src/HOL/Int.thy
author bulwahn
Fri Oct 21 11:17:14 2011 +0200 (2011-10-21)
changeset 45231 d85a2fdc586c
parent 45219 29f6e990674d
child 45532 74b17a0881b3
permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
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(*  Title:      HOL/Int.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
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*)
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header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} 
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theory Int
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imports Equiv_Relations Nat Wellfounded
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uses
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  ("Tools/numeral.ML")
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  ("Tools/numeral_syntax.ML")
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  ("Tools/int_arith.ML")
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begin
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subsection {* The equivalence relation underlying the integers *}
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definition intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" where
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  "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: "0 = Abs_Integ (intrel `` {(0, 0)})"
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definition
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  One_int_def: "1 = Abs_Integ (intrel `` {(1, 0)})"
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definition
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  add_int_def: "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:
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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:  "z - w = z + (-w \<Colon> int)"
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definition
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  mult_int_def: "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:
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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: "(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_on_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,no_atp]  Abs_Integ_inverse [simp,no_atp]
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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 (auto 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 (auto simp add: congruent2_def)
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  thus ?thesis
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    by (simp add: add_int_def UN_UN_split_split_eq
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                  UN_equiv_class2 [OF equiv_intrel equiv_intrel])
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qed
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text{*Congruence property for multiplication*}
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lemma mult_congruent2:
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     "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
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      respects2 intrel"
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apply (rule equiv_intrel [THEN congruent2_commuteI])
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 apply (force simp add: mult_ac, clarify) 
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apply (simp add: congruent_def mult_ac)  
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apply (rename_tac u v w x y z)
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apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
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apply (simp add: mult_ac)
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apply (simp add: add_mult_distrib [symmetric])
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done
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lemma mult:
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     "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
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      Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
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by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
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              UN_equiv_class2 [OF equiv_intrel equiv_intrel])
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text{*The integers form a @{text comm_ring_1}*}
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instance int :: comm_ring_1
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proof
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  fix i j k :: int
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  show "(i + j) + k = i + (j + k)"
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    by (cases i, cases j, cases k) (simp add: add add_assoc)
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  show "i + j = j + i" 
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    by (cases i, cases j) (simp add: add_ac add)
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  show "0 + i = i"
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    by (cases i) (simp add: Zero_int_def add)
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  show "- i + i = 0"
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    by (cases i) (simp add: Zero_int_def minus add)
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  show "i - j = i + - j"
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    by (simp add: diff_int_def)
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  show "(i * j) * k = i * (j * k)"
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    by (cases i, cases j, cases k) (simp add: mult algebra_simps)
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  show "i * j = j * i"
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    by (cases i, cases j) (simp add: mult algebra_simps)
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  show "1 * i = i"
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    by (cases i) (simp add: One_int_def mult)
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  show "(i + j) * k = i * k + j * k"
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    by (cases i, cases j, cases k) (simp add: add mult algebra_simps)
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  show "0 \<noteq> (1::int)"
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    by (simp add: Zero_int_def One_int_def)
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qed
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abbreviation int :: "nat \<Rightarrow> int" where
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  "int \<equiv> of_nat"
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lemma int_def: "int m = Abs_Integ (intrel `` {(m, 0)})"
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by (induct m) (simp_all add: Zero_int_def One_int_def add)
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subsection {* The @{text "\<le>"} Ordering *}
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lemma le:
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  "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
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by (force simp add: le_int_def)
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lemma less:
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  "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
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by (simp add: less_int_def le order_less_le)
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instance int :: linorder
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proof
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  fix i j k :: int
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  show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
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    by (cases i, cases j) (simp add: le)
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  show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
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    by (auto simp add: less_int_def dest: antisym) 
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  show "i \<le> i"
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    by (cases i) (simp add: le)
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  show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
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    by (cases i, cases j, cases k) (simp add: le)
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  show "i \<le> j \<or> j \<le> i"
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    by (cases i, cases j) (simp add: le linorder_linear)
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qed
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instantiation int :: distrib_lattice
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begin
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definition
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  "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by intro_classes
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    (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
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end
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instance int :: ordered_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 ==> int k * i < int k * j"
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apply (induct k)
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apply 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 = int 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 = int 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 :: linordered_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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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 + int (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 of_int :: "int \<Rightarrow> 'a" where
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  "of_int z = the_elem (\<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 (auto simp add: congruent_def) (simp add: algebra_simps of_nat_add [symmetric]
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            del: of_nat_add) 
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  thus ?thesis
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    by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
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qed
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lemma of_int_0 [simp]: "of_int 0 = 0"
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by (simp add: of_int Zero_int_def)
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lemma of_int_1 [simp]: "of_int 1 = 1"
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by (simp add: of_int One_int_def)
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lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
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by (cases w, cases z) (simp add: algebra_simps of_int add)
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lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
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by (cases z) (simp add: algebra_simps of_int minus)
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lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
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by (simp add: diff_minus Groups.diff_minus)
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lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
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apply (cases w, cases z)
nipkow@29667
   316
apply (simp add: algebra_simps of_int mult of_nat_mult)
haftmann@25919
   317
done
haftmann@25919
   318
haftmann@25919
   319
text{*Collapse nested embeddings*}
huffman@44709
   320
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
nipkow@29667
   321
by (induct n) auto
haftmann@25919
   322
haftmann@31015
   323
lemma of_int_power:
haftmann@31015
   324
  "of_int (z ^ n) = of_int z ^ n"
haftmann@31015
   325
  by (induct n) simp_all
haftmann@31015
   326
haftmann@25919
   327
end
haftmann@25919
   328
haftmann@25919
   329
text{*Class for unital rings with characteristic zero.
haftmann@25919
   330
 Includes non-ordered rings like the complex numbers.*}
haftmann@25919
   331
class ring_char_0 = ring_1 + semiring_char_0
haftmann@25919
   332
begin
haftmann@25919
   333
haftmann@25919
   334
lemma of_int_eq_iff [simp]:
haftmann@25919
   335
   "of_int w = of_int z \<longleftrightarrow> w = z"
wenzelm@42676
   336
apply (cases w, cases z)
wenzelm@42676
   337
apply (simp add: of_int)
haftmann@25919
   338
apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
haftmann@25919
   339
apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
haftmann@25919
   340
done
haftmann@25919
   341
haftmann@25919
   342
text{*Special cases where either operand is zero*}
haftmann@36424
   343
lemma of_int_eq_0_iff [simp]:
haftmann@36424
   344
  "of_int z = 0 \<longleftrightarrow> z = 0"
haftmann@36424
   345
  using of_int_eq_iff [of z 0] by simp
haftmann@36424
   346
haftmann@36424
   347
lemma of_int_0_eq_iff [simp]:
haftmann@36424
   348
  "0 = of_int z \<longleftrightarrow> z = 0"
haftmann@36424
   349
  using of_int_eq_iff [of 0 z] by simp
haftmann@25919
   350
haftmann@25919
   351
end
haftmann@25919
   352
haftmann@36424
   353
context linordered_idom
haftmann@36424
   354
begin
haftmann@36424
   355
haftmann@35028
   356
text{*Every @{text linordered_idom} has characteristic zero.*}
haftmann@36424
   357
subclass ring_char_0 ..
haftmann@36424
   358
haftmann@36424
   359
lemma of_int_le_iff [simp]:
haftmann@36424
   360
  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
wenzelm@42676
   361
  by (cases w, cases z)
wenzelm@42676
   362
    (simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add)
haftmann@36424
   363
haftmann@36424
   364
lemma of_int_less_iff [simp]:
haftmann@36424
   365
  "of_int w < of_int z \<longleftrightarrow> w < z"
haftmann@36424
   366
  by (simp add: less_le order_less_le)
haftmann@36424
   367
haftmann@36424
   368
lemma of_int_0_le_iff [simp]:
haftmann@36424
   369
  "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
haftmann@36424
   370
  using of_int_le_iff [of 0 z] by simp
haftmann@36424
   371
haftmann@36424
   372
lemma of_int_le_0_iff [simp]:
haftmann@36424
   373
  "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
haftmann@36424
   374
  using of_int_le_iff [of z 0] by simp
haftmann@36424
   375
haftmann@36424
   376
lemma of_int_0_less_iff [simp]:
haftmann@36424
   377
  "0 < of_int z \<longleftrightarrow> 0 < z"
haftmann@36424
   378
  using of_int_less_iff [of 0 z] by simp
haftmann@36424
   379
haftmann@36424
   380
lemma of_int_less_0_iff [simp]:
haftmann@36424
   381
  "of_int z < 0 \<longleftrightarrow> z < 0"
haftmann@36424
   382
  using of_int_less_iff [of z 0] by simp
haftmann@36424
   383
haftmann@36424
   384
end
haftmann@25919
   385
haftmann@25919
   386
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   387
proof
haftmann@25919
   388
  fix z show "of_int z = id z"
haftmann@25919
   389
    by (cases z) (simp add: of_int add minus int_def diff_minus)
haftmann@25919
   390
qed
haftmann@25919
   391
haftmann@25919
   392
haftmann@25919
   393
subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
haftmann@25919
   394
haftmann@37767
   395
definition nat :: "int \<Rightarrow> nat" where
haftmann@39910
   396
  "nat z = the_elem (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
haftmann@25919
   397
haftmann@25919
   398
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
haftmann@25919
   399
proof -
haftmann@25919
   400
  have "(\<lambda>(x,y). {x-y}) respects intrel"
haftmann@40819
   401
    by (auto simp add: congruent_def)
haftmann@25919
   402
  thus ?thesis
haftmann@25919
   403
    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
haftmann@25919
   404
qed
haftmann@25919
   405
huffman@44709
   406
lemma nat_int [simp]: "nat (int n) = n"
haftmann@25919
   407
by (simp add: nat int_def)
haftmann@25919
   408
huffman@44709
   409
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
wenzelm@42676
   410
by (cases z) (simp add: nat le int_def Zero_int_def)
haftmann@25919
   411
huffman@44709
   412
corollary nat_0_le: "0 \<le> z ==> int (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"
wenzelm@42676
   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
huffman@44709
   437
  assumes "0 \<le> z" and "\<And>m. z = int m \<Longrightarrow> P"
haftmann@25919
   438
  shows P
haftmann@25919
   439
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   440
huffman@44709
   441
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
wenzelm@42676
   442
by (cases w) (simp add: nat le int_def Zero_int_def, arith)
haftmann@25919
   443
huffman@44709
   444
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int 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)
nipkow@29700
   449
apply (simp add: nat le int_def Zero_int_def linorder_not_le[symmetric], arith)
haftmann@25919
   450
done
haftmann@25919
   451
huffman@44709
   452
lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
huffman@44707
   453
  by (cases x, simp add: nat le int_def le_diff_conv)
huffman@44707
   454
huffman@44707
   455
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
huffman@44707
   456
  by (cases x, cases y, simp add: nat le)
huffman@44707
   457
nipkow@29700
   458
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
nipkow@29700
   459
by(simp add: nat_eq_iff) arith
nipkow@29700
   460
haftmann@25919
   461
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   462
by (auto simp add: nat_eq_iff2)
haftmann@25919
   463
haftmann@25919
   464
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   465
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   466
haftmann@25919
   467
lemma nat_add_distrib:
haftmann@25919
   468
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
wenzelm@42676
   469
by (cases z, cases z') (simp add: nat add le Zero_int_def)
haftmann@25919
   470
haftmann@25919
   471
lemma nat_diff_distrib:
haftmann@25919
   472
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
wenzelm@42676
   473
by (cases z, cases z')
wenzelm@42676
   474
  (simp add: nat add minus diff_minus le Zero_int_def)
haftmann@25919
   475
huffman@44709
   476
lemma nat_zminus_int [simp]: "nat (- int n) = 0"
haftmann@25919
   477
by (simp add: int_def minus nat Zero_int_def) 
haftmann@25919
   478
huffman@44709
   479
lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
wenzelm@42676
   480
by (cases z) (simp add: nat less int_def, arith)
haftmann@25919
   481
haftmann@25919
   482
context ring_1
haftmann@25919
   483
begin
haftmann@25919
   484
haftmann@25919
   485
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
haftmann@25919
   486
  by (cases z rule: eq_Abs_Integ)
haftmann@25919
   487
   (simp add: nat le of_int Zero_int_def of_nat_diff)
haftmann@25919
   488
haftmann@25919
   489
end
haftmann@25919
   490
krauss@29779
   491
text {* For termination proofs: *}
krauss@29779
   492
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779
   493
haftmann@25919
   494
haftmann@25919
   495
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   496
huffman@44709
   497
lemma negative_zless_0: "- (int (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   498
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   499
huffman@44709
   500
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
haftmann@25919
   501
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   502
huffman@44709
   503
lemma negative_zle_0: "- int n \<le> 0"
haftmann@25919
   504
by (simp add: minus_le_iff)
haftmann@25919
   505
huffman@44709
   506
lemma negative_zle [iff]: "- int n \<le> int m"
haftmann@25919
   507
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   508
huffman@44709
   509
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
haftmann@25919
   510
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   511
huffman@44709
   512
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
haftmann@25919
   513
by (simp add: int_def le minus Zero_int_def)
haftmann@25919
   514
huffman@44709
   515
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
haftmann@25919
   516
by (simp add: linorder_not_less)
haftmann@25919
   517
huffman@44709
   518
lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   519
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   520
huffman@44709
   521
lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
haftmann@25919
   522
proof -
haftmann@25919
   523
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   524
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   525
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   526
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   527
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   528
    by (simp only: algebra_simps)
haftmann@25919
   529
  finally show ?thesis .
