558
haftmann@25919
  1559
text{*Well-founded segments of the integers*}
haftmann@25919
  1560
haftmann@25919
  1561
definition
haftmann@25919
  1562
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
  1563
where
haftmann@25919
  1564
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
  1565
haftmann@25919
  1566
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1567
proof -
haftmann@25919
  1568
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
  1569
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
  1570
  thus ?thesis 
haftmann@25919
  1571
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1572
qed
haftmann@25919
  1573
haftmann@25919
  1574
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
  1575
by RankFinder.*}
haftmann@25919
  1576
haftmann@25919
  1577
definition
haftmann@25919
  1578
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
  1579
where
haftmann@25919
  1580
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
  1581
haftmann@25919
  1582
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1583
proof -
haftmann@25919
  1584
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
  1585
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
  1586
  thus ?thesis 
haftmann@25919
  1587
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1588
qed
haftmann@25919
  1589
haftmann@25919
  1590
abbreviation
haftmann@25919
  1591
  int :: "nat \<Rightarrow> int"
haftmann@25919
  1592
where
haftmann@25919
  1593
  "int \<equiv> of_nat"
haftmann@25919
  1594
haftmann@25919
  1595
(* `set:int': dummy construction *)
haftmann@25919
  1596
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1597
  fixes i :: int
haftmann@25919
  1598
  assumes ge: "k \<le> i" and
haftmann@25919
  1599
    base: "P k" and
haftmann@25919
  1600
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1601
  shows "P i"
haftmann@25919
  1602
proof -
haftmann@25919
  1603
  { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
  1604
    proof (induct n)
haftmann@25919
  1605
      case 0
haftmann@25919
  1606
      hence "i = k" by arith
haftmann@25919
  1607
      thus "P i" using base by simp
haftmann@25919
  1608
    next
haftmann@25919
  1609
      case (Suc n)
haftmann@25919
  1610
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
  1611
      moreover
haftmann@25919
  1612
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
  1613
      ultimately
haftmann@25919
  1614
      have "P(i - 1)" by(rule Suc.hyps)
haftmann@25919
  1615
      from step[OF ki1 this] show ?case by simp
haftmann@25919
  1616
    qed
haftmann@25919
  1617
  }
haftmann@25919
  1618
  with ge show ?thesis by fast
haftmann@25919
  1619
qed
haftmann@25919
  1620
haftmann@25928
  1621
(* `set:int': dummy construction *)
haftmann@25928
  1622
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
  1623
  assumes gr: "k < (i::int)" and
haftmann@25919
  1624
        base: "P(k+1)" and
haftmann@25919
  1625
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
  1626
  shows "P i"
haftmann@25919
  1627
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
  1628
  using gr apply arith
haftmann@25919
  1629
 apply(rule base)
haftmann@25919
  1630
apply (rule step, simp+)
haftmann@25919
  1631
done
haftmann@25919
  1632
haftmann@25919
  1633
theorem int_le_induct[consumes 1,case_names base step]:
haftmann@25919
  1634
  assumes le: "i \<le> (k::int)" and
haftmann@25919
  1635
        base: "P(k)" and
haftmann@25919
  1636
