src/HOL/Int.thy
author wenzelm
Wed May 04 15:37:39 2011 +0200 (2011-05-04)
changeset 42676 8724f20bf69c
parent 42411 ff997038e8eb
child 43531 cc46a678faaf
permissions -rw-r--r--
proper case_names for int_cases, int_of_nat_induct;
tuned some proofs, eliminating (cases, auto) anti-pattern;
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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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lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})"
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by (induct m) (simp_all add: Zero_int_def One_int_def add)
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subsection {* The @{text "\<le>"} Ordering *}
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lemma le:
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  "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
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by (force simp add: le_int_def)
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lemma less:
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  "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
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by (simp add: less_int_def le order_less_le)
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instance int :: linorder
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proof
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  fix i j k :: int
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  show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
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    by (cases i, cases j) (simp add: le)
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  show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
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    by (auto simp add: less_int_def dest: antisym) 
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  show "i \<le> i"
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    by (cases i) (simp add: le)
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  show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
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    by (cases i, cases j, cases k) (simp add: le)
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  show "i \<le> j \<or> j \<le> i"
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    by (cases i, cases j) (simp add: le linorder_linear)
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qed
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instantiation int :: distrib_lattice
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begin
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definition
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  "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
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definition
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  "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
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instance
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  by intro_classes
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    (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
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end
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instance int :: 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 ==> of_nat k * i < of_nat 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 = of_nat n"
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apply (cases k)
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apply (auto simp add: le add int_def Zero_int_def)
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apply (rule_tac x="x-y" in exI, simp)
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done
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lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"
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apply (cases k)
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apply (simp add: less int_def Zero_int_def)
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apply (rule_tac x="x-y" in exI, simp)
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done
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lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
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apply (drule zero_less_imp_eq_int)
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apply (auto simp add: zmult_zless_mono2_lemma)
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done
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text{*The integers form an ordered integral domain*}
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instance int :: 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 + of_nat (Suc n))"
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apply (cases z, cases w)
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apply (auto simp add: less add int_def)
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apply (rename_tac a b c d) 
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apply (rule_tac x="a+d - Suc(c+b)" in exI) 
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apply arith
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done
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lemmas int_distrib =
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  left_distrib [of "z1::int" "z2" "w", standard]
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  right_distrib [of "w::int" "z1" "z2", standard]
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  left_diff_distrib [of "z1::int" "z2" "w", standard]
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  right_diff_distrib [of "w::int" "z1" "z2", standard]
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subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
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context ring_1
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begin
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definition 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)
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   313
apply (simp add: algebra_simps of_int mult of_nat_mult)
haftmann@25919
   314
done
haftmann@25919
   315
haftmann@25919
   316
text{*Collapse nested embeddings*}
haftmann@25919
   317
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
nipkow@29667
   318
by (induct n) auto
haftmann@25919
   319
haftmann@31015
   320
lemma of_int_power:
haftmann@31015
   321
  "of_int (z ^ n) = of_int z ^ n"
haftmann@31015
   322
  by (induct n) simp_all
haftmann@31015
   323
haftmann@25919
   324
end
haftmann@25919
   325
haftmann@25919
   326
text{*Class for unital rings with characteristic zero.
haftmann@25919
   327
 Includes non-ordered rings like the complex numbers.*}
haftmann@25919
   328
class ring_char_0 = ring_1 + semiring_char_0
haftmann@25919
   329
begin
haftmann@25919
   330
haftmann@25919
   331
lemma of_int_eq_iff [simp]:
haftmann@25919
   332
   "of_int w = of_int z \<longleftrightarrow> w = z"
wenzelm@42676
   333
apply (cases w, cases z)
wenzelm@42676
   334
apply (simp add: of_int)
haftmann@25919
   335
apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
haftmann@25919
   336
apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
haftmann@25919
   337
done
haftmann@25919
   338
haftmann@25919
   339
text{*Special cases where either operand is zero*}
haftmann@36424
   340
lemma of_int_eq_0_iff [simp]:
haftmann@36424
   341
  "of_int z = 0 \<longleftrightarrow> z = 0"
haftmann@36424
   342
  using of_int_eq_iff [of z 0] by simp
haftmann@36424
   343
haftmann@36424
   344
lemma of_int_0_eq_iff [simp]:
haftmann@36424
   345
  "0 = of_int z \<longleftrightarrow> z = 0"
haftmann@36424
   346
  using of_int_eq_iff [of 0 z] by simp
haftmann@25919
   347
haftmann@25919
   348
end
haftmann@25919
   349
haftmann@36424
   350
context linordered_idom
haftmann@36424
   351
begin
haftmann@36424
   352
haftmann@35028
   353
text{*Every @{text linordered_idom} has characteristic zero.*}
haftmann@36424
   354
subclass ring_char_0 ..
haftmann@36424
   355
haftmann@36424
   356
lemma of_int_le_iff [simp]:
haftmann@36424
   357
  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
wenzelm@42676
   358
  by (cases w, cases z)
wenzelm@42676
   359
    (simp add: of_int le minus algebra_simps of_nat_add [symmetric] del: of_nat_add)
haftmann@36424
   360
haftmann@36424
   361
lemma of_int_less_iff [simp]:
haftmann@36424
   362
  "of_int w < of_int z \<longleftrightarrow> w < z"
haftmann@36424
   363
  by (simp add: less_le order_less_le)
haftmann@36424
   364
haftmann@36424
   365
lemma of_int_0_le_iff [simp]:
haftmann@36424
   366
  "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
haftmann@36424
   367
  using of_int_le_iff [of 0 z] by simp
haftmann@36424
   368
haftmann@36424
   369
lemma of_int_le_0_iff [simp]:
haftmann@36424
   370
  "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
haftmann@36424
   371
  using of_int_le_iff [of z 0] by simp
haftmann@36424
   372
haftmann@36424
   373
lemma of_int_0_less_iff [simp]:
haftmann@36424
   374
  "0 < of_int z \<longleftrightarrow> 0 < z"
haftmann@36424
   375
  using of_int_less_iff [of 0 z] by simp
haftmann@36424
   376
haftmann@36424
   377
lemma of_int_less_0_iff [simp]:
haftmann@36424
   378
  "of_int z < 0 \<longleftrightarrow> z < 0"
haftmann@36424
   379
  using of_int_less_iff [of z 0] by simp
haftmann@36424
   380
haftmann@36424
   381
end
haftmann@25919
   382
haftmann@25919
   383
lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919
   384
proof
haftmann@25919
   385
  fix z show "of_int z = id z"
haftmann@25919
   386
    by (cases z) (simp add: of_int add minus int_def diff_minus)
haftmann@25919
   387
qed
haftmann@25919
   388
haftmann@25919
   389
haftmann@25919
   390
subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
haftmann@25919
   391
haftmann@37767
   392
definition nat :: "int \<Rightarrow> nat" where
haftmann@39910
   393
  "nat z = the_elem (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
haftmann@25919
   394
haftmann@25919
   395
lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
haftmann@25919