haftmann@25919
   530
qed
haftmann@25919
   531
huffman@44709
   532
lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
haftmann@25919
   533
by simp
haftmann@25919
   534
huffman@44709
   535
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
haftmann@25919
   536
by simp
haftmann@25919
   537
haftmann@25919
   538
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   539
      It is proved here because attribute @{text arith_split} is not available
haftmann@35050
   540
      in theory @{text Rings}.
haftmann@25919
   541
      But is it really better than just rewriting with @{text abs_if}?*}
blanchet@35828
   542
lemma abs_split [arith_split,no_atp]:
haftmann@35028
   543
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   544
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   545
huffman@44709
   546
lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
haftmann@25919
   547
apply (cases x)
haftmann@25919
   548
apply (auto simp add: le minus Zero_int_def int_def order_less_le)
haftmann@25919
   549
apply (rule_tac x="y - Suc x" in exI, arith)
haftmann@25919
   550
done
haftmann@25919
   551
haftmann@25919
   552
haftmann@25919
   553
subsection {* Cases and induction *}
haftmann@25919
   554
haftmann@25919
   555
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   556
whether an integer is negative or not.*}
haftmann@25919
   557
wenzelm@42676
   558
theorem int_cases [case_names nonneg neg, cases type: int]:
huffman@44709
   559
  "[|!! n. z = int n ==> P;  !! n. z =  - (int (Suc n)) ==> P |] ==> P"
wenzelm@42676
   560
apply (cases "z < 0")
wenzelm@42676
   561
apply (blast dest!: negD)
haftmann@25919
   562
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   563
apply auto
haftmann@25919
   564
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   565
done
haftmann@25919
   566
wenzelm@42676
   567
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
huffman@44709
   568
     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
wenzelm@42676
   569
  by (cases z) auto
haftmann@25919
   570
haftmann@25919
   571
text{*Contributed by Brian Huffman*}
haftmann@25919
   572
theorem int_diff_cases:
huffman@44709
   573
  obtains (diff) m n where "z = int m - int n"
haftmann@25919
   574
apply (cases z rule: eq_Abs_Integ)
haftmann@25919
   575
apply (rule_tac m=x and n=y in diff)
haftmann@37887
   576
apply (simp add: int_def minus add diff_minus)
haftmann@25919
   577
done
haftmann@25919
   578
haftmann@25919
   579
haftmann@25919
   580
subsection {* Binary representation *}
haftmann@25919
   581
haftmann@25919
   582
text {*
haftmann@25919
   583
  This formalization defines binary arithmetic in terms of the integers
haftmann@25919
   584
  rather than using a datatype. This avoids multiple representations (leading
haftmann@25919
   585
  zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
haftmann@25919
   586
  int_of_binary}, for the numerical interpretation.
haftmann@25919
   587
haftmann@25919
   588
  The representation expects that @{text "(m mod 2)"} is 0 or 1,
haftmann@25919
   589
  even if m is negative;
haftmann@25919
   590
  For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
haftmann@25919
   591
  @{text "-5 = (-3)*2 + 1"}.
haftmann@25919
   592
  
haftmann@25919
   593
  This two's complement binary representation derives from the paper 
haftmann@25919
   594
  "An Efficient Representation of Arithmetic for Term Rewriting" by
haftmann@25919
   595
  Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
haftmann@25919
   596
  Springer LNCS 488 (240-251), 1991.
haftmann@25919
   597
*}
haftmann@25919
   598
huffman@28958
   599
subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}
huffman@28958
   600
haftmann@37767
   601
definition Pls :: int where
haftmann@37767
   602
  "Pls = 0"
haftmann@37767
   603
haftmann@37767
   604
definition Min :: int where
haftmann@37767
   605
  "Min = - 1"
haftmann@37767
   606
haftmann@37767
   607
definition Bit0 :: "int \<Rightarrow> int" where
haftmann@37767
   608
  "Bit0 k = k + k"
haftmann@37767
   609
haftmann@37767
   610
definition Bit1 :: "int \<Rightarrow> int" where
haftmann@37767
   611
  "Bit1 k = 1 + k + k"
haftmann@25919
   612
haftmann@29608
   613
class number = -- {* for numeric types: nat, int, real, \dots *}
haftmann@25919
   614
  fixes number_of :: "int \<Rightarrow> 'a"
haftmann@25919
   615
haftmann@25919
   616
use "Tools/numeral.ML"
haftmann@25919
   617
haftmann@25919
   618
syntax
haftmann@25919
   619
  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
haftmann@25919
   620
haftmann@25919
   621
use "Tools/numeral_syntax.ML"
wenzelm@35123
   622
setup Numeral_Syntax.setup
haftmann@25919
   623
haftmann@25919
   624
abbreviation
haftmann@25919
   625
  "Numeral0 \<equiv> number_of Pls"
haftmann@25919
   626
haftmann@25919
   627
abbreviation
huffman@26086
   628
  "Numeral1 \<equiv> number_of (Bit1 Pls)"
haftmann@25919
   629
haftmann@25919
   630
lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
haftmann@25919
   631
  -- {* Unfold all @{text let}s involving constants *}
haftmann@25919
   632
  unfolding Let_def ..
haftmann@25919
   633
haftmann@37767
   634
definition succ :: "int \<Rightarrow> int" where
haftmann@37767
   635
  "succ k = k + 1"
haftmann@37767
   636
haftmann@37767
   637
definition pred :: "int \<Rightarrow> int" where
haftmann@37767
   638
  "pred k = k - 1"
haftmann@25919
   639
haftmann@25919
   640
lemmas
haftmann@25919
   641
  max_number_of [simp] = max_def
huffman@35216
   642
    [of "number_of u" "number_of v", standard]
haftmann@25919
   643
and
haftmann@25919
   644
  min_number_of [simp] = min_def 
huffman@35216
   645
    [of "number_of u" "number_of v", standard]
haftmann@25919
   646
  -- {* unfolding @{text minx} and @{text max} on numerals *}
haftmann@25919
   647
haftmann@25919
   648
lemmas numeral_simps = 
huffman@26086
   649
  succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
haftmann@25919
   650
haftmann@25919
   651
text {* Removal of leading zeroes *}
haftmann@25919
   652
haftmann@31998
   653
lemma Bit0_Pls [simp, code_post]:
huffman@26086
   654
  "Bit0 Pls = Pls"
haftmann@25919
   655
  unfolding numeral_simps by simp
haftmann@25919
   656
haftmann@31998
   657
lemma Bit1_Min [simp, code_post]:
huffman@26086
   658
  "Bit1 Min = Min"
haftmann@25919
   659
  unfolding numeral_simps by simp
haftmann@25919
   660
huffman@26075
   661
lemmas normalize_bin_simps =
huffman@26086
   662
  Bit0_Pls Bit1_Min
huffman@26075
   663
haftmann@25919
   664
huffman@28958
   665
subsubsection {* Successor and predecessor functions *}
huffman@28958
   666
huffman@28958
   667
text {* Successor *}
huffman@28958
   668
huffman@28958
   669
lemma succ_Pls:
huffman@26086
   670
  "succ Pls = Bit1 Pls"
haftmann@25919
   671
  unfolding numeral_simps by simp
haftmann@25919
   672
huffman@28958
   673
lemma succ_Min:
haftmann@25919
   674
  "succ Min = Pls"
haftmann@25919
   675
  unfolding numeral_simps by simp
haftmann@25919
   676
huffman@28958
   677
lemma succ_Bit0:
huffman@26086
   678
  "succ (Bit0 k) = Bit1 k"
haftmann@25919
   679
  unfolding numeral_simps by simp
haftmann@25919
   680
huffman@28958
   681
lemma succ_Bit1:
huffman@26086
   682
  "succ (Bit1 k) = Bit0 (succ k)"
haftmann@25919
   683
  unfolding numeral_simps by simp
haftmann@25919
   684
huffman@28958
   685
lemmas succ_bin_simps [simp] =
huffman@26086
   686
  succ_Pls succ_Min succ_Bit0 succ_Bit1
huffman@26075
   687
huffman@28958
   688
text {* Predecessor *}
huffman@28958
   689
huffman@28958
   690
lemma pred_Pls:
haftmann@25919
   691
  "pred Pls = Min"
haftmann@25919
   692
  unfolding numeral_simps by simp
haftmann@25919
   693
huffman@28958
   694
lemma pred_Min:
huffman@26086
   695
  "pred Min = Bit0 Min"
haftmann@25919
   696
  unfolding numeral_simps by simp
haftmann@25919
   697
huffman@28958
   698
lemma pred_Bit0:
huffman@26086
   699
  "pred (Bit0 k) = Bit1 (pred k)"
haftmann@25919
   700
  unfolding numeral_simps by simp 
haftmann@25919
   701
huffman@28958
   702
lemma pred_Bit1:
huffman@26086
   703
  "pred (Bit1 k) = Bit0 k"
huffman@26086
   704
  unfolding numeral_simps by simp
huffman@26086
   705
huffman@28958
   706
lemmas pred_bin_simps [simp] =
huffman@26086
   707
  pred_Pls pred_Min pred_Bit0 pred_Bit1
huffman@26075
   708
huffman@28958
   709
huffman@28958
   710
subsubsection {* Binary arithmetic *}
huffman@28958
   711
huffman@28958
   712
text {* Addition *}
huffman@28958
   713
huffman@28958
   714
lemma add_Pls:
huffman@28958
   715
  "Pls + k = k"
huffman@28958
   716
  unfolding numeral_simps by simp
huffman@28958
   717
huffman@28958
   718
lemma add_Min:
huffman@28958
   719
  "Min + k = pred k"
huffman@28958
   720
  unfolding numeral_simps by simp
huffman@28958
   721
huffman@28958
   722
lemma add_Bit0_Bit0:
huffman@28958
   723
  "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"
huffman@28958
   724
  unfolding numeral_simps by simp
huffman@28958
   725
huffman@28958
   726
lemma add_Bit0_Bit1:
huffman@28958
   727
  "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"
huffman@28958
   728
  unfolding numeral_simps by simp
huffman@28958
   729
huffman@28958
   730
lemma add_Bit1_Bit0:
huffman@28958
   731
  "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"
huffman@28958
   732
  unfolding numeral_simps by simp
huffman@28958
   733
huffman@28958
   734
lemma add_Bit1_Bit1:
huffman@28958
   735
  "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"
huffman@28958
   736
  unfolding numeral_simps by simp
huffman@28958
   737
huffman@28958
   738
lemma add_Pls_right:
huffman@28958
   739
  "k + Pls = k"
huffman@28958
   740
  unfolding numeral_simps by simp
huffman@28958
   741
huffman@28958
   742
lemma add_Min_right:
huffman@28958
   743
  "k + Min = pred k"
huffman@28958
   744
  unfolding numeral_simps by simp
huffman@28958
   745
huffman@28958
   746
lemmas add_bin_simps [simp] =
huffman@28958
   747
  add_Pls add_Min add_Pls_right add_Min_right
huffman@28958
   748
  add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1
huffman@28958
   749
huffman@28958
   750
text {* Negation *}
huffman@28958
   751
huffman@28958
   752
lemma minus_Pls:
haftmann@25919
   753
  "- Pls = Pls"
huffman@28958
   754
  unfolding numeral_simps by simp
huffman@28958
   755
huffman@28958
   756
lemma minus_Min:
huffman@26086
   757
  "- Min = Bit1 Pls"
huffman@28958
   758
  unfolding numeral_simps by simp
huffman@28958
   759
huffman@28958
   760
lemma minus_Bit0:
huffman@26086
   761
  "- (Bit0 k) = Bit0 (- k)"
huffman@28958
   762
  unfolding numeral_simps by simp
huffman@28958
   763
huffman@28958
   764
lemma minus_Bit1:
huffman@26086
   765
  "- (Bit1 k) = Bit1 (pred (- k))"
huffman@26086
   766
  unfolding numeral_simps by simp
haftmann@25919
   767
huffman@28958
   768
lemmas minus_bin_simps [simp] =
huffman@26086
   769
  minus_Pls minus_Min minus_Bit0 minus_Bit1
huffman@26075
   770
huffman@28958
   771
text {* Subtraction *}
huffman@28958
   772
huffman@29046
   773
lemma diff_bin_simps [simp]:
huffman@29046
   774
  "k - Pls = k"
huffman@29046
   775
  "k - Min = succ k"
huffman@29046
   776
  "Pls - (Bit0 l) = Bit0 (Pls - l)"
huffman@29046
   777
  "Pls - (Bit1 l) = Bit1 (Min - l)"
huffman@29046
   778
  "Min - (Bit0 l) = Bit1 (Min - l)"
huffman@29046
   779
  "Min - (Bit1 l) = Bit0 (Min - l)"
huffman@28958
   780
  "(Bit0 k) - (Bit0 l) = Bit0 (k - l)"
huffman@28958
   781
  "(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)"
huffman@28958
   782
  "(Bit1 k) - (Bit0 l) = Bit1 (k - l)"
huffman@28958
   783
  "(Bit1 k) - (Bit1 l) = Bit0 (k - l)"
huffman@29046
   784
  unfolding numeral_simps by simp_all
huffman@28958
   785
huffman@28958
   786
text {* Multiplication *}
huffman@28958
   787
huffman@28958
   788
lemma mult_Pls:
huffman@28958
   789
  "Pls * w = Pls"
huffman@26086
   790
  unfolding numeral_simps by simp
haftmann@25919
   791
huffman@28958
   792
lemma mult_Min:
haftmann@25919
   793
  "Min * k = - k"
haftmann@25919
   794
  unfolding numeral_simps by simp
haftmann@25919
   795
huffman@28958
   796
lemma mult_Bit0:
huffman@26086
   797
  "(Bit0 k) * l = Bit0 (k * l)"
huffman@26086
   798
  unfolding numeral_simps int_distrib by simp
haftmann@25919
   799
huffman@28958
   800
lemma mult_Bit1:
huffman@26086
   801
  "(Bit1 k) * l = (Bit0 (k * l)) + l"
huffman@28958
   802
  unfolding numeral_simps int_distrib by simp
huffman@28958
   803
huffman@28958
   804
lemmas mult_bin_simps [simp] =
huffman@26086
   805
  mult_Pls mult_Min mult_Bit0 mult_Bit1
huffman@26075
   806
haftmann@25919
   807
huffman@28958
   808
subsubsection {* Binary comparisons *}
huffman@28958
   809
huffman@28958
   810
text {* Preliminaries *}
huffman@28958
   811
huffman@28958
   812
lemma even_less_0_iff:
haftmann@35028
   813
  "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
huffman@28958
   814
proof -
huffman@28958
   815
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)
huffman@28958
   816
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958
   817
    by (simp add: mult_less_0_iff zero_less_two 
huffman@28958
   818
                  order_less_not_sym [OF zero_less_two])
huffman@28958
   819
  finally show ?thesis .