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1637
  shows "P i"
haftmann@25919
  1638
proof -
haftmann@25919
  1639
  { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
  1640
    proof (induct n)
haftmann@25919
  1641
      case 0
haftmann@25919
  1642
      hence "i = k" by arith
haftmann@25919
  1643
      thus "P i" using base by simp
haftmann@25919
  1644
    next
haftmann@25919
  1645
      case (Suc n)
haftmann@25919
  1646
      hence "n = nat(k - (i+1))" by arith
haftmann@25919
  1647
      moreover
haftmann@25919
  1648
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
  1649
      ultimately
haftmann@25919
  1650
      have "P(i+1)" by(rule Suc.hyps)
haftmann@25919
  1651
      from step[OF ki1 this] show ?case by simp
haftmann@25919
  1652
    qed
haftmann@25919
  1653
  }
haftmann@25919
  1654
  with le show ?thesis by fast
haftmann@25919
  1655
qed
haftmann@25919
  1656
haftmann@25919
  1657
theorem int_less_induct [consumes 1,case_names base step]:
haftmann@25919
  1658
  assumes less: "(i::int) < k" and
haftmann@25919
  1659
        base: "P(k - 1)" and
haftmann@25919
  1660
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1661
  shows "P i"
haftmann@25919
  1662
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1663
  using less apply arith
haftmann@25919
  1664
 apply(rule base)
haftmann@25919
  1665
apply (rule step, simp+)
haftmann@25919
  1666
done
haftmann@25919
  1667
haftmann@25919
  1668
subsection{*Intermediate value theorems*}
haftmann@25919
  1669
haftmann@25919
  1670
lemma int_val_lemma:
haftmann@25919
  1671
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1672
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
haftmann@25919
  1673
apply (induct_tac "n", simp)
haftmann@25919
  1674
apply (intro strip)
haftmann@25919
  1675
apply (erule impE, simp)
haftmann@25919
  1676
apply (erule_tac x = n in allE, simp)
haftmann@25919
  1677
apply (case_tac "k = f (n+1) ")
haftmann@25919
  1678
 apply force
haftmann@25919
  1679
apply (erule impE)
haftmann@25919
  1680
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1681
apply (blast intro: le_SucI)
haftmann@25919
  1682
done
haftmann@25919
  1683
haftmann@25919
  1684
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1685
haftmann@25919
  1686
lemma nat_intermed_int_val:
haftmann@25919
  1687
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1688
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1689
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1690
       in int_val_lemma)
haftmann@25919
  1691
apply simp
haftmann@25919
  1692
apply (erule exE)
haftmann@25919
  1693
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1694
done
haftmann@25919
  1695
haftmann@25919
  1696
haftmann@25919
  1697
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1698
haftmann@25919
  1699
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1700
by arith
haftmann@25919
  1701
haftmann@25919
  1702
lemma abs_zmult_eq_1: "(\<bar>m * n\<bar> = 1) ==> \<bar>m\<bar> = (1::int)"
haftmann@25919
  1703
apply (cases "\<bar>n\<bar>=1") 
haftmann@25919
  1704
apply (simp add: abs_mult) 
haftmann@25919
  1705
apply (rule ccontr) 
haftmann@25919
  1706
apply (auto simp add: linorder_neq_iff abs_mult) 
haftmann@25919
  1707
apply (subgoal_tac "2 \<le> \<bar>m\<bar> & 2 \<le> \<bar>n\<bar>")
haftmann@25919
  1708
 prefer 2 apply arith 
haftmann@25919
  1709
apply (subgoal_tac "2*2 \<le> \<bar>m\<bar> * \<bar>n\<bar>", simp) 
haftmann@25919
  1710