   396
proof -
haftmann@25919
   397
  have "(\<lambda>(x,y). {x-y}) respects intrel"
haftmann@40819
   398
    by (auto simp add: congruent_def)
haftmann@25919
   399
  thus ?thesis
haftmann@25919
   400
    by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
haftmann@25919
   401
qed
haftmann@25919
   402
haftmann@25919
   403
lemma nat_int [simp]: "nat (of_nat n) = n"
haftmann@25919
   404
by (simp add: nat int_def)
haftmann@25919
   405
huffman@35216
   406
(* FIXME: duplicates nat_0 *)
haftmann@25919
   407
lemma nat_zero [simp]: "nat 0 = 0"
haftmann@25919
   408
by (simp add: Zero_int_def nat)
haftmann@25919
   409
haftmann@25919
   410
lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
wenzelm@42676
   411
by (cases z) (simp add: nat le int_def Zero_int_def)
haftmann@25919
   412
haftmann@25919
   413
corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
haftmann@25919
   414
by simp
haftmann@25919
   415
haftmann@25919
   416
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
wenzelm@42676
   417
by (cases z) (simp add: nat le Zero_int_def)
haftmann@25919
   418
haftmann@25919
   419
lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
haftmann@25919
   420
apply (cases w, cases z) 
haftmann@25919
   421
apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
haftmann@25919
   422
done
haftmann@25919
   423
haftmann@25919
   424
text{*An alternative condition is @{term "0 \<le> w"} *}
haftmann@25919
   425
corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
haftmann@25919
   426
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919
   427
haftmann@25919
   428
corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
haftmann@25919
   429
by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919
   430
haftmann@25919
   431
lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
haftmann@25919
   432
apply (cases w, cases z) 
haftmann@25919
   433
apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
haftmann@25919
   434
done
haftmann@25919
   435
haftmann@25919
   436
lemma nonneg_eq_int:
haftmann@25919
   437
  fixes z :: int
haftmann@25919
   438
  assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
haftmann@25919
   439
  shows P
haftmann@25919
   440
  using assms by (blast dest: nat_0_le sym)
haftmann@25919
   441
haftmann@25919
   442
lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
wenzelm@42676
   443
by (cases w) (simp add: nat le int_def Zero_int_def, arith)
haftmann@25919
   444
haftmann@25919
   445
corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
haftmann@25919
   446
by (simp only: eq_commute [of m] nat_eq_iff)
haftmann@25919
   447
haftmann@25919
   448
lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
haftmann@25919
   449
apply (cases w)
nipkow@29700
   450
apply (simp add: nat le int_def Zero_int_def linorder_not_le[symmetric], arith)
haftmann@25919
   451
done
haftmann@25919
   452
nipkow@29700
   453
lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
nipkow@29700
   454
by(simp add: nat_eq_iff) arith
nipkow@29700
   455
haftmann@25919
   456
lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919
   457
by (auto simp add: nat_eq_iff2)
haftmann@25919
   458
haftmann@25919
   459
lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919
   460
by (insert zless_nat_conj [of 0], auto)
haftmann@25919
   461
haftmann@25919
   462
lemma nat_add_distrib:
haftmann@25919
   463
     "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
wenzelm@42676
   464
by (cases z, cases z') (simp add: nat add le Zero_int_def)
haftmann@25919
   465
haftmann@25919
   466
lemma nat_diff_distrib:
haftmann@25919
   467
     "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
wenzelm@42676
   468
by (cases z, cases z')
wenzelm@42676
   469
  (simp add: nat add minus diff_minus le Zero_int_def)
haftmann@25919
   470
haftmann@25919
   471
lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
haftmann@25919
   472
by (simp add: int_def minus nat Zero_int_def) 
haftmann@25919
   473
haftmann@25919
   474
lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
wenzelm@42676
   475
by (cases z) (simp add: nat less int_def, arith)
haftmann@25919
   476
haftmann@25919
   477
context ring_1
haftmann@25919
   478
begin
haftmann@25919
   479
haftmann@25919
   480
lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
haftmann@25919
   481
  by (cases z rule: eq_Abs_Integ)
haftmann@25919
   482
   (simp add: nat le of_int Zero_int_def of_nat_diff)
haftmann@25919
   483
haftmann@25919
   484
end
haftmann@25919
   485
krauss@29779
   486
text {* For termination proofs: *}
krauss@29779
   487
lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
krauss@29779
   488
haftmann@25919
   489
haftmann@25919
   490
subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919
   491
haftmann@25919
   492
lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
haftmann@25919
   493
by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919
   494
haftmann@25919
   495
lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
haftmann@25919
   496
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919
   497
haftmann@25919
   498
lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
haftmann@25919
   499
by (simp add: minus_le_iff)
haftmann@25919
   500
haftmann@25919
   501
lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
haftmann@25919
   502
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919
   503
haftmann@25919
   504
lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
haftmann@25919
   505
by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919
   506
haftmann@25919
   507
lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   508
by (simp add: int_def le minus Zero_int_def)
haftmann@25919
   509
haftmann@25919
   510
lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
haftmann@25919
   511
by (simp add: linorder_not_less)
haftmann@25919
   512
haftmann@25919
   513
lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
haftmann@25919
   514
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919
   515
haftmann@25919
   516
lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
haftmann@25919
   517
proof -
haftmann@25919
   518
  have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919
   519
    by (simp only: le_diff_eq add_0_left)
haftmann@25919
   520
  also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919
   521
    by (auto elim: zero_le_imp_eq_int)
haftmann@25919
   522
  also have "\<dots> = (\<exists>n. z = w + of_nat n)"
nipkow@29667
   523
    by (simp only: algebra_simps)
haftmann@25919
   524
  finally show ?thesis .
haftmann@25919
   525
qed
haftmann@25919
   526
haftmann@25919
   527
lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
haftmann@25919
   528
by simp
haftmann@25919
   529
haftmann@25919
   530
lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
haftmann@25919
   531
by simp
haftmann@25919
   532
haftmann@25919
   533
text{*This version is proved for all ordered rings, not just integers!
haftmann@25919
   534
      It is proved here because attribute @{text arith_split} is not available
haftmann@35050
   535
      in theory @{text Rings}.
haftmann@25919
   536
      But is it really better than just rewriting with @{text abs_if}?*}
blanchet@35828
   537
lemma abs_split [arith_split,no_atp]:
haftmann@35028
   538
     "P(abs(a::'a::linordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919
   539
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919
   540
haftmann@25919
   541
lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
haftmann@25919
   542
apply (cases x)
haftmann@25919
   543
apply (auto simp add: le minus Zero_int_def int_def order_less_le)
haftmann@25919
   544
apply (rule_tac x="y - Suc x" in exI, arith)
haftmann@25919
   545
done
haftmann@25919
   546
haftmann@25919
   547
haftmann@25919
   548
subsection {* Cases and induction *}
haftmann@25919
   549
haftmann@25919
   550
text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919
   551
whether an integer is negative or not.*}
haftmann@25919
   552
wenzelm@42676
   553
theorem int_cases [case_names nonneg neg, cases type: int]:
haftmann@25919
   554
  "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"
wenzelm@42676
   555
apply (cases "z < 0")
wenzelm@42676
   556
apply (blast dest!: negD)
haftmann@25919
   557
apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919
   558
apply auto
haftmann@25919
   559
apply (blast dest: nat_0_le [THEN sym])
haftmann@25919
   560
done
haftmann@25919
   561
wenzelm@42676
   562
theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
haftmann@25919
   563
     "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"
wenzelm@42676
   564
  by (cases z) auto
haftmann@25919
   565
haftmann@25919
   566
text{*Contributed by Brian Huffman*}
haftmann@25919
   567
theorem int_diff_cases:
haftmann@25919
   568
  obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
haftmann@25919
   569
apply (cases z rule: eq_Abs_Integ)
haftmann@25919
   570
apply (rule_tac m=x and n=y in diff)
haftmann@37887
   571
apply (simp add: int_def minus add diff_minus)
haftmann@25919
   572
done
haftmann@25919
   573
haftmann@25919
   574
haftmann@25919
   575
subsection {* Binary representation *}
haftmann@25919
   576
haftmann@25919
   577
text {*
haftmann@25919
   578
  This formalization defines binary arithmetic in terms of the integers
haftmann@25919
   579
  rather than using a datatype. This avoids multiple representations (leading
haftmann@25919
   580
  zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
haftmann@25919
   581
  int_of_binary}, for the numerical interpretation.
haftmann@25919
   582
haftmann@25919
   583
  The representation expects that @{text "(m mod 2)"} is 0 or 1,
haftmann@25919
   584
  even if m is negative;
haftmann@25919
   585
  For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
haftmann@25919
   586
  @{text "-5 = (-3)*2 + 1"}.
haftmann@25919
   587
  
haftmann@25919
   588
  This two's complement binary representation derives from the paper 
haftmann@25919
   589
  "An Efficient Representation of Arithmetic for Term Rewriting" by
haftmann@25919
   590
  Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
haftmann@25919
   591
  Springer LNCS 488 (240-251), 1991.