huffman@28958
   820
qed
huffman@28958
   821
huffman@28958
   822
lemma le_imp_0_less: 
huffman@28958
   823
  assumes le: "0 \<le> z"
huffman@28958
   824
  shows "(0::int) < 1 + z"
huffman@28958
   825
proof -
huffman@28958
   826
  have "0 \<le> z" by fact
huffman@28958
   827
  also have "... < z + 1" by (rule less_add_one) 
huffman@28958
   828
  also have "... = 1 + z" by (simp add: add_ac)
huffman@28958
   829
  finally show "0 < 1 + z" .
huffman@28958
   830
qed
huffman@28958
   831
huffman@28958
   832
lemma odd_less_0_iff:
huffman@28958
   833
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
   834
proof (cases z)
huffman@28958
   835
  case (nonneg n)
huffman@28958
   836
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
huffman@28958
   837
                             le_imp_0_less [THEN order_less_imp_le])  
huffman@28958
   838
next
huffman@28958
   839
  case (neg n)
huffman@30079
   840
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079
   841
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   842
qed
huffman@28958
   843
huffman@28985
   844
lemma bin_less_0_simps:
huffman@28958
   845
  "Pls < 0 \<longleftrightarrow> False"
huffman@28958
   846
  "Min < 0 \<longleftrightarrow> True"
huffman@28958
   847
  "Bit0 w < 0 \<longleftrightarrow> w < 0"
huffman@28958
   848
  "Bit1 w < 0 \<longleftrightarrow> w < 0"
huffman@28958
   849
  unfolding numeral_simps
huffman@28958
   850
  by (simp_all add: even_less_0_iff odd_less_0_iff)
huffman@28958
   851
huffman@28958
   852
lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)"
huffman@28958
   853
  by simp
huffman@28958
   854
huffman@28958
   855
lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l"
huffman@28958
   856
  unfolding numeral_simps
huffman@28958
   857
  proof
huffman@28958
   858
    have "k - 1 < k" by simp
huffman@28958
   859
    also assume "k \<le> l"
huffman@28958
   860
    finally show "k - 1 < l" .
huffman@28958
   861
  next
huffman@28958
   862
    assume "k - 1 < l"
huffman@28958
   863
    hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle)
huffman@28958
   864
    thus "k \<le> l" by simp
huffman@28958
   865
  qed
huffman@28958
   866
huffman@28958
   867
lemma succ_pred: "succ (pred x) = x"
huffman@28958
   868
  unfolding numeral_simps by simp
huffman@28958
   869
huffman@28958
   870
text {* Less-than *}
huffman@28958
   871
huffman@28958
   872
lemma less_bin_simps [simp]:
huffman@28958
   873
  "Pls < Pls \<longleftrightarrow> False"
huffman@28958
   874
  "Pls < Min \<longleftrightarrow> False"
huffman@28958
   875
  "Pls < Bit0 k \<longleftrightarrow> Pls < k"
huffman@28958
   876
  "Pls < Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   877
  "Min < Pls \<longleftrightarrow> True"
huffman@28958
   878
  "Min < Min \<longleftrightarrow> False"
huffman@28958
   879
  "Min < Bit0 k \<longleftrightarrow> Min < k"
huffman@28958
   880
  "Min < Bit1 k \<longleftrightarrow> Min < k"
huffman@28958
   881
  "Bit0 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958
   882
  "Bit0 k < Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   883
  "Bit1 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958
   884
  "Bit1 k < Min \<longleftrightarrow> k < Min"
huffman@28958
   885
  "Bit0 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   886
  "Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   887
  "Bit1 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   888
  "Bit1 k < Bit1 l \<longleftrightarrow> k < l"
huffman@28958
   889
  unfolding le_iff_pred_less
huffman@28958
   890
    less_bin_lemma [of Pls]
huffman@28958
   891
    less_bin_lemma [of Min]
huffman@28958
   892
    less_bin_lemma [of "k"]
huffman@28958
   893
    less_bin_lemma [of "Bit0 k"]
huffman@28958
   894
    less_bin_lemma [of "Bit1 k"]
huffman@28958
   895
    less_bin_lemma [of "pred Pls"]
huffman@28958
   896
    less_bin_lemma [of "pred k"]
huffman@28985
   897
  by (simp_all add: bin_less_0_simps succ_pred)
huffman@28958
   898
huffman@28958
   899
text {* Less-than-or-equal *}
huffman@28958
   900
huffman@28958
   901
lemma le_bin_simps [simp]:
huffman@28958
   902
  "Pls \<le> Pls \<longleftrightarrow> True"
huffman@28958
   903
  "Pls \<le> Min \<longleftrightarrow> False"
huffman@28958
   904
  "Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   905
  "Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   906
  "Min \<le> Pls \<longleftrightarrow> True"
huffman@28958
   907
  "Min \<le> Min \<longleftrightarrow> True"
huffman@28958
   908
  "Min \<le> Bit0 k \<longleftrightarrow> Min < k"
huffman@28958
   909
  "Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k"
huffman@28958
   910
  "Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls"
huffman@28958
   911
  "Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   912
  "Bit1 k \<le> Pls \<longleftrightarrow> k < Pls"
huffman@28958
   913
  "Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   914
  "Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l"
huffman@28958
   915
  "Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   916
  "Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   917
  "Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   918
  unfolding not_less [symmetric]
huffman@28958
   919
  by (simp_all add: not_le)
huffman@28958
   920
huffman@28958
   921
text {* Equality *}
huffman@28958
   922
huffman@28958
   923
lemma eq_bin_simps [simp]:
huffman@28958
   924
  "Pls = Pls \<longleftrightarrow> True"
huffman@28958
   925
  "Pls = Min \<longleftrightarrow> False"
huffman@28958
   926
  "Pls = Bit0 l \<longleftrightarrow> Pls = l"
huffman@28958
   927
  "Pls = Bit1 l \<longleftrightarrow> False"
huffman@28958
   928
  "Min = Pls \<longleftrightarrow> False"
huffman@28958
   929
  "Min = Min \<longleftrightarrow> True"
huffman@28958
   930
  "Min = Bit0 l \<longleftrightarrow> False"
huffman@28958
   931
  "Min = Bit1 l \<longleftrightarrow> Min = l"
huffman@28958
   932
  "Bit0 k = Pls \<longleftrightarrow> k = Pls"
huffman@28958
   933
  "Bit0 k = Min \<longleftrightarrow> False"
huffman@28958
   934
  "Bit1 k = Pls \<longleftrightarrow> False"
huffman@28958
   935
  "Bit1 k = Min \<longleftrightarrow> k = Min"
huffman@28958
   936
  "Bit0 k = Bit0 l \<longleftrightarrow> k = l"
huffman@28958
   937
  "Bit0 k = Bit1 l \<longleftrightarrow> False"
huffman@28958
   938
  "Bit1 k = Bit0 l \<longleftrightarrow> False"
huffman@28958
   939
  "Bit1 k = Bit1 l \<longleftrightarrow> k = l"
huffman@28958
   940
  unfolding order_eq_iff [where 'a=int]
huffman@28958
   941
  by (simp_all add: not_less)
huffman@28958
   942
huffman@28958
   943
haftmann@25919
   944
subsection {* Converting Numerals to Rings: @{term number_of} *}
haftmann@25919
   945
haftmann@25919
   946
class number_ring = number + comm_ring_1 +
haftmann@25919
   947
  assumes number_of_eq: "number_of k = of_int k"
haftmann@25919
   948
huffman@43531
   949
class number_semiring = number + comm_semiring_1 +
huffman@44709
   950
  assumes number_of_int: "number_of (int n) = of_nat n"
huffman@43531
   951
huffman@43531
   952
instance number_ring \<subseteq> number_semiring
huffman@43531
   953
proof
huffman@44709
   954
  fix n show "number_of (int n) = (of_nat n :: 'a)"
huffman@43531
   955
    unfolding number_of_eq by (rule of_int_of_nat_eq)
huffman@43531
   956
qed
huffman@43531
   957
haftmann@25919
   958
text {* self-embedding of the integers *}
haftmann@25919
   959
haftmann@25919
   960
instantiation int :: number_ring
haftmann@25919
   961
begin
haftmann@25919
   962
haftmann@37767
   963
definition
haftmann@37767
   964
  int_number_of_def: "number_of w = (of_int w \<Colon> int)"
haftmann@25919
   965
haftmann@28724
   966
instance proof
haftmann@28724
   967
qed (simp only: int_number_of_def)
haftmann@25919
   968
haftmann@25919
   969
end
haftmann@25919
   970
haftmann@25919
   971
lemma number_of_is_id:
haftmann@25919
   972
  "number_of (k::int) = k"
haftmann@25919
   973
  unfolding int_number_of_def by simp
haftmann@25919
   974
haftmann@25919
   975
lemma number_of_succ:
haftmann@25919
   976
  "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
haftmann@25919
   977
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   978
haftmann@25919
   979
lemma number_of_pred:
haftmann@25919
   980
  "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
haftmann@25919
   981
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   982
haftmann@25919
   983
lemma number_of_minus:
haftmann@25919
   984
  "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
huffman@28958
   985
  unfolding number_of_eq by (rule of_int_minus)
haftmann@25919
   986
haftmann@25919
   987
lemma number_of_add:
haftmann@25919
   988
  "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
huffman@28958
   989
  unfolding number_of_eq by (rule of_int_add)
huffman@28958
   990
huffman@28958
   991
lemma number_of_diff:
huffman@28958
   992
  "number_of (v - w) = (number_of v - number_of w::'a::number_ring)"
huffman@28958
   993
  unfolding number_of_eq by (rule of_int_diff)
haftmann@25919
   994
haftmann@25919
   995
lemma number_of_mult:
haftmann@25919
   996
  "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
huffman@28958
   997
  unfolding number_of_eq by (rule of_int_mult)
haftmann@25919
   998
haftmann@25919
   999
text {*
haftmann@25919
  1000
  The correctness of shifting.
haftmann@25919
  1001
  But it doesn't seem to give a measurable speed-up.
haftmann@25919
  1002
*}
haftmann@25919
  1003
huffman@26086
  1004
lemma double_number_of_Bit0:
huffman@26086
  1005
  "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"
haftmann@25919
  1006
  unfolding number_of_eq numeral_simps left_distrib by simp
haftmann@25919
  1007
haftmann@25919
  1008
text {*
haftmann@25919
  1009
  Converting numerals 0 and 1 to their abstract versions.
haftmann@25919
  1010
*}
haftmann@25919
  1011
huffman@43531
  1012
lemma semiring_numeral_0_eq_0:
huffman@43531
  1013
  "Numeral0 = (0::'a::number_semiring)"
huffman@43531
  1014
  using number_of_int [where 'a='a and n=0]
huffman@43531
  1015
  unfolding numeral_simps by simp
huffman@43531
  1016
huffman@43531
  1017
lemma semiring_numeral_1_eq_1:
huffman@43531
  1018
  "Numeral1 = (1::'a::number_semiring)"
huffman@43531
  1019
  using number_of_int [where 'a='a and n=1]
huffman@43531
  1020
  unfolding numeral_simps by simp
huffman@43531
  1021
haftmann@32272
  1022
lemma numeral_0_eq_0 [simp, code_post]:
haftmann@25919
  1023
  "Numeral0 = (0::'a::number_ring)"
huffman@43531
  1024
  by (rule semiring_numeral_0_eq_0)
haftmann@25919
  1025
haftmann@32272
  1026
lemma numeral_1_eq_1 [simp, code_post]:
haftmann@25919
  1027
  "Numeral1 = (1::'a::number_ring)"
huffman@43531
  1028
  by (rule semiring_numeral_1_eq_1)
haftmann@25919
  1029
haftmann@25919
  1030
text {*
haftmann@25919
  1031
  Special-case simplification for small constants.
haftmann@25919
  1032
*}
haftmann@25919
  1033
haftmann@25919
  1034
text{*
haftmann@25919
  1035
  Unary minus for the abstract constant 1. Cannot be inserted
haftmann@25919
  1036
  as a simprule until later: it is @{text number_of_Min} re-oriented!