apply (rule mult_mono, auto) 
haftmann@25919
  1711
done
haftmann@25919
  1712
haftmann@25919
  1713
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1714
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1715
haftmann@25919
  1716
lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)"
haftmann@25919
  1717
apply (auto dest: pos_zmult_eq_1_iff_lemma) 
haftmann@25919
  1718
apply (simp add: mult_commute [of m]) 
haftmann@25919
  1719
apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1720
done
haftmann@25919
  1721
haftmann@25919
  1722
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1723
apply (rule iffI) 
haftmann@25919
  1724
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919
  1725
 apply (simp add: mult_commute [of m]) 
haftmann@25919
  1726
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1727
done
haftmann@25919
  1728
haftmann@25919
  1729
(* Could be simplified but Presburger only becomes available too late *)
haftmann@25919
  1730
lemma infinite_UNIV_int: "~finite(UNIV::int set)"
haftmann@25919
  1731
proof
haftmann@25919
  1732
  assume "finite(UNIV::int set)"
haftmann@25919
  1733
  moreover have "~(EX i::int. 2*i = 1)"
haftmann@25919
  1734
    by (auto simp: pos_zmult_eq_1_iff)
haftmann@25919
  1735
  ultimately show False using finite_UNIV_inj_surj[of "%n::int. n+n"]
haftmann@25919
  1736
    by (simp add:inj_on_def surj_def) (blast intro:sym)
haftmann@25919
  1737
qed
haftmann@25919
  1738
haftmann@25919
  1739
haftmann@25961
  1740
subsection{*Integer Powers*} 
haftmann@25961
  1741
haftmann@25961
  1742
instantiation int :: recpower
haftmann@25961
  1743
begin
haftmann@25961
  1744
haftmann@25961
  1745
primrec power_int where
haftmann@25961
  1746
  "p ^ 0 = (1\<Colon>int)"
haftmann@25961
  1747
  | "p ^ (Suc n) = (p\<Colon>int) * (p ^ n)"
haftmann@25961
  1748
haftmann@25961
  1749
instance proof
haftmann@25961
  1750
  fix z :: int
haftmann@25961
  1751
  fix n :: nat
haftmann@25961
  1752
  show "z ^ 0 = 1" by simp
haftmann@25961
  1753
  show "z ^ Suc n = z * (z ^ n)" by simp
haftmann@25961
  1754
qed
haftmann@25961
  1755
haftmann@25961
  1756
end
haftmann@25961
  1757
haftmann@25961
  1758
lemma zpower_zadd_distrib: "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
haftmann@25961
  1759
  by (rule Power.power_add)
haftmann@25961
  1760
haftmann@25961
  1761
lemma zpower_zpower: "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@25961
  1762
  by (rule Power.power_mult [symmetric])
haftmann@25961
  1763
haftmann@25961
  1764
lemma zero_less_zpower_abs_iff [simp]:
haftmann@25961
  1765
  "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
haftmann@25961
  1766
  by (induct n) (auto simp add: zero_less_mult_iff)
haftmann@25961
  1767
haftmann@25961
  1768
lemma zero_le_zpower_abs [simp]: "(0::int) \<le> abs x ^ n"
haftmann@25961
  1769
  by (induct n) (auto simp add: zero_le_mult_iff)
haftmann@25961
  1770
haftmann@25961
  1771
lemma of_int_power:
haftmann@25961
  1772
  "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})"
haftmann@25961
  1773
  by (induct n) (simp_all add: power_Suc)
haftmann@25961
  1774
haftmann@25961
  1775
lemma int_power: "int (m^n) = (int m) ^ n"
haftmann@25961
  1776
  by (rule of_nat_power)
haftmann@25961
  1777
haftmann@25961
  1778
lemmas zpower_int = int_power [symmetric]
haftmann@25961
  1779
haftmann@25919
  1780
subsection {* Configuration of the code generator *}
haftmann@25919
  1781
haftmann@25919
  1782
instance int :: eq ..