haftmann@25919
   592
*}
haftmann@25919
   593
huffman@28958
   594
subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}
huffman@28958
   595
haftmann@37767
   596
definition Pls :: int where
haftmann@37767
   597
  "Pls = 0"
haftmann@37767
   598
haftmann@37767
   599
definition Min :: int where
haftmann@37767
   600
  "Min = - 1"
haftmann@37767
   601
haftmann@37767
   602
definition Bit0 :: "int \<Rightarrow> int" where
haftmann@37767
   603
  "Bit0 k = k + k"
haftmann@37767
   604
haftmann@37767
   605
definition Bit1 :: "int \<Rightarrow> int" where
haftmann@37767
   606
  "Bit1 k = 1 + k + k"
haftmann@25919
   607
haftmann@29608
   608
class number = -- {* for numeric types: nat, int, real, \dots *}
haftmann@25919
   609
  fixes number_of :: "int \<Rightarrow> 'a"
haftmann@25919
   610
haftmann@25919
   611
use "Tools/numeral.ML"
haftmann@25919
   612
haftmann@25919
   613
syntax
haftmann@25919
   614
  "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
haftmann@25919
   615
haftmann@25919
   616
use "Tools/numeral_syntax.ML"
wenzelm@35123
   617
setup Numeral_Syntax.setup
haftmann@25919
   618
haftmann@25919
   619
abbreviation
haftmann@25919
   620
  "Numeral0 \<equiv> number_of Pls"
haftmann@25919
   621
haftmann@25919
   622
abbreviation
huffman@26086
   623
  "Numeral1 \<equiv> number_of (Bit1 Pls)"
haftmann@25919
   624
haftmann@25919
   625
lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
haftmann@25919
   626
  -- {* Unfold all @{text let}s involving constants *}
haftmann@25919
   627
  unfolding Let_def ..
haftmann@25919
   628
haftmann@37767
   629
definition succ :: "int \<Rightarrow> int" where
haftmann@37767
   630
  "succ k = k + 1"
haftmann@37767
   631
haftmann@37767
   632
definition pred :: "int \<Rightarrow> int" where
haftmann@37767
   633
  "pred k = k - 1"
haftmann@25919
   634
haftmann@25919
   635
lemmas
haftmann@25919
   636
  max_number_of [simp] = max_def
huffman@35216
   637
    [of "number_of u" "number_of v", standard]
haftmann@25919
   638
and
haftmann@25919
   639
  min_number_of [simp] = min_def 
huffman@35216
   640
    [of "number_of u" "number_of v", standard]
haftmann@25919
   641
  -- {* unfolding @{text minx} and @{text max} on numerals *}
haftmann@25919
   642
haftmann@25919
   643
lemmas numeral_simps = 
huffman@26086
   644
  succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
haftmann@25919
   645
haftmann@25919
   646
text {* Removal of leading zeroes *}
haftmann@25919
   647
haftmann@31998
   648
lemma Bit0_Pls [simp, code_post]:
huffman@26086
   649
  "Bit0 Pls = Pls"
haftmann@25919
   650
  unfolding numeral_simps by simp
haftmann@25919
   651
haftmann@31998
   652
lemma Bit1_Min [simp, code_post]:
huffman@26086
   653
  "Bit1 Min = Min"
haftmann@25919
   654
  unfolding numeral_simps by simp
haftmann@25919
   655
huffman@26075
   656
lemmas normalize_bin_simps =
huffman@26086
   657
  Bit0_Pls Bit1_Min
huffman@26075
   658
haftmann@25919
   659
huffman@28958
   660
subsubsection {* Successor and predecessor functions *}
huffman@28958
   661
huffman@28958
   662
text {* Successor *}
huffman@28958
   663
huffman@28958
   664
lemma succ_Pls:
huffman@26086
   665
  "succ Pls = Bit1 Pls"
haftmann@25919
   666
  unfolding numeral_simps by simp
haftmann@25919
   667
huffman@28958
   668
lemma succ_Min:
haftmann@25919
   669
  "succ Min = Pls"
haftmann@25919
   670
  unfolding numeral_simps by simp
haftmann@25919
   671
huffman@28958
   672
lemma succ_Bit0:
huffman@26086
   673
  "succ (Bit0 k) = Bit1 k"
haftmann@25919
   674
  unfolding numeral_simps by simp
haftmann@25919
   675
huffman@28958
   676
lemma succ_Bit1:
huffman@26086
   677
  "succ (Bit1 k) = Bit0 (succ k)"
haftmann@25919
   678
  unfolding numeral_simps by simp
haftmann@25919
   679
huffman@28958
   680
lemmas succ_bin_simps [simp] =
huffman@26086
   681
  succ_Pls succ_Min succ_Bit0 succ_Bit1
huffman@26075
   682
huffman@28958
   683
text {* Predecessor *}
huffman@28958
   684
huffman@28958
   685
lemma pred_Pls:
haftmann@25919
   686
  "pred Pls = Min"
haftmann@25919
   687
  unfolding numeral_simps by simp
haftmann@25919
   688
huffman@28958
   689
lemma pred_Min:
huffman@26086
   690
  "pred Min = Bit0 Min"
haftmann@25919
   691
  unfolding numeral_simps by simp
haftmann@25919
   692
huffman@28958
   693
lemma pred_Bit0:
huffman@26086
   694
  "pred (Bit0 k) = Bit1 (pred k)"
haftmann@25919
   695
  unfolding numeral_simps by simp 
haftmann@25919
   696
huffman@28958
   697
lemma pred_Bit1:
huffman@26086
   698
  "pred (Bit1 k) = Bit0 k"
huffman@26086
   699
  unfolding numeral_simps by simp
huffman@26086
   700
huffman@28958
   701
lemmas pred_bin_simps [simp] =
huffman@26086
   702
  pred_Pls pred_Min pred_Bit0 pred_Bit1
huffman@26075
   703
huffman@28958
   704
huffman@28958
   705
subsubsection {* Binary arithmetic *}
huffman@28958
   706
huffman@28958
   707
text {* Addition *}
huffman@28958
   708
huffman@28958
   709
lemma add_Pls:
huffman@28958
   710
  "Pls + k = k"
huffman@28958
   711
  unfolding numeral_simps by simp
huffman@28958
   712
huffman@28958
   713
lemma add_Min:
huffman@28958
   714
  "Min + k = pred k"
huffman@28958
   715
  unfolding numeral_simps by simp
huffman@28958
   716
huffman@28958
   717
lemma add_Bit0_Bit0:
huffman@28958
   718
  "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"
huffman@28958
   719
  unfolding numeral_simps by simp
huffman@28958
   720
huffman@28958
   721
lemma add_Bit0_Bit1:
huffman@28958
   722
  "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"
huffman@28958
   723
  unfolding numeral_simps by simp
huffman@28958
   724
huffman@28958
   725
lemma add_Bit1_Bit0:
huffman@28958
   726
  "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"
huffman@28958
   727
  unfolding numeral_simps by simp
huffman@28958
   728
huffman@28958
   729
lemma add_Bit1_Bit1:
huffman@28958
   730
  "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"
huffman@28958
   731
  unfolding numeral_simps by simp
huffman@28958
   732
huffman@28958
   733
lemma add_Pls_right:
huffman@28958
   734
  "k + Pls = k"
huffman@28958
   735
  unfolding numeral_simps by simp
huffman@28958
   736
huffman@28958
   737
lemma add_Min_right:
huffman@28958
   738
  "k + Min = pred k"
huffman@28958
   739
  unfolding numeral_simps by simp
huffman@28958
   740
huffman@28958
   741
lemmas add_bin_simps [simp] =
huffman@28958
   742
  add_Pls add_Min add_Pls_right add_Min_right
huffman@28958
   743
  add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1
huffman@28958
   744
huffman@28958
   745
text {* Negation *}
huffman@28958
   746
huffman@28958
   747
lemma minus_Pls:
haftmann@25919
   748
  "- Pls = Pls"
huffman@28958
   749
  unfolding numeral_simps by simp
huffman@28958
   750
huffman@28958
   751
lemma minus_Min:
huffman@26086
   752
  "- Min = Bit1 Pls"
huffman@28958
   753
  unfolding numeral_simps by simp
huffman@28958
   754
huffman@28958
   755
lemma minus_Bit0:
huffman@26086
   756
  "- (Bit0 k) = Bit0 (- k)"
huffman@28958
   757
  unfolding numeral_simps by simp
huffman@28958
   758
huffman@28958
   759
lemma minus_Bit1:
huffman@26086
   760
  "- (Bit1 k) = Bit1 (pred (- k))"
huffman@26086
   761
  unfolding numeral_simps by simp
haftmann@25919
   762
huffman@28958
   763
lemmas minus_bin_simps [simp] =
huffman@26086
   764
  minus_Pls minus_Min minus_Bit0 minus_Bit1
huffman@26075
   765
huffman@28958
   766
text {* Subtraction *}
huffman@28958
   767
huffman@29046
   768
lemma diff_bin_simps [simp]:
huffman@29046
   769
  "k - Pls = k"
huffman@29046
   770
  "k - Min = succ k"
huffman@29046
   771
  "Pls - (Bit0 l) = Bit0 (Pls - l)"
huffman@29046
   772
  "Pls - (Bit1 l) = Bit1 (Min - l)"
huffman@29046
   773
  "Min - (Bit0 l) = Bit1 (Min - l)"
huffman@29046
   774
  "Min - (Bit1 l) = Bit0 (Min - l)"
huffman@28958