haftmann@25919
  1037
*}
haftmann@25919
  1038
haftmann@25919
  1039
lemma numeral_m1_eq_minus_1:
haftmann@25919
  1040
  "(-1::'a::number_ring) = - 1"
haftmann@25919
  1041
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1042
haftmann@25919
  1043
lemma mult_minus1 [simp]:
haftmann@25919
  1044
  "-1 * z = -(z::'a::number_ring)"
haftmann@25919
  1045
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1046
haftmann@25919
  1047
lemma mult_minus1_right [simp]:
haftmann@25919
  1048
  "z * -1 = -(z::'a::number_ring)"
haftmann@25919
  1049
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1050
haftmann@25919
  1051
(*Negation of a coefficient*)
haftmann@25919
  1052
lemma minus_number_of_mult [simp]:
haftmann@25919
  1053
   "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
haftmann@25919
  1054
   unfolding number_of_eq by simp
haftmann@25919
  1055
haftmann@25919
  1056
text {* Subtraction *}
haftmann@25919
  1057
haftmann@25919
  1058
lemma diff_number_of_eq:
haftmann@25919
  1059
  "number_of v - number_of w =
haftmann@25919
  1060
    (number_of (v + uminus w)::'a::number_ring)"
haftmann@25919
  1061
  unfolding number_of_eq by simp
haftmann@25919
  1062
haftmann@25919
  1063
lemma number_of_Pls:
haftmann@25919
  1064
  "number_of Pls = (0::'a::number_ring)"
haftmann@25919
  1065
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1066
haftmann@25919
  1067
lemma number_of_Min:
haftmann@25919
  1068
  "number_of Min = (- 1::'a::number_ring)"
haftmann@25919
  1069
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1070
huffman@26086
  1071
lemma number_of_Bit0:
huffman@26086
  1072
  "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086
  1073
  unfolding number_of_eq numeral_simps by simp
huffman@26086
  1074
huffman@26086
  1075
lemma number_of_Bit1:
huffman@26086
  1076
  "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086
  1077
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1078
haftmann@25919
  1079
huffman@28958
  1080
subsubsection {* Equality of Binary Numbers *}
haftmann@25919
  1081
haftmann@25919
  1082
text {* First version by Norbert Voelker *}
haftmann@25919
  1083
haftmann@36716
  1084
definition (*for simplifying equalities*) iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool" where
haftmann@25919
  1085
  "iszero z \<longleftrightarrow> z = 0"
haftmann@25919
  1086
haftmann@25919
  1087
lemma iszero_0: "iszero 0"
haftmann@36716
  1088
  by (simp add: iszero_def)
haftmann@36716
  1089
haftmann@36716
  1090
lemma iszero_Numeral0: "iszero (Numeral0 :: 'a::number_ring)"
haftmann@36716
  1091
  by (simp add: iszero_0)
haftmann@36716
  1092
haftmann@36716
  1093
lemma not_iszero_1: "\<not> iszero 1"
haftmann@36716
  1094
  by (simp add: iszero_def)
haftmann@36716
  1095
haftmann@36716
  1096
lemma not_iszero_Numeral1: "\<not> iszero (Numeral1 :: 'a::number_ring)"
haftmann@36716
  1097
  by (simp add: not_iszero_1)
haftmann@25919
  1098
huffman@35216
  1099
lemma eq_number_of_eq [simp]:
haftmann@25919
  1100
  "((number_of x::'a::number_ring) = number_of y) =
haftmann@36716
  1101
     iszero (number_of (x + uminus y) :: 'a)"
nipkow@29667
  1102
unfolding iszero_def number_of_add number_of_minus
nipkow@29667
  1103
by (simp add: algebra_simps)
haftmann@25919
  1104
haftmann@25919
  1105
lemma iszero_number_of_Pls:
haftmann@25919
  1106
  "iszero ((number_of Pls)::'a::number_ring)"
nipkow@29667
  1107
unfolding iszero_def numeral_0_eq_0 ..
haftmann@25919
  1108
haftmann@25919
  1109
lemma nonzero_number_of_Min:
haftmann@25919
  1110
  "~ iszero ((number_of Min)::'a::number_ring)"
nipkow@29667
  1111
unfolding iszero_def numeral_m1_eq_minus_1 by simp
haftmann@25919
  1112
haftmann@25919
  1113
huffman@28958
  1114
subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919
  1115
haftmann@25919
  1116
lemmas double_eq_0_iff = double_zero
haftmann@25919
  1117
haftmann@25919
  1118
lemma odd_nonzero:
haftmann@33296
  1119
  "1 + z + z \<noteq> (0::int)"
wenzelm@42676
  1120
proof (cases z)
haftmann@25919
  1121
  case (nonneg n)
haftmann@25919
  1122
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
  1123
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919
  1124
    by (auto simp add: add_assoc) 
haftmann@25919
  1125
next
haftmann@25919
  1126
  case (neg n)
haftmann@25919
  1127
  show ?thesis
haftmann@25919
  1128
  proof
haftmann@25919
  1129
    assume eq: "1 + z + z = 0"
huffman@44709
  1130
    have "(0::int) < 1 + (int n + int n)"
haftmann@25919
  1131
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
  1132
    also have "... = - (1 + z + z)" 
haftmann@25919
  1133
      by (simp add: neg add_assoc [symmetric]) 
haftmann@25919
  1134
    also have "... = 0" by (simp add: eq) 
haftmann@25919
  1135
    finally have "0<0" ..
haftmann@25919
  1136
    thus False by blast
haftmann@25919
  1137
  qed
haftmann@25919
  1138
qed
haftmann@25919
  1139
huffman@26086
  1140
lemma iszero_number_of_Bit0:
huffman@26086
  1141
  "iszero (number_of (Bit0 w)::'a) = 
huffman@26086
  1142
   iszero (number_of w::'a::{ring_char_0,number_ring})"
haftmann@25919
  1143
proof -
haftmann@25919
  1144
  have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
haftmann@25919
  1145
  proof -
haftmann@25919
  1146
    assume eq: "of_int w + of_int w = (0::'a)"
haftmann@25919
  1147
    then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
haftmann@25919
  1148
    then have "w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1149
    then show "w = 0" by (simp only: double_eq_0_iff)
haftmann@25919
  1150
  qed
huffman@26086
  1151
  thus ?thesis
huffman@26086
  1152
    by (auto simp add: iszero_def number_of_eq numeral_simps)
huffman@26086
  1153
qed
huffman@26086
  1154
huffman@26086
  1155
lemma iszero_number_of_Bit1:
huffman@26086
  1156
  "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"
huffman@26086
  1157
proof -
huffman@26086
  1158
  have "1 + of_int w + of_int w \<noteq> (0::'a)"
haftmann@25919
  1159
  proof
haftmann@25919
  1160
    assume eq: "1 + of_int w + of_int w = (0::'a)"
haftmann@25919
  1161
    hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp 
haftmann@25919
  1162
    hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1163
    with odd_nonzero show False by blast
haftmann@25919
  1164
  qed
huffman@26086
  1165
  thus ?thesis
huffman@26086
  1166
    by (auto simp add: iszero_def number_of_eq numeral_simps)
haftmann@25919
  1167
qed
haftmann@25919
  1168
huffman@35216
  1169
lemmas iszero_simps [simp] =
huffman@28985
  1170
  iszero_0 not_iszero_1
huffman@28985
  1171
  iszero_number_of_Pls nonzero_number_of_Min
huffman@28985
  1172
  iszero_number_of_Bit0 iszero_number_of_Bit1
huffman@28985
  1173
(* iszero_number_of_Pls would never normally be used
huffman@28985
  1174
   because its lhs simplifies to "iszero 0" *)
haftmann@25919
  1175
huffman@28958
  1176
subsubsection {* The Less-Than Relation *}
haftmann@25919
  1177
haftmann@25919
  1178
lemma double_less_0_iff:
haftmann@35028
  1179
  "(a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
  1180
proof -
haftmann@25919
  1181
  have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
haftmann@25919
  1182
  also have "... = (a < 0)"
haftmann@25919
  1183
    by (simp add: mult_less_0_iff zero_less_two 
haftmann@25919
  1184
                  order_less_not_sym [OF zero_less_two]) 
haftmann@25919
  1185
  finally show ?thesis .
haftmann@25919
  1186
qed
haftmann@25919
  1187
haftmann@25919
  1188
lemma odd_less_0:
haftmann@33296
  1189
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
  1190
proof (cases z)
haftmann@25919
  1191
  case (nonneg n)
wenzelm@42676
  1192
  then show ?thesis
wenzelm@42676
  1193
    by (simp add: linorder_not_less add_assoc add_increasing
wenzelm@42676
  1194
      le_imp_0_less [THEN order_less_imp_le])
haftmann@25919
  1195
next
haftmann@25919
  1196
  case (neg n)
wenzelm@42676
  1197
  then show ?thesis
wenzelm@42676
  1198
    by (simp del: of_nat_Suc of_nat_add of_nat_1
wenzelm@42676
  1199
      add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
haftmann@25919
  1200
qed
haftmann@25919
  1201
haftmann@25919
  1202
text {* Less-Than or Equals *}
haftmann@25919
  1203
haftmann@25919
  1204
text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
haftmann@25919
  1205
haftmann@25919
  1206
lemmas le_number_of_eq_not_less =
haftmann@25919
  1207
  linorder_not_less [of "number_of w" "number_of v", symmetric, 
haftmann@25919
  1208
  standard]
haftmann@25919
  1209
haftmann@25919
  1210
haftmann@25919
  1211
text {* Absolute value (@{term abs}) *}
haftmann@25919
  1212
haftmann@25919
  1213
lemma abs_number_of:
haftmann@35028
  1214
  "abs(number_of x::'a::{linordered_idom,number_ring}) =
haftmann@25919
  1215
   (if number_of x < (0::'a) then -number_of x else number_of x)"
haftmann@25919
  1216
  by (simp add: abs_if)
haftmann@25919
  1217
haftmann@25919
  1218
haftmann@25919
  1219
text {* Re-orientation of the equation nnn=x *}
haftmann@25919
  1220
haftmann@25919
  1221
lemma number_of_reorient:
haftmann@25919
  1222
  "(number_of w = x) = (x = number_of w)"
haftmann@25919
  1223
  by auto
haftmann@25919
  1224
haftmann@25919
  1225
huffman@28958
  1226
subsubsection {* Simplification of arithmetic operations on integer constants. *}
haftmann@25919
  1227
haftmann@25919
  1228
lemmas arith_extra_simps [standard, simp] =
haftmann@25919
  1229
  number_of_add [symmetric]
huffman@28958
  1230
  number_of_minus [symmetric]
huffman@28958
  1231
  numeral_m1_eq_minus_1 [symmetric]
haftmann@25919
  1232
  number_of_mult [symmetric]
haftmann@25919
  1233
  diff_number_of_eq abs_number_of 
haftmann@25919
  1234
haftmann@25919
  1235
text {*
haftmann@25919
  1236
  For making a minimal simpset, one must include these default simprules.
haftmann@25919
  1237
  Also include @{text simp_thms}.