haftmann@25919
  1783
haftmann@25919
  1784
code_datatype Pls Min Bit "number_of \<Colon> int \<Rightarrow> int"
haftmann@25919
  1785
haftmann@25919
  1786
definition
haftmann@25919
  1787
  int_aux :: "nat \<Rightarrow> int \<Rightarrow> int" where
haftmann@25928
  1788
  [code func del]: "int_aux = of_nat_aux"
haftmann@25919
  1789
haftmann@25928
  1790
lemmas int_aux_code = of_nat_aux_code [where ?'a = int, simplified int_aux_def [symmetric], code]
haftmann@25919
  1791
haftmann@25919
  1792
lemma [code, code unfold, code inline del]:
haftmann@25928
  1793
  "of_nat n = int_aux n 0"
haftmann@25928
  1794
  by (simp add: int_aux_def of_nat_aux_def)
haftmann@25919
  1795
haftmann@25919
  1796
definition
haftmann@25919
  1797
  nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
haftmann@25919
  1798
  "nat_aux i n = nat i + n"
haftmann@25919
  1799
haftmann@25919
  1800
lemma [code]:
haftmann@25919
  1801
  "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
haftmann@25919
  1802
  by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
haftmann@25919
  1803
    dest: zless_imp_add1_zle)
haftmann@25919
  1804
haftmann@25919
  1805
lemma [code]: "nat i = nat_aux i 0"
haftmann@25919
  1806
  by (simp add: nat_aux_def)
haftmann@25919
  1807
haftmann@25928
  1808
hide (open) const int_aux nat_aux
haftmann@25928
  1809
haftmann@25919
  1810
lemma zero_is_num_zero [code func, code inline, symmetric, code post]:
haftmann@25919
  1811
  "(0\<Colon>int) = Numeral0" 
haftmann@25919
  1812
  by simp
haftmann@25919
  1813
haftmann@25919
  1814
lemma one_is_num_one [code func, code inline, symmetric, code post]:
haftmann@25919
  1815
  "(1\<Colon>int) = Numeral1" 
haftmann@25961
  1816
  by simp
haftmann@25919
  1817
haftmann@25919
  1818
code_modulename SML
haftmann@25928
  1819
  Int Integer
haftmann@25919
  1820
haftmann@25919
  1821
code_modulename OCaml
haftmann@25928
  1822
  Int Integer
haftmann@25919
  1823
haftmann@25919
  1824
code_modulename Haskell
haftmann@25928
  1825
  Int Integer
haftmann@25919
  1826
haftmann@25919
  1827
types_code
haftmann@25919
  1828
  "int" ("int")
haftmann@25919
  1829
attach (term_of) {*
haftmann@25919
  1830
val term_of_int = HOLogic.mk_number HOLogic.intT;
haftmann@25919
  1831
*}
haftmann@25919
  1832
attach (test) {*
haftmann@25919
  1833
fun gen_int i =
haftmann@25919
  1834
  let val j = one_of [~1, 1] * random_range 0 i
haftmann@25919
  1835
  in (j, fn () => term_of_int j) end;
haftmann@25919
  1836
*}
haftmann@25919
  1837
haftmann@25919
  1838
setup {*
haftmann@25919
  1839
let
haftmann@25919
  1840
haftmann@25919
  1841
fun strip_number_of (@{term "Int.number_of :: int => int"} $ t) = t
haftmann@25919
  1842
  | strip_number_of t = t;
haftmann@25919
  1843
haftmann@25919
  1844
fun numeral_codegen thy defs gr dep module b t =
haftmann@25919
  1845
  let val i = HOLogic.dest_numeral (strip_number_of t)
haftmann@25919
  1846
  in
haftmann@25919
  1847
    SOME (fst (Codegen.invoke_tycodegen thy defs dep module false (gr, HOLogic.intT)),
haftmann@25919
  1848
      Pretty.str (string_of_int i))
haftmann@25919
  1849
  end handle TERM _ => NONE;
haftmann@25919
  1850
haftmann@25919
  1851
in
haftmann@25919
  1852
haftmann@25919
  1853
Codegen.add_codegen "numeral_codegen" numeral_codegen
haftmann@25919
  1854
haftmann@25919
  1855
end
haftmann@25919
  1856
*}
haftmann@25919
  1857
haftmann@25919
  1858
consts_code
haftmann@25919
  1859
  "number_of :: int \<Rightarrow> int"    ("(_)")
haftmann@25919
  1860
  "0 :: int"                   ("0")
haftmann@25919
  1861
  "1 :: int"                   ("1")
haftmann@25919
  1862
  "uminus :: int => int"       ("~")
haftmann@25919
  1863
  "op + :: int => int => int"  ("(_ +/ _)")
haftmann@25919
  1864
  "op * :: int => int => int"  ("(_ */ _)")
haftmann@25919
  1865
  "op \<le> :: int => int => bool" ("(_ <=/ _)")
haftmann@25919
  1866
  "op < :: int => int => bool" ("(_ </ _)")
haftmann@25919
  1867
haftmann@25919
  1868
quickcheck_params [default_type = int]
haftmann@25919
  1869
haftmann@25919