   775
  "(Bit0 k) - (Bit0 l) = Bit0 (k - l)"
huffman@28958
   776
  "(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)"
huffman@28958
   777
  "(Bit1 k) - (Bit0 l) = Bit1 (k - l)"
huffman@28958
   778
  "(Bit1 k) - (Bit1 l) = Bit0 (k - l)"
huffman@29046
   779
  unfolding numeral_simps by simp_all
huffman@28958
   780
huffman@28958
   781
text {* Multiplication *}
huffman@28958
   782
huffman@28958
   783
lemma mult_Pls:
huffman@28958
   784
  "Pls * w = Pls"
huffman@26086
   785
  unfolding numeral_simps by simp
haftmann@25919
   786
huffman@28958
   787
lemma mult_Min:
haftmann@25919
   788
  "Min * k = - k"
haftmann@25919
   789
  unfolding numeral_simps by simp
haftmann@25919
   790
huffman@28958
   791
lemma mult_Bit0:
huffman@26086
   792
  "(Bit0 k) * l = Bit0 (k * l)"
huffman@26086
   793
  unfolding numeral_simps int_distrib by simp
haftmann@25919
   794
huffman@28958
   795
lemma mult_Bit1:
huffman@26086
   796
  "(Bit1 k) * l = (Bit0 (k * l)) + l"
huffman@28958
   797
  unfolding numeral_simps int_distrib by simp
huffman@28958
   798
huffman@28958
   799
lemmas mult_bin_simps [simp] =
huffman@26086
   800
  mult_Pls mult_Min mult_Bit0 mult_Bit1
huffman@26075
   801
haftmann@25919
   802
huffman@28958
   803
subsubsection {* Binary comparisons *}
huffman@28958
   804
huffman@28958
   805
text {* Preliminaries *}
huffman@28958
   806
huffman@28958
   807
lemma even_less_0_iff:
haftmann@35028
   808
  "a + a < 0 \<longleftrightarrow> a < (0::'a::linordered_idom)"
huffman@28958
   809
proof -
huffman@28958
   810
  have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)
huffman@28958
   811
  also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958
   812
    by (simp add: mult_less_0_iff zero_less_two 
huffman@28958
   813
                  order_less_not_sym [OF zero_less_two])
huffman@28958
   814
  finally show ?thesis .
huffman@28958
   815
qed
huffman@28958
   816
huffman@28958
   817
lemma le_imp_0_less: 
huffman@28958
   818
  assumes le: "0 \<le> z"
huffman@28958
   819
  shows "(0::int) < 1 + z"
huffman@28958
   820
proof -
huffman@28958
   821
  have "0 \<le> z" by fact
huffman@28958
   822
  also have "... < z + 1" by (rule less_add_one) 
huffman@28958
   823
  also have "... = 1 + z" by (simp add: add_ac)
huffman@28958
   824
  finally show "0 < 1 + z" .
huffman@28958
   825
qed
huffman@28958
   826
huffman@28958
   827
lemma odd_less_0_iff:
huffman@28958
   828
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
   829
proof (cases z)
huffman@28958
   830
  case (nonneg n)
huffman@28958
   831
  thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
huffman@28958
   832
                             le_imp_0_less [THEN order_less_imp_le])  
huffman@28958
   833
next
huffman@28958
   834
  case (neg n)
huffman@30079
   835
  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
huffman@30079
   836
    add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
huffman@28958
   837
qed
huffman@28958
   838
huffman@28985
   839
lemma bin_less_0_simps:
huffman@28958
   840
  "Pls < 0 \<longleftrightarrow> False"
huffman@28958
   841
  "Min < 0 \<longleftrightarrow> True"
huffman@28958
   842
  "Bit0 w < 0 \<longleftrightarrow> w < 0"
huffman@28958
   843
  "Bit1 w < 0 \<longleftrightarrow> w < 0"
huffman@28958
   844
  unfolding numeral_simps
huffman@28958
   845
  by (simp_all add: even_less_0_iff odd_less_0_iff)
huffman@28958
   846
huffman@28958
   847
lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)"
huffman@28958
   848
  by simp
huffman@28958
   849
huffman@28958
   850
lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l"
huffman@28958
   851
  unfolding numeral_simps
huffman@28958
   852
  proof
huffman@28958
   853
    have "k - 1 < k" by simp
huffman@28958
   854
    also assume "k \<le> l"
huffman@28958
   855
    finally show "k - 1 < l" .
huffman@28958
   856
  next
huffman@28958
   857
    assume "k - 1 < l"
huffman@28958
   858
    hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle)
huffman@28958
   859
    thus "k \<le> l" by simp
huffman@28958
   860
  qed
huffman@28958
   861
huffman@28958
   862
lemma succ_pred: "succ (pred x) = x"
huffman@28958
   863
  unfolding numeral_simps by simp
huffman@28958
   864
huffman@28958
   865
text {* Less-than *}
huffman@28958
   866
huffman@28958
   867
lemma less_bin_simps [simp]:
huffman@28958
   868
  "Pls < Pls \<longleftrightarrow> False"
huffman@28958
   869
  "Pls < Min \<longleftrightarrow> False"
huffman@28958
   870
  "Pls < Bit0 k \<longleftrightarrow> Pls < k"
huffman@28958
   871
  "Pls < Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   872
  "Min < Pls \<longleftrightarrow> True"
huffman@28958
   873
  "Min < Min \<longleftrightarrow> False"
huffman@28958
   874
  "Min < Bit0 k \<longleftrightarrow> Min < k"
huffman@28958
   875
  "Min < Bit1 k \<longleftrightarrow> Min < k"
huffman@28958
   876
  "Bit0 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958
   877
  "Bit0 k < Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   878
  "Bit1 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958
   879
  "Bit1 k < Min \<longleftrightarrow> k < Min"
huffman@28958
   880
  "Bit0 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   881
  "Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   882
  "Bit1 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   883
  "Bit1 k < Bit1 l \<longleftrightarrow> k < l"
huffman@28958
   884
  unfolding le_iff_pred_less
huffman@28958
   885
    less_bin_lemma [of Pls]
huffman@28958
   886
    less_bin_lemma [of Min]
huffman@28958
   887
    less_bin_lemma [of "k"]
huffman@28958
   888
    less_bin_lemma [of "Bit0 k"]
huffman@28958
   889
    less_bin_lemma [of "Bit1 k"]
huffman@28958
   890
    less_bin_lemma [of "pred Pls"]
huffman@28958
   891
    less_bin_lemma [of "pred k"]
huffman@28985
   892
  by (simp_all add: bin_less_0_simps succ_pred)
huffman@28958
   893
huffman@28958
   894
text {* Less-than-or-equal *}
huffman@28958
   895
huffman@28958
   896
lemma le_bin_simps [simp]:
huffman@28958
   897
  "Pls \<le> Pls \<longleftrightarrow> True"
huffman@28958
   898
  "Pls \<le> Min \<longleftrightarrow> False"
huffman@28958
   899
  "Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   900
  "Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958
   901
  "Min \<le> Pls \<longleftrightarrow> True"
huffman@28958
   902
  "Min \<le> Min \<longleftrightarrow> True"
huffman@28958
   903
  "Min \<le> Bit0 k \<longleftrightarrow> Min < k"
huffman@28958
   904
  "Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k"
huffman@28958
   905
  "Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls"
huffman@28958
   906
  "Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   907
  "Bit1 k \<le> Pls \<longleftrightarrow> k < Pls"
huffman@28958
   908
  "Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958
   909
  "Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l"
huffman@28958
   910
  "Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   911
  "Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l"
huffman@28958
   912
  "Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958
   913
  unfolding not_less [symmetric]
huffman@28958
   914
  by (simp_all add: not_le)
huffman@28958
   915
huffman@28958
   916
text {* Equality *}
huffman@28958
   917
huffman@28958
   918
lemma eq_bin_simps [simp]:
huffman@28958
   919
  "Pls = Pls \<longleftrightarrow> True"
huffman@28958
   920
  "Pls = Min \<longleftrightarrow> False"
huffman@28958
   921
  "Pls = Bit0 l \<longleftrightarrow> Pls = l"
huffman@28958
   922
  "Pls = Bit1 l \<longleftrightarrow> False"
huffman@28958
   923
  "Min = Pls \<longleftrightarrow> False"
huffman@28958
   924
  "Min = Min \<longleftrightarrow> True"
huffman@28958
   925