haftmann@25919
  1238
*}
haftmann@25919
  1239
haftmann@25919
  1240
lemmas arith_simps = 
huffman@26075
  1241
  normalize_bin_simps pred_bin_simps succ_bin_simps
huffman@26075
  1242
  add_bin_simps minus_bin_simps mult_bin_simps
haftmann@25919
  1243
  abs_zero abs_one arith_extra_simps
haftmann@25919
  1244
haftmann@25919
  1245
text {* Simplification of relational operations *}
haftmann@25919
  1246
huffman@28962
  1247
lemma less_number_of [simp]:
haftmann@35028
  1248
  "(number_of x::'a::{linordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
huffman@28962
  1249
  unfolding number_of_eq by (rule of_int_less_iff)
huffman@28962
  1250
huffman@28962
  1251
lemma le_number_of [simp]:
haftmann@35028
  1252
  "(number_of x::'a::{linordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
huffman@28962
  1253
  unfolding number_of_eq by (rule of_int_le_iff)
huffman@28962
  1254
huffman@28967
  1255
lemma eq_number_of [simp]:
huffman@28967
  1256
  "(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y"
huffman@28967
  1257
  unfolding number_of_eq by (rule of_int_eq_iff)
huffman@28967
  1258
huffman@35216
  1259
lemmas rel_simps =
huffman@28962
  1260
  less_number_of less_bin_simps
huffman@28962
  1261
  le_number_of le_bin_simps
huffman@28988
  1262
  eq_number_of_eq eq_bin_simps
huffman@29039
  1263
  iszero_simps
haftmann@25919
  1264
haftmann@25919
  1265
huffman@28958
  1266
subsubsection {* Simplification of arithmetic when nested to the right. *}
haftmann@25919
  1267
haftmann@25919
  1268
lemma add_number_of_left [simp]:
haftmann@25919
  1269
  "number_of v + (number_of w + z) =
haftmann@25919
  1270
   (number_of(v + w) + z::'a::number_ring)"
haftmann@25919
  1271
  by (simp add: add_assoc [symmetric])
haftmann@25919
  1272
haftmann@25919
  1273
lemma mult_number_of_left [simp]:
haftmann@25919
  1274
  "number_of v * (number_of w * z) =
haftmann@25919
  1275
   (number_of(v * w) * z::'a::number_ring)"
haftmann@25919
  1276
  by (simp add: mult_assoc [symmetric])
haftmann@25919
  1277
haftmann@25919
  1278
lemma add_number_of_diff1:
haftmann@25919
  1279
  "number_of v + (number_of w - c) = 
haftmann@25919
  1280
  number_of(v + w) - (c::'a::number_ring)"
huffman@35216
  1281
  by (simp add: diff_minus)
haftmann@25919
  1282
haftmann@25919
  1283
lemma add_number_of_diff2 [simp]:
haftmann@25919
  1284
  "number_of v + (c - number_of w) =
haftmann@25919
  1285
   number_of (v + uminus w) + (c::'a::number_ring)"
nipkow@29667
  1286
by (simp add: algebra_simps diff_number_of_eq [symmetric])
haftmann@25919
  1287
haftmann@25919
  1288
haftmann@30652
  1289
haftmann@30652
  1290
haftmann@25919
  1291
subsection {* The Set of Integers *}
haftmann@25919
  1292
haftmann@25919
  1293
context ring_1
haftmann@25919
  1294
begin
haftmann@25919
  1295
haftmann@30652
  1296
definition Ints  :: "'a set" where
haftmann@37767
  1297
  "Ints = range of_int"
haftmann@25919
  1298
haftmann@25919
  1299
notation (xsymbols)
haftmann@25919
  1300
  Ints  ("\<int>")
haftmann@25919
  1301
huffman@35634
  1302
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
  1303
  by (simp add: Ints_def)
huffman@35634
  1304
huffman@35634
  1305
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@35634
  1306
apply (simp add: Ints_def)
huffman@35634
  1307
apply (rule range_eqI)
huffman@35634
  1308
apply (rule of_int_of_nat_eq [symmetric])
huffman@35634
  1309
done
huffman@35634
  1310
haftmann@25919
  1311
lemma Ints_0 [simp]: "0 \<in> \<int>"
haftmann@25919
  1312
apply (simp add: Ints_def)
haftmann@25919
  1313
apply (rule range_eqI)
haftmann@25919
  1314
apply (rule of_int_0 [symmetric])
haftmann@25919
  1315
done
haftmann@25919
  1316
haftmann@25919
  1317
lemma Ints_1 [simp]: "1 \<in> \<int>"
haftmann@25919
  1318
apply (simp add: Ints_def)
haftmann@25919
  1319
apply (rule range_eqI)
haftmann@25919
  1320
apply (rule of_int_1 [symmetric])
haftmann@25919
  1321
done
haftmann@25919
  1322
haftmann@25919
  1323
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
  1324
apply (auto simp add: Ints_def)
haftmann@25919
  1325
apply (rule range_eqI)
haftmann@25919
  1326
apply (rule of_int_add [symmetric])
haftmann@25919
  1327
done
haftmann@25919
  1328
haftmann@25919
  1329
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
  1330
apply (auto simp add: Ints_def)
haftmann@25919
  1331
apply (rule range_eqI)
haftmann@25919
  1332
apply (rule of_int_minus [symmetric])
haftmann@25919
  1333
done
haftmann@25919
  1334
huffman@35634
  1335
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634
  1336
apply (auto simp add: Ints_def)
huffman@35634
  1337
apply (rule range_eqI)
huffman@35634
  1338
apply (rule of_int_diff [symmetric])
huffman@35634
  1339
done
huffman@35634
  1340
haftmann@25919
  1341
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
  1342
apply (auto simp add: Ints_def)
haftmann@25919
  1343
apply (rule range_eqI)
haftmann@25919
  1344
apply (rule of_int_mult [symmetric])
haftmann@25919
  1345
done
haftmann@25919
  1346
huffman@35634
  1347
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634
  1348
by (induct n) simp_all
huffman@35634
  1349
haftmann@25919
  1350
lemma Ints_cases [cases set: Ints]:
haftmann@25919
  1351
  assumes "q \<in> \<int>"
haftmann@25919
  1352
  obtains (of_int) z where "q = of_int z"
haftmann@25919
  1353
  unfolding Ints_def
haftmann@25919
  1354
proof -
haftmann@25919
  1355
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
  1356
  then obtain z where "q = of_int z" ..
haftmann@25919
  1357
  then show thesis ..
haftmann@25919
  1358
qed
haftmann@25919
  1359
haftmann@25919
  1360
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
  1361
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
  1362
  by (rule Ints_cases) auto
haftmann@25919
  1363
haftmann@25919
  1364
end
haftmann@25919
  1365
haftmann@25919
  1366
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
  1367
haftmann@25919
  1368
lemma Ints_double_eq_0_iff:
haftmann@25919
  1369
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1370
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
  1371
proof -
haftmann@25919
  1372
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1373
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1374
  show ?thesis
haftmann@25919
  1375
  proof
haftmann@25919
  1376
    assume "a = 0"
haftmann@25919
  1377
    thus "a + a = 0" by simp
haftmann@25919
  1378
  next
haftmann@25919
  1379
    assume eq: "a + a = 0"
haftmann@25919
  1380
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1381
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1382
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
  1383
    thus "a = 0" by (simp add: a)
haftmann@25919
  1384
  qed
haftmann@25919
  1385
qed
haftmann@25919
  1386
haftmann@25919
  1387
lemma Ints_odd_nonzero:
haftmann@25919
  1388
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1389
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
  1390
proof -
haftmann@25919
  1391
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1392
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1393
  show ?thesis
haftmann@25919
  1394
  proof
haftmann@25919
  1395
    assume eq: "1 + a + a = 0"
haftmann@25919
  1396
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1397
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1398
    with odd_nonzero show False by blast
haftmann@25919
  1399
  qed
haftmann@25919
  1400
qed 
haftmann@25919
  1401
huffman@35634
  1402
lemma Ints_number_of [simp]:
haftmann@25919
  1403
  "(number_of w :: 'a::number_ring) \<in> Ints"
haftmann@25919
  1404
  unfolding number_of_eq Ints_def by simp
haftmann@25919
  1405
huffman@35634
  1406
lemma Nats_number_of [simp]:
huffman@35634
  1407
  "Int.Pls \<le> w \<Longrightarrow> (number_of w :: 'a::number_ring) \<in> Nats"
huffman@35634
  1408
unfolding Int.Pls_def number_of_eq
huffman@35634
  1409
by (simp only: of_nat_nat [symmetric] of_nat_in_Nats)
huffman@35634
  1410
haftmann@25919
  1411
lemma Ints_odd_less_0: 
haftmann@25919
  1412
  assumes in_Ints: "a \<in> Ints"
haftmann@35028
  1413
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
  1414
proof -
haftmann@25919
  1415
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1416
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1417
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
  1418
    by (simp add: a)
haftmann@25919
  1419
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
haftmann@25919
  1420
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
  1421
  finally show ?thesis .
haftmann@25919
  1422
qed
haftmann@25919
  1423
haftmann@25919
  1424
haftmann@25919
  1425
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
  1426
haftmann@25919
  1427
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
  1428
  apply (cases "finite A")
haftmann@25919
  1429
  apply (erule finite_induct, auto)
haftmann@25919
  1430
  done
haftmann@25919
  1431
haftmann@25919
  1432
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
  1433
  apply (cases "finite A")
haftmann@25919
  1434
  apply (erule finite_induct, auto)
haftmann@25919
  1435
  done
haftmann@25919
  1436
haftmann@25919
  1437
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
  1438
  apply (cases "finite A")
haftmann@25919
  1439
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
  1440
  done
haftmann@25919
  1441
haftmann@25919
  1442
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
  1443
  apply (cases "finite A")
haftmann@25919
  1444
  apply (erule finite_induct, auto)
haftmann@25919
  1445
  done
haftmann@25919
  1446
haftmann@25919
  1447
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
  1448
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
  1449
haftmann@25919
  1450
haftmann@25919
  1451
subsection{*Inequality Reasoning for the Arithmetic Simproc*}
haftmann@25919
  1452
haftmann@25919
  1453
lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
haftmann@25919
  1454
by simp 
haftmann@25919
  1455
haftmann@25919
  1456
lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
haftmann@25919
  1457
by simp
haftmann@25919
  1458
haftmann@25919
  1459
lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
haftmann@25919
  1460
by simp 
haftmann@25919
  1461
haftmann@25919
  1462
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
haftmann@25919
  1463
by simp
haftmann@25919
  1464
haftmann@25919
  1465
lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
haftmann@25919
  1466
by simp
haftmann@25919
  1467
haftmann@25919
  1468
lemma inverse_numeral_1:
haftmann@25919
  1469
  "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
haftmann@25919
  1470
by simp
haftmann@25919
  1471
haftmann@25919
  1472
text{*Theorem lists for the cancellation simprocs. The use of binary numerals
haftmann@25919
  1473
for 0 and 1 reduces the number of special cases.*}
haftmann@25919
  1474
haftmann@25919
  1475
lemmas add_0s = add_numeral_0 add_numeral_0_right
haftmann@25919
  1476
lemmas mult_1s = mult_numeral_1 mult_numeral_1_right 
haftmann@25919
  1477
                 mult_minus1 mult_minus1_right
haftmann@25919
  1478
haftmann@25919
  1479
haftmann@25919
  1480
subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
haftmann@25919
  1481
haftmann@25919
  1482
text{*Arithmetic computations are defined for binary literals, which leaves 0
haftmann@25919
  1483
and 1 as special cases. Addition already has rules for 0, but not 1.
haftmann@25919
  1484
Multiplication and unary minus already have rules for both 0 and 1.*}
haftmann@25919
  1485
haftmann@25919
  1486
haftmann@25919
  1487
lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
haftmann@25919
  1488
by simp
haftmann@25919
  1489
haftmann@25919
  1490
haftmann@25919
  1491
lemmas add_number_of_eq = number_of_add [symmetric]
haftmann@25919
  1492
haftmann@25919
  1493
text{*Allow 1 on either or both sides*}
huffman@43531
  1494
lemma semiring_one_add_one_is_two: "1 + 1 = (2::'a::number_semiring)"
huffman@43531
  1495
  using number_of_int [where 'a='a and n="Suc (Suc 0)"]
huffman@43531
  1496
  by (simp add: numeral_simps)
huffman@43531
  1497
haftmann@25919
  1498
lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
huffman@43531
  1499
by (rule semiring_one_add_one_is_two)
haftmann@25919
  1500
haftmann@25919
  1501
lemmas add_special =
haftmann@25919
  1502
    one_add_one_is_two
haftmann@25919
  1503
    binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1504
    binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1505
haftmann@25919
  1506
text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
haftmann@25919
  1507
lemmas diff_special =
haftmann@25919
  1508
    binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1509
    binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1510
haftmann@25919
  1511
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1512
lemmas eq_special =
haftmann@25919
  1513
    binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
haftmann@25919
  1514
    binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1515
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
haftmann@25919
  1516
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1517
haftmann@25919
  1518
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1519
lemmas less_special =
huffman@28984
  1520
  binop_eq [of "op <", OF less_number_of numeral_0_eq_0 refl, standard]
huffman@28984
  1521
  binop_eq [of "op <", OF less_number_of numeral_1_eq_1 refl, standard]
huffman@28984
  1522
  binop_eq [of "op <", OF less_number_of refl numeral_0_eq_0, standard]
huffman@28984
  1523
  binop_eq [of "op <", OF less_number_of refl numeral_1_eq_1, standard]
haftmann@25919
  1524
haftmann@25919
  1525
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1526
lemmas le_special =
huffman@28984
  1527
    binop_eq [of "op \<le>", OF le_number_of numeral_0_eq_0 refl, standard]
huffman@28984
  1528
    binop_eq [of "op \<le>", OF le_number_of numeral_1_eq_1 refl, standard]
huffman@28984
  1529
    binop_eq [of "op \<le>", OF le_number_of refl numeral_0_eq_0, standard]
huffman@28984
  1530
    binop_eq [of "op \<le>", OF le_number_of refl numeral_1_eq_1, standard]
haftmann@25919
  1531
haftmann@25919
  1532
lemmas arith_special[simp] = 
haftmann@25919
  1533
       add_special diff_special eq_special less_special le_special
haftmann@25919
  1534
haftmann@25919
  1535
haftmann@25919
  1536
text {* Legacy theorems *}
haftmann@25919
  1537
haftmann@25919
  1538
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
  1539
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@25919
  1540
huffman@30802
  1541
subsection {* Setting up simplification procedures *}
huffman@30802
  1542
huffman@30802
  1543
lemmas int_arith_rules =
huffman@30802
  1544
  neg_le_iff_le numeral_0_eq_0 numeral_1_eq_1
huffman@30802
  1545
  minus_zero diff_minus left_minus right_minus
huffman@45219
  1546
  mult_zero_left mult_zero_right mult_1_left mult_1_right
huffman@30802
  1547
  mult_minus_left mult_minus_right
huffman@30802
  1548
  minus_add_distrib minus_minus mult_assoc
huffman@30802
  1549
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
huffman@30802
  1550
  of_int_0 of_int_1 of_int_add of_int_mult
huffman@30802
  1551
haftmann@28952
  1552
use "Tools/int_arith.ML"
haftmann@30496
  1553
declaration {* K Int_Arith.setup *}
haftmann@25919
  1554
wenzelm@43595
  1555
simproc_setup fast_arith ("(m::'a::{linordered_idom,number_ring}) < n" |
wenzelm@43595
  1556
  "(m::'a::{linordered_idom,number_ring}) <= n" |
wenzelm@43595
  1557
  "(m::'a::{linordered_idom,number_ring}) = n") =
wenzelm@43595
  1558
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
  1559
huffman@31024
  1560
setup {*
wenzelm@33523
  1561
  Reorient_Proc.add
haftmann@31065
  1562
    (fn Const (@{const_name number_of}, _) $ _ => true | _ => false)
huffman@31024
  1563
*}
huffman@31024
  1564
wenzelm@33523
  1565
simproc_setup reorient_numeral ("number_of w = x") = Reorient_Proc.proc
huffman@31024
  1566
haftmann@25919
  1567
haftmann@25919
  1568
subsection{*Lemmas About Small Numerals*}
haftmann@25919
  1569
haftmann@25919
  1570
lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
haftmann@25919
  1571
proof -
haftmann@25919
  1572
  have "(of_int -1 :: 'a) = of_int (- 1)" by simp
haftmann@25919
  1573
  also have "... = - of_int 1" by (simp only: of_int_minus)
haftmann@25919
  1574
  also have "... = -1" by simp
haftmann@25919
  1575
  finally show ?thesis .