  1870
(*setup continues in theory Presburger*)
haftmann@25919
  1871
haftmann@25919
  1872
hide (open) const Pls Min B0 B1 succ pred
haftmann@25919
  1873
haftmann@25919
  1874
haftmann@25919
  1875
subsection {* Legacy theorems *}
haftmann@25919
  1876
haftmann@25919
  1877
lemmas zminus_zminus = minus_minus [of "z::int", standard]
haftmann@25919
  1878
lemmas zminus_0 = minus_zero [where 'a=int]
haftmann@25919
  1879
lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
haftmann@25919
  1880
lemmas zadd_commute = add_commute [of "z::int" "w", standard]
haftmann@25919
  1881
lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919
  1882
lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]
haftmann@25919
  1883
lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
haftmann@25919
  1884
lemmas zmult_ac = OrderedGroup.mult_ac
haftmann@25919
  1885
lemmas zadd_0 = OrderedGroup.add_0_left [of "z::int", standard]
haftmann@25919
  1886
lemmas zadd_0_right = OrderedGroup.add_0_left [of "z::int", standard]
haftmann@25919
  1887
lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]
haftmann@25919
  1888
lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]
haftmann@25919
  1889
lemmas zmult_commute = mult_commute [of "z::int" "w", standard]
haftmann@25919
  1890
lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919
  1891
lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919
  1892
lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919
  1893
lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919
  1894
lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919
  1895
haftmann@25919
  1896
lemmas int_distrib =
haftmann@25919
  1897
  zadd_zmult_distrib zadd_zmult_distrib2
haftmann@25919
  1898
  zdiff_zmult_distrib zdiff_zmult_distrib2
haftmann@25919
  1899
haftmann@25919
  1900
lemmas zmult_1 = mult_1_left [of "z::int", standard]
haftmann@25919
  1901
lemmas zmult_1_right = mult_1_right [of "z::int", standard]
haftmann@25919
  1902
haftmann@25919
  1903
lemmas zle_refl = order_refl [of "w::int", standard]
haftmann@25919
  1904
lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]
haftmann@25919
  1905
lemmas zle_anti_sym = order_antisym [of "z::int" "w", standard]
haftmann@25919
  1906
lemmas zle_linear = linorder_linear [of "z::int" "w", standard]
haftmann@25919
  1907
lemmas zless_linear = linorder_less_linear [where 'a = int]
haftmann@25919
  1908
haftmann@25919
  1909
lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]
haftmann@25919
  1910
lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]
haftmann@25919
  1911
lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]
haftmann@25919
  1912
haftmann@25919
  1913
lemmas int_0_less_1 = zero_less_one [where 'a=int]
haftmann@25919
  1914
lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
haftmann@25919
  1915
haftmann@25919
  1916
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  1917
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  1918
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  1919
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
haftmann@25919
  1920
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  1921
lemmas zless_int = of_nat_less_iff [where 'a=int]
haftmann@25919
  1922
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
haftmann@25919
  1923
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  1924
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
haftmann@25919
  1925
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  1926
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  1927
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  1928
lemmas int_Suc = of_nat_Suc [where 'a=int]
haftmann@25919
  1929
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
haftmann@25919
  1930
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  1931
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
haftmann@25919
  1932
lemmas zless_le = less_int_def
haftmann@25919
  1933
lemmas int_eq_of_nat = TrueI
haftmann@25919
  1934
haftmann@25919
  1935
end