  "Min = Bit0 l \<longleftrightarrow> False"
huffman@28958
   926
  "Min = Bit1 l \<longleftrightarrow> Min = l"
huffman@28958
   927
  "Bit0 k = Pls \<longleftrightarrow> k = Pls"
huffman@28958
   928
  "Bit0 k = Min \<longleftrightarrow> False"
huffman@28958
   929
  "Bit1 k = Pls \<longleftrightarrow> False"
huffman@28958
   930
  "Bit1 k = Min \<longleftrightarrow> k = Min"
huffman@28958
   931
  "Bit0 k = Bit0 l \<longleftrightarrow> k = l"
huffman@28958
   932
  "Bit0 k = Bit1 l \<longleftrightarrow> False"
huffman@28958
   933
  "Bit1 k = Bit0 l \<longleftrightarrow> False"
huffman@28958
   934
  "Bit1 k = Bit1 l \<longleftrightarrow> k = l"
huffman@28958
   935
  unfolding order_eq_iff [where 'a=int]
huffman@28958
   936
  by (simp_all add: not_less)
huffman@28958
   937
huffman@28958
   938
haftmann@25919
   939
subsection {* Converting Numerals to Rings: @{term number_of} *}
haftmann@25919
   940
haftmann@25919
   941
class number_ring = number + comm_ring_1 +
haftmann@25919
   942
  assumes number_of_eq: "number_of k = of_int k"
haftmann@25919
   943
haftmann@25919
   944
text {* self-embedding of the integers *}
haftmann@25919
   945
haftmann@25919
   946
instantiation int :: number_ring
haftmann@25919
   947
begin
haftmann@25919
   948
haftmann@37767
   949
definition
haftmann@37767
   950
  int_number_of_def: "number_of w = (of_int w \<Colon> int)"
haftmann@25919
   951
haftmann@28724
   952
instance proof
haftmann@28724
   953
qed (simp only: int_number_of_def)
haftmann@25919
   954
haftmann@25919
   955
end
haftmann@25919
   956
haftmann@25919
   957
lemma number_of_is_id:
haftmann@25919
   958
  "number_of (k::int) = k"
haftmann@25919
   959
  unfolding int_number_of_def by simp
haftmann@25919
   960
haftmann@25919
   961
lemma number_of_succ:
haftmann@25919
   962
  "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
haftmann@25919
   963
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   964
haftmann@25919
   965
lemma number_of_pred:
haftmann@25919
   966
  "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
haftmann@25919
   967
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
   968
haftmann@25919
   969
lemma number_of_minus:
haftmann@25919
   970
  "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
huffman@28958
   971
  unfolding number_of_eq by (rule of_int_minus)
haftmann@25919
   972
haftmann@25919
   973
lemma number_of_add:
haftmann@25919
   974
  "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
huffman@28958
   975
  unfolding number_of_eq by (rule of_int_add)
huffman@28958
   976
huffman@28958
   977
lemma number_of_diff:
huffman@28958
   978
  "number_of (v - w) = (number_of v - number_of w::'a::number_ring)"
huffman@28958
   979
  unfolding number_of_eq by (rule of_int_diff)
haftmann@25919
   980
haftmann@25919
   981
lemma number_of_mult:
haftmann@25919
   982
  "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
huffman@28958
   983
  unfolding number_of_eq by (rule of_int_mult)
haftmann@25919
   984
haftmann@25919
   985
text {*
haftmann@25919
   986
  The correctness of shifting.
haftmann@25919
   987
  But it doesn't seem to give a measurable speed-up.
haftmann@25919
   988
*}
haftmann@25919
   989
huffman@26086
   990
lemma double_number_of_Bit0:
huffman@26086
   991
  "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"
haftmann@25919
   992
  unfolding number_of_eq numeral_simps left_distrib by simp
haftmann@25919
   993
haftmann@25919
   994
text {*
haftmann@25919
   995
  Converting numerals 0 and 1 to their abstract versions.
haftmann@25919
   996
*}
haftmann@25919
   997
haftmann@32272
   998
lemma numeral_0_eq_0 [simp, code_post]:
haftmann@25919
   999
  "Numeral0 = (0::'a::number_ring)"
haftmann@25919
  1000
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1001
haftmann@32272
  1002
lemma numeral_1_eq_1 [simp, code_post]:
haftmann@25919
  1003
  "Numeral1 = (1::'a::number_ring)"
haftmann@25919
  1004
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1005
haftmann@25919
  1006
text {*
haftmann@25919
  1007
  Special-case simplification for small constants.
haftmann@25919
  1008
*}
haftmann@25919
  1009
haftmann@25919
  1010
text{*
haftmann@25919
  1011
  Unary minus for the abstract constant 1. Cannot be inserted
haftmann@25919
  1012
  as a simprule until later: it is @{text number_of_Min} re-oriented!
haftmann@25919
  1013
*}
haftmann@25919
  1014
haftmann@25919
  1015
lemma numeral_m1_eq_minus_1:
haftmann@25919
  1016
  "(-1::'a::number_ring) = - 1"
haftmann@25919
  1017
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1018
haftmann@25919
  1019
lemma mult_minus1 [simp]:
haftmann@25919
  1020
  "-1 * z = -(z::'a::number_ring)"
haftmann@25919
  1021
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1022
haftmann@25919
  1023
lemma mult_minus1_right [simp]:
haftmann@25919
  1024
  "z * -1 = -(z::'a::number_ring)"
haftmann@25919
  1025
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1026
haftmann@25919
  1027
(*Negation of a coefficient*)
haftmann@25919
  1028
lemma minus_number_of_mult [simp]:
haftmann@25919
  1029
   "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
haftmann@25919
  1030
   unfolding number_of_eq by simp
haftmann@25919
  1031
haftmann@25919
  1032
text {* Subtraction *}
haftmann@25919
  1033
haftmann@25919
  1034
lemma diff_number_of_eq:
haftmann@25919
  1035
  "number_of v - number_of w =
haftmann@25919
  1036
    (number_of (v + uminus w)::'a::number_ring)"
haftmann@25919
  1037
  unfolding number_of_eq by simp
haftmann@25919
  1038
haftmann@25919
  1039
lemma number_of_Pls:
haftmann@25919
  1040
  "number_of Pls = (0::'a::number_ring)"
haftmann@25919
  1041
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1042
haftmann@25919
  1043
lemma number_of_Min:
haftmann@25919
  1044
  "number_of Min = (- 1::'a::number_ring)"
haftmann@25919
  1045
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1046
huffman@26086
  1047
lemma number_of_Bit0:
huffman@26086
  1048
  "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086
  1049
  unfolding number_of_eq numeral_simps by simp
huffman@26086
  1050
huffman@26086
  1051
lemma number_of_Bit1:
huffman@26086
  1052
  "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086
  1053
  unfolding number_of_eq numeral_simps by simp
haftmann@25919
  1054
haftmann@25919
  1055
huffman@28958
  1056
subsubsection {* Equality of Binary Numbers *}
haftmann@25919
  1057
haftmann@25919
  1058
text {* First version by Norbert Voelker *}
haftmann@25919
  1059
haftmann@36716
  1060
definition (*for simplifying equalities*) iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool" where
haftmann@25919
  1061
  "iszero z \<longleftrightarrow> z = 0"
haftmann@25919
  1062
haftmann@25919
  1063
lemma iszero_0: "iszero 0"
haftmann@36716
  1064
  by (simp add: iszero_def)
haftmann@36716
  1065
haftmann@36716
  1066
lemma iszero_Numeral0: "iszero (Numeral0 :: 'a::number_ring)"
haftmann@36716
  1067
  by (simp add: iszero_0)
haftmann@36716
  1068
haftmann@36716
  1069
lemma not_iszero_1: "\<not> iszero 1"
haftmann@36716
  1070
  by (simp add: iszero_def)
haftmann@36716
  1071
haftmann@36716
  1072
lemma not_iszero_Numeral1: "\<not> iszero (Numeral1 :: 'a::number_ring)"
haftmann@36716
  1073
  by (simp add: not_iszero_1)
haftmann@25919
  1074
huffman@35216
  1075
lemma eq_number_of_eq [simp]:
haftmann@25919
  1076
  "((number_of x::'a::number_ring) = number_of y) =
haftmann@36716
  1077
     iszero (number_of (x + uminus y) :: 'a)"
nipkow@29667
  1078
unfolding iszero_def number_of_add number_of_minus
nipkow@29667
  1079
by (simp add: algebra_simps)
haftmann@25919
  1080
haftmann@25919
  1081
lemma iszero_number_of_Pls:
haftmann@25919
  1082
  "iszero ((number_of Pls)::'a::number_ring)"
nipkow@29667
  1083
unfolding iszero_def numeral_0_eq_0 ..