haftmann@25919
  1576
qed
haftmann@25919
  1577
haftmann@35028
  1578
lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{linordered_idom,number_ring})"
haftmann@25919
  1579
by (simp add: abs_if)
haftmann@25919
  1580
haftmann@25919
  1581
lemma abs_power_minus_one [simp]:
haftmann@35028
  1582
  "abs(-1 ^ n) = (1::'a::{linordered_idom,number_ring})"
haftmann@25919
  1583
by (simp add: power_abs)
haftmann@25919
  1584
huffman@30000
  1585
lemma of_int_number_of_eq [simp]:
haftmann@25919
  1586
     "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
haftmann@25919
  1587
by (simp add: number_of_eq) 
haftmann@25919
  1588
haftmann@25919
  1589
text{*Lemmas for specialist use, NOT as default simprules*}
huffman@43531
  1590
(* TODO: see if semiring duplication can be removed without breaking proofs *)
huffman@43531
  1591
lemma semiring_mult_2: "2 * z = (z+z::'a::number_semiring)"
huffman@43531
  1592
unfolding semiring_one_add_one_is_two [symmetric] left_distrib by simp
huffman@43531
  1593
huffman@43531
  1594
lemma semiring_mult_2_right: "z * 2 = (z+z::'a::number_semiring)"
huffman@43531
  1595
by (subst mult_commute, rule semiring_mult_2)
huffman@43531
  1596
haftmann@25919
  1597
lemma mult_2: "2 * z = (z+z::'a::number_ring)"
huffman@43531
  1598
by (rule semiring_mult_2)
haftmann@25919
  1599
haftmann@25919
  1600
lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
huffman@43531
  1601
by (rule semiring_mult_2_right)
haftmann@25919
  1602
haftmann@25919
  1603
haftmann@25919
  1604
subsection{*More Inequality Reasoning*}
haftmann@25919
  1605
haftmann@25919
  1606
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
  1607
by arith
haftmann@25919
  1608
haftmann@25919
  1609
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
  1610
by arith
haftmann@25919
  1611
haftmann@25919
  1612
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
  1613
by arith
haftmann@25919
  1614
haftmann@25919
  1615
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
  1616
by arith
haftmann@25919
  1617
haftmann@25919
  1618
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
  1619
by arith
haftmann@25919
  1620
haftmann@25919
  1621
huffman@28958
  1622
subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919
  1623
haftmann@25919
  1624
text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
haftmann@25919
  1625
  @{term "w + - z"}*}
haftmann@25919
  1626
declare Zero_int_def [symmetric, simp]
haftmann@25919
  1627
declare One_int_def [symmetric, simp]
haftmann@25919
  1628
haftmann@25919
  1629
lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
haftmann@25919
  1630
huffman@44695
  1631
lemma nat_0 [simp]: "nat 0 = 0"
haftmann@25919
  1632
by (simp add: nat_eq_iff)
haftmann@25919
  1633
haftmann@25919
  1634
lemma nat_1: "nat 1 = Suc 0"
haftmann@25919
  1635
by (subst nat_eq_iff, simp)
haftmann@25919
  1636
haftmann@25919
  1637
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@25919
  1638
by (subst nat_eq_iff, simp)
haftmann@25919
  1639
haftmann@25919
  1640
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
  1641
apply (insert zless_nat_conj [of 1 z])
haftmann@25919
  1642
apply (auto simp add: nat_1)
haftmann@25919
  1643
done
haftmann@25919
  1644
haftmann@25919
  1645
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
  1646
      z is an integer literal.*}
haftmann@25919
  1647
lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
haftmann@25919
  1648
haftmann@25919
  1649
lemma split_nat [arith_split]:
huffman@44709
  1650
  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
  1651
  (is "?P = (?L & ?R)")
haftmann@25919
  1652
proof (cases "i < 0")
haftmann@25919
  1653
  case True thus ?thesis by auto
haftmann@25919
  1654
next
haftmann@25919
  1655
  case False
haftmann@25919
  1656
  have "?P = ?L"
haftmann@25919
  1657
  proof
haftmann@25919
  1658
    assume ?P thus ?L using False by clarsimp
haftmann@25919
  1659
  next
haftmann@25919
  1660
    assume ?L thus ?P using False by simp
haftmann@25919
  1661
  qed
haftmann@25919
  1662
  with False show ?thesis by simp
haftmann@25919
  1663
qed
haftmann@25919
  1664
haftmann@25919
  1665
context ring_1
haftmann@25919
  1666
begin
haftmann@25919
  1667
blanchet@33056
  1668
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
  1669
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1670
proof (cases "k < 0")
haftmann@25919
  1671
  case True then have "0 \<le> - k" by simp
haftmann@25919
  1672
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1673
  with True show ?thesis by simp
haftmann@25919
  1674
next
haftmann@25919
  1675
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
  1676
qed
haftmann@25919
  1677
haftmann@25919
  1678
end
haftmann@25919
  1679
haftmann@25919
  1680
lemma nat_mult_distrib:
haftmann@25919
  1681
  fixes z z' :: int
haftmann@25919
  1682
  assumes "0 \<le> z"
haftmann@25919
  1683
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1684
proof (cases "0 \<le> z'")
haftmann@25919
  1685
  case False with assms have "z * z' \<le> 0"
haftmann@25919
  1686
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1687
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1688
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1689
  ultimately show ?thesis by simp
haftmann@25919
  1690
next
haftmann@25919
  1691
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1692
  show ?thesis
haftmann@25919
  1693
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1694
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1695
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1696
qed
haftmann@25919
  1697
haftmann@25919
  1698
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
  1699
apply (rule trans)
haftmann@25919
  1700
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
  1701
done
haftmann@25919
  1702
haftmann@25919
  1703
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
  1704
apply (cases "z=0 | w=0")
haftmann@25919
  1705
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
  1706
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1707
done
haftmann@25919
  1708
haftmann@25919
  1709
haftmann@25919
  1710
subsection "Induction principles for int"
haftmann@25919
  1711
haftmann@25919
  1712
text{*Well-founded segments of the integers*}
haftmann@25919
  1713
haftmann@25919
  1714
definition
haftmann@25919
  1715
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
  1716
where
haftmann@25919
  1717
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
  1718
haftmann@25919
  1719
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1720
proof -
haftmann@25919
  1721
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
  1722
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
  1723
  thus ?thesis 
haftmann@25919
  1724
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1725
qed
haftmann@25919
  1726
haftmann@25919
  1727
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
  1728
by RankFinder.*}
haftmann@25919
  1729
haftmann@25919
  1730
definition
haftmann@25919
  1731
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
  1732
where
haftmann@25919
  1733
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
  1734
haftmann@25919
  1735
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1736
proof -
haftmann@25919
  1737
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
  1738
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
  1739
  thus ?thesis 
haftmann@25919
  1740
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1741
qed
haftmann@25919
  1742
haftmann@25919
  1743
(* `set:int': dummy construction *)
haftmann@25919
  1744
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1745
  fixes i :: int
haftmann@25919
  1746
  assumes ge: "k \<le> i" and
haftmann@25919
  1747
    base: "P k" and
haftmann@25919
  1748
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1749
  shows "P i"
haftmann@25919
  1750
proof -
wenzelm@42676
  1751
  { fix n
wenzelm@42676
  1752
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
  1753
    proof (induct n)
haftmann@25919
  1754
      case 0
haftmann@25919
  1755
      hence "i = k" by arith
haftmann@25919
  1756
      thus "P i" using base by simp
haftmann@25919
  1757
    next
haftmann@25919
  1758
      case (Suc n)
haftmann@25919
  1759
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
  1760
      moreover
haftmann@25919
  1761
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
  1762
      ultimately
wenzelm@42676
  1763
      have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676
  1764
      from step [OF ki1 this] show ?case by simp
haftmann@25919
  1765
    qed
haftmann@25919
  1766
  }
haftmann@25919
  1767
  with ge show ?thesis by fast
haftmann@25919
  1768
qed
haftmann@25919
  1769
haftmann@25928
  1770
(* `set:int': dummy construction *)
haftmann@25928
  1771
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
  1772
  assumes gr: "k < (i::int)" and
haftmann@25919
  1773
        base: "P(k+1)" and
haftmann@25919
  1774
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
  1775
  shows "P i"
haftmann@25919
  1776
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
  1777
  using gr apply arith
haftmann@25919
  1778
 apply(rule base)
haftmann@25919
  1779
apply (rule step, simp+)
haftmann@25919
  1780
done
haftmann@25919
  1781
wenzelm@42676
  1782
theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919
  1783
  assumes le: "i \<le> (k::int)" and
haftmann@25919
  1784
        base: "P(k)" and
haftmann@25919
  1785
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1786
  shows "P i"
haftmann@25919
  1787
proof -
wenzelm@42676
  1788
  { fix n
wenzelm@42676
  1789
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
  1790
    proof (induct n)
haftmann@25919
  1791
      case 0
haftmann@25919
  1792
      hence "i = k" by arith
haftmann@25919
  1793
      thus "P i" using base by simp
haftmann@25919
  1794
    next
haftmann@25919
  1795
      case (Suc n)
wenzelm@42676
  1796
      hence "n = nat (k - (i + 1))" by arith
haftmann@25919
  1797
      moreover
haftmann@25919
  1798
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
  1799
      ultimately
wenzelm@42676
  1800
      have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919
  1801
      from step[OF ki1 this] show ?case by simp
haftmann@25919
  1802
    qed
haftmann@25919
  1803
  }
haftmann@25919
  1804
  with le show ?thesis by fast
haftmann@25919
  1805
qed
haftmann@25919
  1806
wenzelm@42676
  1807
theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919
  1808
  assumes less: "(i::int) < k" and
haftmann@25919
  1809
        base: "P(k - 1)" and
haftmann@25919
  1810
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1811
  shows "P i"
haftmann@25919
  1812
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1813
  using less apply arith
haftmann@25919
  1814
 apply(rule base)
haftmann@25919
  1815
apply (rule step, simp+)
haftmann@25919
  1816
done
haftmann@25919
  1817
haftmann@36811
  1818
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1819
  fixes k :: int
haftmann@36801
  1820
  assumes base: "P k"
haftmann@36801
  1821
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1822
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1823
  shows "P i"
haftmann@36801
  1824
proof -
haftmann@36801
  1825
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1826
  then show ?thesis
wenzelm@42676
  1827
  proof
wenzelm@42676
  1828
    assume "i \<ge> k"
wenzelm@42676
  1829
    then show ?thesis using base
haftmann@36801
  1830
      by (rule int_ge_induct) (fact step1)
haftmann@36801
  1831
  next
wenzelm@42676
  1832
    assume "i \<le> k"
wenzelm@42676
  1833
    then show ?thesis using base
haftmann@36801
  1834
      by (rule int_le_induct) (fact step2)
haftmann@36801
  1835
  qed
haftmann@36801
  1836
qed
haftmann@36801
  1837
haftmann@25919
  1838
subsection{*Intermediate value theorems*}
haftmann@25919
  1839
haftmann@25919
  1840
lemma int_val_lemma:
haftmann@25919
  1841
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1842
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079
  1843
unfolding One_nat_def
wenzelm@42676
  1844
apply (induct n)
wenzelm@42676
  1845
apply simp
haftmann@25919
  1846
apply (intro strip)
haftmann@25919
  1847
apply (erule impE, simp)
haftmann@25919
  1848
apply (erule_tac x = n in allE, simp)
huffman@30079
  1849
apply (case_tac "k = f (Suc n)")
haftmann@27106
  1850
apply force
haftmann@25919
  1851
apply (erule impE)
haftmann@25919
  1852
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1853
apply (blast intro: le_SucI)
haftmann@25919
  1854
done
haftmann@25919
  1855
haftmann@25919
  1856
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1857
haftmann@25919
  1858
lemma nat_intermed_int_val:
haftmann@25919
  1859
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1860
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1861
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1862
       in int_val_lemma)
huffman@30079
  1863
unfolding One_nat_def
haftmann@25919
  1864
apply simp
haftmann@25919
  1865
apply (erule exE)
haftmann@25919
  1866
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1867
done
haftmann@25919
  1868
haftmann@25919
  1869
haftmann@25919
  1870
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1871
haftmann@25919
  1872
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1873
by arith
haftmann@25919
  1874
paulson@34055
  1875
lemma abs_zmult_eq_1:
paulson@34055
  1876
  assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055
  1877
  shows "\<bar>m\<bar> = (1::int)"
paulson@34055
  1878
proof -
paulson@34055
  1879
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055
  1880
    by auto
paulson@34055
  1881
  have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055
  1882
  proof
paulson@34055
  1883
    assume "2 \<le> \<bar>m\<bar>"
paulson@34055
  1884
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
paulson@34055
  1885
      by (simp add: mult_mono 0) 
paulson@34055
  1886
    also have "... = \<bar>m*n\<bar>" 
paulson@34055
  1887
      by (simp add: abs_mult)
paulson@34055
  1888
    also have "... = 1"
paulson@34055
  1889
      by (simp add: mn)
paulson@34055
  1890
    finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055
  1891
    thus "False" using 0
paulson@34055
  1892
      by auto
paulson@34055
  1893
  qed
paulson@34055
  1894
  thus ?thesis using 0
paulson@34055
  1895
    by auto
paulson@34055
  1896
qed
haftmann@25919
  1897
haftmann@25919
  1898
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1899
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1900
boehmes@35815
  1901
lemma pos_zmult_eq_1_iff:
boehmes@35815
  1902
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815
  1903
proof -
boehmes@35815
  1904
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1905
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1906
qed
haftmann@25919
  1907
haftmann@25919
  1908
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1909
apply (rule iffI) 
haftmann@25919
  1910
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919
  1911
 apply (simp add: mult_commute [of m]) 
haftmann@25919
  1912
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1913
done
haftmann@25919
  1914
haftmann@33296
  1915
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1916
proof
haftmann@33296
  1917
  assume "finite (UNIV::int set)"
haftmann@33296
  1918
  moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1919
    by (rule injI) simp
haftmann@33296
  1920
  ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1921
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1922
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1923
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1924
qed
haftmann@25919
  1925
haftmann@25919
  1926
haftmann@30652
  1927
subsection {* Further theorems on numerals *}
haftmann@30652
  1928
haftmann@30652
  1929
subsubsection{*Special Simplification for Constants*}
haftmann@30652
  1930
haftmann@30652
  1931
text{*These distributive laws move literals inside sums and differences.*}
haftmann@30652
  1932
haftmann@30652
  1933
lemmas left_distrib_number_of [simp] =
haftmann@30652
  1934
  left_distrib [of _ _ "number_of v", standard]
haftmann@30652
  1935
haftmann@30652
  1936
lemmas right_distrib_number_of [simp] =
haftmann@30652
  1937
  right_distrib [of "number_of v", standard]
haftmann@30652
  1938
haftmann@30652
  1939
lemmas left_diff_distrib_number_of [simp] =
haftmann@30652
  1940
  left_diff_distrib [of _ _ "number_of v", standard]
haftmann@30652
  1941
haftmann@30652
  1942
lemmas right_diff_distrib_number_of [simp] =
haftmann@30652
  1943
  right_diff_distrib [of "number_of v", standard]
haftmann@30652
  1944
haftmann@30652
  1945
text{*These are actually for fields, like real: but where else to put them?*}
haftmann@30652
  1946
blanchet@35828
  1947
lemmas zero_less_divide_iff_number_of [simp, no_atp] =
haftmann@30652
  1948
  zero_less_divide_iff [of "number_of w", standard]
haftmann@30652
  1949
blanchet@35828
  1950
lemmas divide_less_0_iff_number_of [simp, no_atp] =
haftmann@30652
  1951
  divide_less_0_iff [of "number_of w", standard]
haftmann@30652
  1952
blanchet@35828
  1953
lemmas zero_le_divide_iff_number_of [simp, no_atp] =
haftmann@30652
  1954
  zero_le_divide_iff [of "number_of w", standard]
haftmann@30652
  1955
blanchet@35828
  1956
lemmas divide_le_0_iff_number_of [simp, no_atp] =
haftmann@30652
  1957
  divide_le_0_iff [of "number_of w", standard]
haftmann@30652
  1958
haftmann@30652
  1959
haftmann@30652
  1960
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
haftmann@30652
  1961
  strange, but then other simprocs simplify the quotient.*}
haftmann@30652
  1962
haftmann@30652
  1963
lemmas inverse_eq_divide_number_of [simp] =
haftmann@30652
  1964
  inverse_eq_divide [of "number_of w", standard]
haftmann@30652
  1965
haftmann@30652
  1966
text {*These laws simplify inequalities, moving unary minus from a term
haftmann@30652
  1967
into the literal.*}
haftmann@30652
  1968
blanchet@35828
  1969
lemmas less_minus_iff_number_of [simp, no_atp] =
haftmann@30652
  1970
  less_minus_iff [of "number_of v", standard]
haftmann@30652
  1971
blanchet@35828
  1972
lemmas le_minus_iff_number_of [simp, no_atp] =
haftmann@30652
  1973
  le_minus_iff [of "number_of v", standard]
haftmann@30652
  1974
blanchet@35828
  1975
lemmas equation_minus_iff_number_of [simp, no_atp] =
haftmann@30652
  1976
  equation_minus_iff [of "number_of v", standard]
haftmann@30652
  1977
blanchet@35828
  1978
lemmas minus_less_iff_number_of [simp, no_atp] =
haftmann@30652
  1979
  minus_less_iff [of _ "number_of v", standard]
haftmann@30652
  1980
blanchet@35828
  1981
lemmas minus_le_iff_number_of [simp, no_atp] =
haftmann@30652
  1982
  minus_le_iff [of _ "number_of v", standard]
haftmann@30652
  1983
blanchet@35828
  1984
lemmas minus_equation_iff_number_of [simp, no_atp] =
haftmann@30652
  1985
  minus_equation_iff [of _ "number_of v", standard]
haftmann@30652
  1986
haftmann@30652
  1987
haftmann@30652
  1988
text{*To Simplify Inequalities Where One Side is the Constant 1*}
haftmann@30652
  1989
blanchet@35828
  1990
lemma less_minus_iff_1 [simp,no_atp]:
haftmann@35028
  1991
  fixes b::"'b::{linordered_idom,number_ring}"
haftmann@30652
  1992
  shows "(1 < - b) = (b < -1)"
haftmann@30652
  1993
by auto
haftmann@30652
  1994
blanchet@35828
  1995
lemma le_minus_iff_1 [simp,no_atp]:
haftmann@35028
  1996
  fixes b::"'b::{linordered_idom,number_ring}"
haftmann@30652
  1997
  shows "(1 \<le> - b) = (b \<le> -1)"
haftmann@30652
  1998
by auto
haftmann@30652
  1999
blanchet@35828
  2000
lemma equation_minus_iff_1 [simp,no_atp]:
haftmann@30652
  2001
  fixes b::"'b::number_ring"
haftmann@30652
  2002
  shows "(1 = - b) = (b = -1)"
haftmann@30652
  2003
by (subst equation_minus_iff, auto)
haftmann@30652
  2004
blanchet@35828
  2005
lemma minus_less_iff_1 [simp,no_atp]:
haftmann@35028
  2006
  fixes a::"'b::{linordered_idom,number_ring}"
haftmann@30652
  2007
  shows "(- a < 1) = (-1 < a)"
haftmann@30652
  2008
by auto
haftmann@30652
  2009
blanchet@35828
  2010
lemma minus_le_iff_1 [simp,no_atp]:
haftmann@35028
  2011
  fixes a::"'b::{linordered_idom,number_ring}"
haftmann@30652
  2012
  shows "(- a \<le> 1) = (-1 \<le> a)"
haftmann@30652
  2013
by auto
haftmann@30652
  2014
blanchet@35828
  2015
lemma minus_equation_iff_1 [simp,no_atp]:
haftmann@30652
  2016
  fixes a::"'b::number_ring"
haftmann@30652
  2017
  shows "(- a = 1) = (a = -1)"
haftmann@30652
  2018
by (subst minus_equation_iff, auto)
haftmann@30652
  2019
haftmann@30652
  2020
haftmann@30652
  2021
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
haftmann@30652
  2022
blanchet@35828
  2023
lemmas mult_less_cancel_left_number_of [simp, no_atp] =
haftmann@30652
  2024
  mult_less_cancel_left [of "number_of v", standard]
haftmann@30652
  2025
blanchet@35828
  2026
lemmas mult_less_cancel_right_number_of [simp, no_atp] =
haftmann@30652
  2027
  mult_less_cancel_right [of _ "number_of v", standard]
haftmann@30652
  2028
blanchet@35828
  2029
lemmas mult_le_cancel_left_number_of [simp, no_atp] =
haftmann@30652
  2030
  mult_le_cancel_left [of "number_of v", standard]
haftmann@30652
  2031
blanchet@35828
  2032
lemmas mult_le_cancel_right_number_of [simp, no_atp] =
haftmann@30652
  2033
  mult_le_cancel_right [of _ "number_of v", standard]
haftmann@30652
  2034
haftmann@30652
  2035
haftmann@30652
  2036
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
haftmann@30652
  2037
haftmann@30652
  2038
lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
haftmann@30652
  2039
lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
haftmann@30652
  2040
lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
haftmann@30652
  2041
lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
haftmann@30652
  2042
lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
haftmann@30652
  2043
lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
haftmann@30652
  2044
haftmann@30652
  2045
haftmann@30652
  2046
subsubsection{*Optional Simplification Rules Involving Constants*}
haftmann@30652
  2047
haftmann@30652
  2048
text{*Simplify quotients that are compared with a literal constant.*}
haftmann@30652
  2049
haftmann@30652
  2050
lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
haftmann@30652
  2051
lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
haftmann@30652
  2052
lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
haftmann@30652
  2053
lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
haftmann@30652
  2054
lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
haftmann@30652
  2055
lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
haftmann@30652
  2056
haftmann@30652
  2057
haftmann@30652
  2058
text{*Not good as automatic simprules because they cause case splits.*}
haftmann@30652
  2059
lemmas divide_const_simps =
haftmann@30652
  2060
  le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
haftmann@30652
  2061
  divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
haftmann@30652
  2062
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  2063
haftmann@30652
  2064
text{*Division By @{text "-1"}*}
haftmann@30652
  2065
haftmann@30652
  2066
lemma divide_minus1 [simp]:
haftmann@36409
  2067
     "x/-1 = -(x::'a::{field_inverse_zero, number_ring})"
haftmann@30652
  2068
by simp
haftmann@30652
  2069
haftmann@30652
  2070
lemma minus1_divide [simp]:
haftmann@36409
  2071
     "-1 / (x::'a::{field_inverse_zero, number_ring}) = - (1/x)"
huffman@35216
  2072
by (simp add: divide_inverse)
haftmann@30652
  2073
haftmann@30652
  2074
lemma half_gt_zero_iff:
haftmann@36409
  2075
     "(0 < r/2) = (0 < (r::'a::{linordered_field_inverse_zero,number_ring}))"
haftmann@30652
  2076
by auto
haftmann@30652
  2077
haftmann@30652
  2078
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]
haftmann@30652
  2079
haftmann@36719
  2080
lemma divide_Numeral1:
haftmann@36719
  2081
  "(x::'a::{field, number_ring}) / Numeral1 = x"
haftmann@36719
  2082
  by simp
haftmann@36719
  2083
haftmann@36719
  2084
lemma divide_Numeral0:
haftmann@36719
  2085
  "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0"
haftmann@36719
  2086
  by simp
haftmann@36719
  2087
haftmann@30652
  2088
haftmann@33320
  2089
subsection {* The divides relation *}
haftmann@33320
  2090
nipkow@33657
  2091
lemma zdvd_antisym_nonneg:
nipkow@33657
  2092
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320
  2093
  apply (simp add: dvd_def, auto)
nipkow@33657
  2094
  apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  2095
  done
haftmann@33320
  2096
nipkow@33657
  2097
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
haftmann@33320
  2098
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657
  2099
proof cases
nipkow@33657
  2100
  assume "a = 0" with assms show ?thesis by simp
nipkow@33657
  2101
next
nipkow@33657
  2102
  assume "a \<noteq> 0"
haftmann@33320
  2103
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
haftmann@33320
  2104
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
haftmann@33320
  2105
  from k k' have "a = a*k*k'" by simp
haftmann@33320
  2106
  with mult_cancel_left1[where c="a" and b="k*k'"]
haftmann@33320
  2107
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
haftmann@33320
  2108
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320
  2109
  thus ?thesis using k k' by auto
haftmann@33320
  2110
qed
haftmann@33320
  2111
haftmann@33320
  2112
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
haftmann@33320
  2113
  apply (subgoal_tac "m = n + (m - n)")
haftmann@33320
  2114
   apply (erule ssubst)
haftmann@33320
  2115
   apply (blast intro: dvd_add, simp)
haftmann@33320
  2116
  done
haftmann@33320
  2117
haftmann@33320
  2118
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@33320
  2119
apply (rule iffI)
haftmann@33320
  2120
 apply (erule_tac [2] dvd_add)
haftmann@33320
  2121
 apply (subgoal_tac "n = (n + k * m) - k * m")
haftmann@33320
  2122
  apply (erule ssubst)
haftmann@33320
  2123
  apply (erule dvd_diff)
haftmann@33320
  2124
  apply(simp_all)
haftmann@33320
  2125
done
haftmann@33320
  2126
haftmann@33320
  2127
lemma dvd_imp_le_int:
haftmann@33320
  2128
  fixes d i :: int
haftmann@33320
  2129
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  2130
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  2131
proof -
haftmann@33320
  2132
  from `d dvd i` obtain k where "i = d * k" ..