haftmann@25919
  1084
haftmann@25919
  1085
lemma nonzero_number_of_Min:
haftmann@25919
  1086
  "~ iszero ((number_of Min)::'a::number_ring)"
nipkow@29667
  1087
unfolding iszero_def numeral_m1_eq_minus_1 by simp
haftmann@25919
  1088
haftmann@25919
  1089
huffman@28958
  1090
subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919
  1091
haftmann@25919
  1092
lemmas double_eq_0_iff = double_zero
haftmann@25919
  1093
haftmann@25919
  1094
lemma odd_nonzero:
haftmann@33296
  1095
  "1 + z + z \<noteq> (0::int)"
wenzelm@42676
  1096
proof (cases z)
haftmann@25919
  1097
  case (nonneg n)
haftmann@25919
  1098
  have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919
  1099
  thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919
  1100
    by (auto simp add: add_assoc) 
haftmann@25919
  1101
next
haftmann@25919
  1102
  case (neg n)
haftmann@25919
  1103
  show ?thesis
haftmann@25919
  1104
  proof
haftmann@25919
  1105
    assume eq: "1 + z + z = 0"
haftmann@25919
  1106
    have "(0::int) < 1 + (of_nat n + of_nat n)"
haftmann@25919
  1107
      by (simp add: le_imp_0_less add_increasing) 
haftmann@25919
  1108
    also have "... = - (1 + z + z)" 
haftmann@25919
  1109
      by (simp add: neg add_assoc [symmetric]) 
haftmann@25919
  1110
    also have "... = 0" by (simp add: eq) 
haftmann@25919
  1111
    finally have "0<0" ..
haftmann@25919
  1112
    thus False by blast
haftmann@25919
  1113
  qed
haftmann@25919
  1114
qed
haftmann@25919
  1115
huffman@26086
  1116
lemma iszero_number_of_Bit0:
huffman@26086
  1117
  "iszero (number_of (Bit0 w)::'a) = 
huffman@26086
  1118
   iszero (number_of w::'a::{ring_char_0,number_ring})"
haftmann@25919
  1119
proof -
haftmann@25919
  1120
  have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
haftmann@25919
  1121
  proof -
haftmann@25919
  1122
    assume eq: "of_int w + of_int w = (0::'a)"
haftmann@25919
  1123
    then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
haftmann@25919
  1124
    then have "w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1125
    then show "w = 0" by (simp only: double_eq_0_iff)
haftmann@25919
  1126
  qed
huffman@26086
  1127
  thus ?thesis
huffman@26086
  1128
    by (auto simp add: iszero_def number_of_eq numeral_simps)
huffman@26086
  1129
qed
huffman@26086
  1130
huffman@26086
  1131
lemma iszero_number_of_Bit1:
huffman@26086
  1132
  "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"
huffman@26086
  1133
proof -
huffman@26086
  1134
  have "1 + of_int w + of_int w \<noteq> (0::'a)"
haftmann@25919
  1135
  proof
haftmann@25919
  1136
    assume eq: "1 + of_int w + of_int w = (0::'a)"
haftmann@25919
  1137
    hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp 
haftmann@25919
  1138
    hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1139
    with odd_nonzero show False by blast
haftmann@25919
  1140
  qed
huffman@26086
  1141
  thus ?thesis
huffman@26086
  1142
    by (auto simp add: iszero_def number_of_eq numeral_simps)
haftmann@25919
  1143
qed
haftmann@25919
  1144
huffman@35216
  1145
lemmas iszero_simps [simp] =
huffman@28985
  1146
  iszero_0 not_iszero_1
huffman@28985
  1147
  iszero_number_of_Pls nonzero_number_of_Min
huffman@28985
  1148
  iszero_number_of_Bit0 iszero_number_of_Bit1
huffman@28985
  1149
(* iszero_number_of_Pls would never normally be used
huffman@28985
  1150
   because its lhs simplifies to "iszero 0" *)
haftmann@25919
  1151
huffman@28958
  1152
subsubsection {* The Less-Than Relation *}
haftmann@25919
  1153
haftmann@25919
  1154
lemma double_less_0_iff:
haftmann@35028
  1155
  "(a + a < 0) = (a < (0::'a::linordered_idom))"
haftmann@25919
  1156
proof -
haftmann@25919
  1157
  have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
haftmann@25919
  1158
  also have "... = (a < 0)"
haftmann@25919
  1159
    by (simp add: mult_less_0_iff zero_less_two 
haftmann@25919
  1160
                  order_less_not_sym [OF zero_less_two]) 
haftmann@25919
  1161
  finally show ?thesis .
haftmann@25919
  1162
qed
haftmann@25919
  1163
haftmann@25919
  1164
lemma odd_less_0:
haftmann@33296
  1165
  "(1 + z + z < 0) = (z < (0::int))"
wenzelm@42676
  1166
proof (cases z)
haftmann@25919
  1167
  case (nonneg n)
wenzelm@42676
  1168
  then show ?thesis
wenzelm@42676
  1169
    by (simp add: linorder_not_less add_assoc add_increasing
wenzelm@42676
  1170
      le_imp_0_less [THEN order_less_imp_le])
haftmann@25919
  1171
next
haftmann@25919
  1172
  case (neg n)
wenzelm@42676
  1173
  then show ?thesis
wenzelm@42676
  1174
    by (simp del: of_nat_Suc of_nat_add of_nat_1
wenzelm@42676
  1175
      add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
haftmann@25919
  1176
qed
haftmann@25919
  1177
haftmann@25919
  1178
text {* Less-Than or Equals *}
haftmann@25919
  1179
haftmann@25919
  1180
text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
haftmann@25919
  1181
haftmann@25919
  1182
lemmas le_number_of_eq_not_less =
haftmann@25919
  1183
  linorder_not_less [of "number_of w" "number_of v", symmetric, 
haftmann@25919
  1184
  standard]
haftmann@25919
  1185
haftmann@25919
  1186
haftmann@25919
  1187
text {* Absolute value (@{term abs}) *}
haftmann@25919
  1188
haftmann@25919
  1189
lemma abs_number_of:
haftmann@35028
  1190
  "abs(number_of x::'a::{linordered_idom,number_ring}) =
haftmann@25919
  1191
   (if number_of x < (0::'a) then -number_of x else number_of x)"
haftmann@25919
  1192
  by (simp add: abs_if)
haftmann@25919
  1193
haftmann@25919
  1194
haftmann@25919
  1195
text {* Re-orientation of the equation nnn=x *}
haftmann@25919
  1196
haftmann@25919
  1197
lemma number_of_reorient:
haftmann@25919
  1198
  "(number_of w = x) = (x = number_of w)"
haftmann@25919
  1199
  by auto
haftmann@25919
  1200
haftmann@25919
  1201
huffman@28958
  1202
subsubsection {* Simplification of arithmetic operations on integer constants. *}
haftmann@25919
  1203
haftmann@25919
  1204
lemmas arith_extra_simps [standard, simp] =
haftmann@25919
  1205
  number_of_add [symmetric]
huffman@28958
  1206
  number_of_minus [symmetric]
huffman@28958
  1207
  numeral_m1_eq_minus_1 [symmetric]
haftmann@25919
  1208
  number_of_mult [symmetric]
haftmann@25919
  1209
  diff_number_of_eq abs_number_of 
haftmann@25919
  1210
haftmann@25919
  1211
text {*
haftmann@25919
  1212
  For making a minimal simpset, one must include these default simprules.