haftmann@33320
  2133
  with `i \<noteq> 0` have "k \<noteq> 0" by auto
haftmann@33320
  2134
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  2135
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
haftmann@33320
  2136
  with `i = d * k` show ?thesis by (simp add: abs_mult)
haftmann@33320
  2137
qed
haftmann@33320
  2138
haftmann@33320
  2139
lemma zdvd_not_zless:
haftmann@33320
  2140
  fixes m n :: int
haftmann@33320
  2141
  assumes "0 < m" and "m < n"
haftmann@33320
  2142
  shows "\<not> n dvd m"
haftmann@33320
  2143
proof
haftmann@33320
  2144
  from assms have "0 < n" by auto
haftmann@33320
  2145
  assume "n dvd m" then obtain k where k: "m = n * k" ..
haftmann@33320
  2146
  with `0 < m` have "0 < n * k" by auto
haftmann@33320
  2147
  with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
haftmann@33320
  2148
  with k `0 < n` `m < n` have "n * k < n * 1" by simp
haftmann@33320
  2149
  with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
haftmann@33320
  2150
qed
haftmann@33320
  2151
haftmann@33320
  2152
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320
  2153
  shows "m dvd n"
haftmann@33320
  2154
proof-
haftmann@33320
  2155
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320
  2156
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@33320
  2157
    with h have False by (simp add: mult_assoc)}
haftmann@33320
  2158
  hence "n = m * h" by blast
haftmann@33320
  2159
  thus ?thesis by simp
haftmann@33320
  2160
qed
haftmann@33320
  2161
haftmann@33320
  2162
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320
  2163
proof -
haftmann@33320
  2164
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320
  2165
  proof -
haftmann@33320
  2166
    fix k
haftmann@33320
  2167
    assume A: "int y = int x * k"
wenzelm@42676
  2168
    then show "x dvd y"
wenzelm@42676
  2169
    proof (cases k)
wenzelm@42676
  2170
      case (nonneg n)
wenzelm@42676
  2171
      with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
haftmann@33320
  2172
      then show ?thesis ..
haftmann@33320
  2173
    next
wenzelm@42676
  2174
      case (neg n)
wenzelm@42676
  2175
      with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320
  2176
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320
  2177
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320
  2178
      finally have "- int (x * Suc n) = int y" ..
haftmann@33320
  2179
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320
  2180
    qed
haftmann@33320
  2181
  qed
haftmann@33320
  2182
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  2183
qed
haftmann@33320
  2184
wenzelm@42676
  2185
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320
  2186
proof
haftmann@33320
  2187
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320
  2188
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320
  2189
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676
  2190
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320
  2191
next
haftmann@33320
  2192
  assume "\<bar>x\<bar>=1"
haftmann@33320
  2193
  then have "x = 1 \<or> x = -1" by auto
haftmann@33320
  2194
  then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320
  2195
qed
haftmann@33320
  2196
haftmann@33320
  2197
lemma zdvd_mult_cancel1: 
haftmann@33320
  2198
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320
  2199
proof
haftmann@33320
  2200
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@42676
  2201
    by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320
  2202
next
haftmann@33320
  2203
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320
  2204
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320
  2205
qed
haftmann@33320
  2206
haftmann@33320
  2207
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@33320
  2208
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  2209
haftmann@33320
  2210
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@33320
  2211
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  2212
haftmann@33320
  2213
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320
  2214
  by (auto simp add: dvd_int_iff)
haftmann@33320
  2215
haftmann@33341
  2216
lemma eq_nat_nat_iff:
haftmann@33341
  2217
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  2218
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  2219
haftmann@33341
  2220
lemma nat_power_eq:
haftmann@33341
  2221
  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  2222
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  2223
haftmann@33320
  2224
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676
  2225
  apply (cases n)
haftmann@33320
  2226
  apply (auto simp add: dvd_int_iff)
wenzelm@42676
  2227
  apply (cases z)
haftmann@33320
  2228
  apply (auto simp add: dvd_imp_le)
haftmann@33320
  2229
  done
haftmann@33320
  2230
haftmann@36749
  2231
lemma zdvd_period:
haftmann@36749
  2232
  fixes a d :: int
haftmann@36749
  2233
  assumes "a dvd d"
haftmann@36749
  2234
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749
  2235
proof -
haftmann@36749
  2236
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  2237
  show ?thesis
wenzelm@42676
  2238
  proof
haftmann@36749
  2239
    assume "a dvd (x + t)"
haftmann@36749
  2240
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  2241
    then have "x = a * l - t" by simp
haftmann@36749
  2242
    with `d = a * k` show "a dvd x + c * d + t" by simp
haftmann@36749
  2243
  next
haftmann@36749
  2244
    assume "a dvd x + c * d + t"
haftmann@36749
  2245
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  2246
    then have "x = a * l - c * d - t" by simp
haftmann@36749
  2247
    with `d = a * k` show "a dvd (x + t)" by simp
haftmann@36749
  2248
  qed
haftmann@36749
  2249
qed
haftmann@36749
  2250
haftmann@33320
  2251
haftmann@25919
  2252
subsection {* Configuration of the code generator *}
haftmann@25919
  2253
haftmann@26507
  2254
code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"
haftmann@26507
  2255
haftmann@28562
  2256
lemmas pred_succ_numeral_code [code] =
haftmann@26507
  2257
  pred_bin_simps succ_bin_simps
haftmann@26507
  2258
haftmann@28562
  2259
lemmas plus_numeral_code [code] =
haftmann@26507
  2260
  add_bin_simps
haftmann@26507
  2261
  arith_extra_simps(1) [where 'a = int]
haftmann@26507
  2262
haftmann@28562
  2263
lemmas minus_numeral_code [code] =
haftmann@26507
  2264
  minus_bin_simps
haftmann@26507
  2265
  arith_extra_simps(2) [where 'a = int]
haftmann@26507
  2266
  arith_extra_simps(5) [where 'a = int]
haftmann@26507
  2267
haftmann@28562
  2268
lemmas times_numeral_code [code] =
haftmann@26507
  2269
  mult_bin_simps
haftmann@26507
  2270
  arith_extra_simps(4) [where 'a = int]
haftmann@26507
  2271
haftmann@38857
  2272
instantiation int :: equal
haftmann@26507
  2273
begin
haftmann@26507
  2274
haftmann@37767
  2275
definition
haftmann@38857
  2276
  "HOL.equal k l \<longleftrightarrow> k - l = (0\<Colon>int)"
haftmann@38857
  2277
haftmann@38857
  2278
instance by default (simp add: equal_int_def)
haftmann@26507
  2279
haftmann@26507
  2280
end
haftmann@26507
  2281
haftmann@28562
  2282
lemma eq_number_of_int_code [code]:
haftmann@38857
  2283
  "HOL.equal (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> HOL.equal k l"
haftmann@38857
  2284
  unfolding equal_int_def number_of_is_id ..
haftmann@26507
  2285
haftmann@28562
  2286
lemma eq_int_code [code]:
haftmann@38857
  2287
  "HOL.equal Int.Pls Int.Pls \<longleftrightarrow> True"
haftmann@38857
  2288
  "HOL.equal Int.Pls Int.Min \<longleftrightarrow> False"
haftmann@38857
  2289
  "HOL.equal Int.Pls (Int.Bit0 k2) \<longleftrightarrow> HOL.equal Int.Pls k2"
haftmann@38857
  2290
  "HOL.equal Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@38857
  2291
  "HOL.equal Int.Min Int.Pls \<longleftrightarrow> False"
haftmann@38857
  2292
  "HOL.equal Int.Min Int.Min \<longleftrightarrow> True"
haftmann@38857
  2293
  "HOL.equal Int.Min (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@38857
  2294
  "HOL.equal Int.Min (Int.Bit1 k2) \<longleftrightarrow> HOL.equal Int.Min k2"
haftmann@38857
  2295
  "HOL.equal (Int.Bit0 k1) Int.Pls \<longleftrightarrow> HOL.equal k1 Int.Pls"
haftmann@38857
  2296
  "HOL.equal (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False"
haftmann@38857
  2297
  "HOL.equal (Int.Bit0 k1) Int.Min \<longleftrightarrow> False"
haftmann@38857
  2298
  "HOL.equal (Int.Bit1 k1) Int.Min \<longleftrightarrow> HOL.equal k1 Int.Min"
haftmann@38857
  2299
  "HOL.equal (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> HOL.equal k1 k2"
haftmann@38857
  2300
  "HOL.equal (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@38857
  2301
  "HOL.equal (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@38857
  2302
  "HOL.equal (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> HOL.equal k1 k2"
haftmann@38857
  2303
  unfolding equal_eq by simp_all
haftmann@25919
  2304
haftmann@28351
  2305
lemma eq_int_refl [code nbe]:
haftmann@38857
  2306
  "HOL.equal (k::int) k \<longleftrightarrow> True"
haftmann@38857
  2307
  by (rule equal_refl)
haftmann@28351
  2308
haftmann@28562
  2309
lemma less_eq_number_of_int_code [code]:
haftmann@26507
  2310
  "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
haftmann@26507
  2311
  unfolding number_of_is_id ..
haftmann@26507
  2312
haftmann@28562
  2313
lemma less_eq_int_code [code]:
haftmann@26507
  2314
  "Int.Pls \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507
  2315
  "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
haftmann@26507
  2316
  "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  2317
  "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  2318
  "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507
  2319
  "Int.Min \<le> Int.Min \<longleftrightarrow> True"
haftmann@26507
  2320
  "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  2321
  "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"
haftmann@26507
  2322
  "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
haftmann@26507
  2323
  "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  2324
  "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  2325
  "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  2326
  "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  2327
  "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  2328
  "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  2329
  "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
huffman@28958
  2330
  by simp_all
haftmann@26507
  2331
haftmann@28562
  2332
lemma less_number_of_int_code [code]:
haftmann@26507
  2333
  "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
haftmann@26507
  2334
  unfolding number_of_is_id ..
haftmann@26507
  2335
haftmann@28562
  2336
lemma less_int_code [code]:
haftmann@26507
  2337
  "Int.Pls < Int.Pls \<longleftrightarrow> False"
haftmann@26507
  2338
  "Int.Pls < Int.Min \<longleftrightarrow> False"
haftmann@26507
  2339
  "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"
haftmann@26507
  2340
  "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  2341
  "Int.Min < Int.Pls \<longleftrightarrow> True"
haftmann@26507
  2342
  "Int.Min < Int.Min \<longleftrightarrow> False"
haftmann@26507
  2343
  "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  2344
  "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  2345
  "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  2346
  "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  2347
  "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  2348
  "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"
haftmann@26507
  2349
  "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  2350
  "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  2351
  "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  2352
  "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"
huffman@28958
  2353
  by simp_all
haftmann@25919
  2354
haftmann@25919
  2355
definition
haftmann@25919
  2356
  nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
haftmann@25919
  2357
  "nat_aux i n = nat i + n"
haftmann@25919
  2358
haftmann@25919
  2359
lemma [code]:
haftmann@25919
  2360
  "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
haftmann@25919
  2361
  by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
haftmann@25919
  2362
    dest: zless_imp_add1_zle)
haftmann@25919
  2363
haftmann@25919
  2364
lemma [code]: "nat i = nat_aux i 0"
haftmann@25919
  2365
  by (simp add: nat_aux_def)
haftmann@25919
  2366
wenzelm@36176
  2367
hide_const (open) nat_aux
haftmann@25928
  2368
haftmann@32069
  2369
lemma zero_is_num_zero [code, code_unfold_post]:
haftmann@25919
  2370
  "(0\<Colon>int) = Numeral0" 
haftmann@25919
  2371
  by simp
haftmann@25919
  2372
haftmann@32069
  2373
lemma one_is_num_one [code, code_unfold_post]:
haftmann@25919
  2374
  "(1\<Colon>int) = Numeral1" 
haftmann@25961
  2375
  by simp
haftmann@25919
  2376
haftmann@25919
  2377
code_modulename SML
haftmann@33364
  2378
  Int Arith
haftmann@25919
  2379
haftmann@25919
  2380
code_modulename OCaml
haftmann@33364
  2381
  Int Arith
haftmann@25919
  2382
haftmann@25919
  2383
code_modulename Haskell
haftmann@33364
  2384
  Int Arith
haftmann@25919
  2385
haftmann@25919
  2386
quickcheck_params [default_type = int]
haftmann@25919
  2387
wenzelm@36176
  2388
hide_const (open) Pls Min Bit0 Bit1 succ pred
haftmann@25919
  2389
haftmann@25919
  2390
haftmann@25919
  2391
subsection {* Legacy theorems *}
haftmann@25919
  2392
haftmann@25919
  2393
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  2394
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  2395
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  2396
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
haftmann@25919
  2397
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  2398
lemmas zless_int = of_nat_less_iff [where 'a=int]
haftmann@25919
  2399
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
haftmann@25919
  2400
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  2401
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
haftmann@25919
  2402
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  2403
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  2404
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  2405
lemmas int_Suc = of_nat_Suc [where 'a=int]
haftmann@25919
  2406
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
haftmann@25919
  2407
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  2408
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
haftmann@30960
  2409
haftmann@31015
  2410
lemma zpower_zpower:
haftmann@31015
  2411
  "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@31015
  2412
  by (rule power_mult [symmetric])
haftmann@31015
  2413
haftmann@31015
  2414
lemma int_power:
haftmann@31015
  2415
  "int (m ^ n) = int m ^ n"
haftmann@31015
  2416
  by (rule of_nat_power)
haftmann@31015
  2417
haftmann@31015
  2418
lemmas zpower_int = int_power [symmetric]
haftmann@31015
  2419
haftmann@25919
  2420
end