haftmann@25919
  1213
  Also include @{text simp_thms}.
haftmann@25919
  1214
*}
haftmann@25919
  1215
haftmann@25919
  1216
lemmas arith_simps = 
huffman@26075
  1217
  normalize_bin_simps pred_bin_simps succ_bin_simps
huffman@26075
  1218
  add_bin_simps minus_bin_simps mult_bin_simps
haftmann@25919
  1219
  abs_zero abs_one arith_extra_simps
haftmann@25919
  1220
haftmann@25919
  1221
text {* Simplification of relational operations *}
haftmann@25919
  1222
huffman@28962
  1223
lemma less_number_of [simp]:
haftmann@35028
  1224
  "(number_of x::'a::{linordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
huffman@28962
  1225
  unfolding number_of_eq by (rule of_int_less_iff)
huffman@28962
  1226
huffman@28962
  1227
lemma le_number_of [simp]:
haftmann@35028
  1228
  "(number_of x::'a::{linordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
huffman@28962
  1229
  unfolding number_of_eq by (rule of_int_le_iff)
huffman@28962
  1230
huffman@28967
  1231
lemma eq_number_of [simp]:
huffman@28967
  1232
  "(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y"
huffman@28967
  1233
  unfolding number_of_eq by (rule of_int_eq_iff)
huffman@28967
  1234
huffman@35216
  1235
lemmas rel_simps =
huffman@28962
  1236
  less_number_of less_bin_simps
huffman@28962
  1237
  le_number_of le_bin_simps
huffman@28988
  1238
  eq_number_of_eq eq_bin_simps
huffman@29039
  1239
  iszero_simps
haftmann@25919
  1240
haftmann@25919
  1241
huffman@28958
  1242
subsubsection {* Simplification of arithmetic when nested to the right. *}
haftmann@25919
  1243
haftmann@25919
  1244
lemma add_number_of_left [simp]:
haftmann@25919
  1245
  "number_of v + (number_of w + z) =
haftmann@25919
  1246
   (number_of(v + w) + z::'a::number_ring)"
haftmann@25919
  1247
  by (simp add: add_assoc [symmetric])
haftmann@25919
  1248
haftmann@25919
  1249
lemma mult_number_of_left [simp]:
haftmann@25919
  1250
  "number_of v * (number_of w * z) =
haftmann@25919
  1251
   (number_of(v * w) * z::'a::number_ring)"
haftmann@25919
  1252
  by (simp add: mult_assoc [symmetric])
haftmann@25919
  1253
haftmann@25919
  1254
lemma add_number_of_diff1:
haftmann@25919
  1255
  "number_of v + (number_of w - c) = 
haftmann@25919
  1256
  number_of(v + w) - (c::'a::number_ring)"
huffman@35216
  1257
  by (simp add: diff_minus)
haftmann@25919
  1258
haftmann@25919
  1259
lemma add_number_of_diff2 [simp]:
haftmann@25919
  1260
  "number_of v + (c - number_of w) =
haftmann@25919
  1261
   number_of (v + uminus w) + (c::'a::number_ring)"
nipkow@29667
  1262
by (simp add: algebra_simps diff_number_of_eq [symmetric])
haftmann@25919
  1263
haftmann@25919
  1264
haftmann@30652
  1265
haftmann@30652
  1266
haftmann@25919
  1267
subsection {* The Set of Integers *}
haftmann@25919
  1268
haftmann@25919
  1269
context ring_1
haftmann@25919
  1270
begin
haftmann@25919
  1271
haftmann@30652
  1272
definition Ints  :: "'a set" where
haftmann@37767
  1273
  "Ints = range of_int"
haftmann@25919
  1274
haftmann@25919
  1275
notation (xsymbols)
haftmann@25919
  1276
  Ints  ("\<int>")
haftmann@25919
  1277
huffman@35634
  1278
lemma Ints_of_int [simp]: "of_int z \<in> \<int>"
huffman@35634
  1279
  by (simp add: Ints_def)
huffman@35634
  1280
huffman@35634
  1281
lemma Ints_of_nat [simp]: "of_nat n \<in> \<int>"
huffman@35634
  1282
apply (simp add: Ints_def)
huffman@35634
  1283
apply (rule range_eqI)
huffman@35634
  1284
apply (rule of_int_of_nat_eq [symmetric])
huffman@35634
  1285
done
huffman@35634
  1286
haftmann@25919
  1287
lemma Ints_0 [simp]: "0 \<in> \<int>"
haftmann@25919
  1288
apply (simp add: Ints_def)
haftmann@25919
  1289
apply (rule range_eqI)
haftmann@25919
  1290
apply (rule of_int_0 [symmetric])
haftmann@25919
  1291
done
haftmann@25919
  1292
haftmann@25919
  1293
lemma Ints_1 [simp]: "1 \<in> \<int>"
haftmann@25919
  1294
apply (simp add: Ints_def)
haftmann@25919
  1295
apply (rule range_eqI)
haftmann@25919
  1296
apply (rule of_int_1 [symmetric])
haftmann@25919
  1297
done
haftmann@25919
  1298
haftmann@25919
  1299
lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919
  1300
apply (auto simp add: Ints_def)
haftmann@25919
  1301
apply (rule range_eqI)
haftmann@25919
  1302
apply (rule of_int_add [symmetric])
haftmann@25919
  1303
done
haftmann@25919
  1304
haftmann@25919
  1305
lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919
  1306
apply (auto simp add: Ints_def)
haftmann@25919
  1307
apply (rule range_eqI)
haftmann@25919
  1308
apply (rule of_int_minus [symmetric])
haftmann@25919
  1309
done
haftmann@25919
  1310
huffman@35634
  1311
lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
huffman@35634
  1312
apply (auto simp add: Ints_def)
huffman@35634
  1313
apply (rule range_eqI)
huffman@35634
  1314
apply (rule of_int_diff [symmetric])
huffman@35634
  1315
done
huffman@35634
  1316
haftmann@25919
  1317
lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919
  1318
apply (auto simp add: Ints_def)
haftmann@25919
  1319
apply (rule range_eqI)
haftmann@25919
  1320
apply (rule of_int_mult [symmetric])
haftmann@25919
  1321
done
haftmann@25919
  1322
huffman@35634
  1323
lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
huffman@35634
  1324
by (induct n) simp_all
huffman@35634
  1325
haftmann@25919
  1326
lemma Ints_cases [cases set: Ints]:
haftmann@25919
  1327
  assumes "q \<in> \<int>"
haftmann@25919
  1328
  obtains (of_int) z where "q = of_int z"
haftmann@25919
  1329
  unfolding Ints_def
haftmann@25919
  1330
proof -
haftmann@25919
  1331
  from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919
  1332
  then obtain z where "q = of_int z" ..
haftmann@25919
  1333
  then show thesis ..
haftmann@25919
  1334
qed
haftmann@25919
  1335
haftmann@25919
  1336
lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919
  1337
  "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919
  1338
  by (rule Ints_cases) auto
haftmann@25919
  1339
haftmann@25919
  1340
end
haftmann@25919
  1341
haftmann@25919
  1342
text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919
  1343
haftmann@25919
  1344
lemma Ints_double_eq_0_iff:
haftmann@25919
  1345
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1346
  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919
  1347
proof -
haftmann@25919
  1348
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1349
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1350
  show ?thesis
haftmann@25919
  1351
  proof
haftmann@25919
  1352
    assume "a = 0"
haftmann@25919
  1353
    thus "a + a = 0" by simp
haftmann@25919
  1354
  next
haftmann@25919
  1355
    assume eq: "a + a = 0"
haftmann@25919
  1356
    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1357
    hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1358
    hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919
  1359
    thus "a = 0" by (simp add: a)
haftmann@25919
  1360
  qed
haftmann@25919
  1361
qed
haftmann@25919
  1362
haftmann@25919
  1363
lemma Ints_odd_nonzero:
haftmann@25919
  1364
  assumes in_Ints: "a \<in> Ints"
haftmann@25919
  1365
  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919
  1366
proof -
haftmann@25919
  1367
  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919
  1368
  then obtain z where a: "a = of_int z" ..
haftmann@25919
  1369
  show ?thesis
haftmann@25919
  1370
  proof
haftmann@25919
  1371
    assume eq: "1 + a + a = 0"
haftmann@25919
  1372
    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919
  1373
    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919
  1374
    with odd_nonzero show False by blast
haftmann@25919
  1375
  qed
haftmann@25919
  1376
qed 
haftmann@25919
  1377
huffman@35634
  1378
lemma Ints_number_of [simp]:
haftmann@25919
  1379
  "(number_of w :: 'a::number_ring) \<in> Ints"
haftmann@25919
  1380
  unfolding number_of_eq Ints_def by simp
haftmann@25919
  1381
huffman@35634
  1382
lemma Nats_number_of [simp]:
huffman@35634
  1383
  "Int.Pls \<le> w \<Longrightarrow> (number_of w :: 'a::number_ring) \<in> Nats"
huffman@35634
  1384
unfolding Int.Pls_def number_of_eq
huffman@35634
  1385
by (simp only: of_nat_nat [symmetric] of_nat_in_Nats)
huffman@35634
  1386
haftmann@25919
  1387
lemma Ints_odd_less_0: 
haftmann@25919
  1388
  assumes in_Ints: "a \<in> Ints"
haftmann@35028
  1389
  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
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
  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919
  1394
    by (simp add: a)
haftmann@25919
  1395
  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
haftmann@25919
  1396
  also have "... = (a < 0)" by (simp add: a)
haftmann@25919
  1397
  finally show ?thesis .
haftmann@25919
  1398
qed
haftmann@25919
  1399
haftmann@25919
  1400
haftmann@25919
  1401
subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919
  1402
haftmann@25919
  1403
lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919
  1404
  apply (cases "finite A")
haftmann@25919
  1405
  apply (erule finite_induct, auto)
haftmann@25919
  1406
  done
haftmann@25919
  1407
haftmann@25919
  1408
lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919
  1409
  apply (cases "finite A")
haftmann@25919
  1410
  apply (erule finite_induct, auto)
haftmann@25919
  1411
  done
haftmann@25919
  1412
haftmann@25919
  1413
lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919
  1414
  apply (cases "finite A")
haftmann@25919
  1415
  apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919
  1416
  done
haftmann@25919
  1417
haftmann@25919
  1418
lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919
  1419
  apply (cases "finite A")
haftmann@25919
  1420
  apply (erule finite_induct, auto)
haftmann@25919
  1421
  done
haftmann@25919
  1422
haftmann@25919
  1423
lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919
  1424
lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919
  1425
haftmann@25919
  1426
haftmann@25919
  1427
subsection{*Inequality Reasoning for the Arithmetic Simproc*}
haftmann@25919
  1428
haftmann@25919
  1429
lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
haftmann@25919
  1430
by simp 
haftmann@25919
  1431
haftmann@25919
  1432
lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
haftmann@25919
  1433
by simp
haftmann@25919
  1434
haftmann@25919
  1435
lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
haftmann@25919
  1436
by simp 
haftmann@25919
  1437
haftmann@25919
  1438
lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
haftmann@25919
  1439
by simp
haftmann@25919
  1440
haftmann@25919
  1441
lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
haftmann@25919
  1442
by simp
haftmann@25919
  1443
haftmann@25919
  1444
lemma inverse_numeral_1:
haftmann@25919
  1445
  "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
haftmann@25919
  1446
by simp
haftmann@25919
  1447
haftmann@25919
  1448
text{*Theorem lists for the cancellation simprocs. The use of binary numerals
haftmann@25919
  1449
for 0 and 1 reduces the number of special cases.*}
haftmann@25919
  1450
haftmann@25919
  1451
lemmas add_0s = add_numeral_0 add_numeral_0_right
haftmann@25919
  1452
lemmas mult_1s = mult_numeral_1 mult_numeral_1_right 
haftmann@25919
  1453
                 mult_minus1 mult_minus1_right
haftmann@25919
  1454
haftmann@25919
  1455
haftmann@25919
  1456
subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
haftmann@25919
  1457
haftmann@25919
  1458
text{*Arithmetic computations are defined for binary literals, which leaves 0
haftmann@25919
  1459
and 1 as special cases. Addition already has rules for 0, but not 1.
haftmann@25919
  1460
Multiplication and unary minus already have rules for both 0 and 1.*}
haftmann@25919
  1461
haftmann@25919
  1462
haftmann@25919
  1463
lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
haftmann@25919
  1464
by simp
haftmann@25919
  1465
haftmann@25919
  1466
haftmann@25919
  1467
lemmas add_number_of_eq = number_of_add [symmetric]
haftmann@25919
  1468
haftmann@25919
  1469
text{*Allow 1 on either or both sides*}
haftmann@25919
  1470
lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
huffman@35216
  1471
by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric])
haftmann@25919
  1472
haftmann@25919
  1473
lemmas add_special =
haftmann@25919
  1474
    one_add_one_is_two
haftmann@25919
  1475
    binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1476
    binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1477
haftmann@25919
  1478
text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
haftmann@25919
  1479
lemmas diff_special =
haftmann@25919
  1480
    binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1481
    binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1482
haftmann@25919
  1483
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1484
lemmas eq_special =
haftmann@25919
  1485
    binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
haftmann@25919
  1486
    binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919
  1487
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
haftmann@25919
  1488
    binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919
  1489
haftmann@25919
  1490
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1491
lemmas less_special =
huffman@28984
  1492
  binop_eq [of "op <", OF less_number_of numeral_0_eq_0 refl, standard]
huffman@28984
  1493
  binop_eq [of "op <", OF less_number_of numeral_1_eq_1 refl, standard]
huffman@28984
  1494
  binop_eq [of "op <", OF less_number_of refl numeral_0_eq_0, standard]
huffman@28984
  1495
  binop_eq [of "op <", OF less_number_of refl numeral_1_eq_1, standard]
haftmann@25919
  1496
haftmann@25919
  1497
text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919
  1498
lemmas le_special =
huffman@28984
  1499
    binop_eq [of "op \<le>", OF le_number_of numeral_0_eq_0 refl, standard]
huffman@28984
  1500
    binop_eq [of "op \<le>", OF le_number_of numeral_1_eq_1 refl, standard]
huffman@28984
  1501
    binop_eq [of "op \<le>", OF le_number_of refl numeral_0_eq_0, standard]
huffman@28984
  1502
    binop_eq [of "op \<le>", OF le_number_of refl numeral_1_eq_1, standard]
haftmann@25919
  1503
haftmann@25919
  1504
lemmas arith_special[simp] = 
haftmann@25919
  1505
       add_special diff_special eq_special less_special le_special
haftmann@25919
  1506
haftmann@25919
  1507
haftmann@25919
  1508
text {* Legacy theorems *}
haftmann@25919
  1509
haftmann@25919
  1510
lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919
  1511
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@25919
  1512
huffman@30802
  1513
subsection {* Setting up simplification procedures *}
huffman@30802
  1514
huffman@30802
  1515
lemmas int_arith_rules =
huffman@30802
  1516
  neg_le_iff_le numeral_0_eq_0 numeral_1_eq_1
huffman@30802
  1517
  minus_zero diff_minus left_minus right_minus
boehmes@36076
  1518
  mult_zero_left mult_zero_right mult_Bit1 mult_1_left mult_1_right
huffman@30802
  1519
  mult_minus_left mult_minus_right
huffman@30802
  1520
  minus_add_distrib minus_minus mult_assoc
huffman@30802
  1521
  of_nat_0 of_nat_1 of_nat_Suc of_nat_add of_nat_mult
huffman@30802
  1522
  of_int_0 of_int_1 of_int_add of_int_mult
huffman@30802
  1523
haftmann@28952
  1524
use "Tools/int_arith.ML"
haftmann@31100
  1525
setup {* Int_Arith.global_setup *}
haftmann@30496
  1526
declaration {* K Int_Arith.setup *}
haftmann@25919
  1527
huffman@31024
  1528
setup {*
wenzelm@33523
  1529
  Reorient_Proc.add
haftmann@31065
  1530
    (fn Const (@{const_name number_of}, _) $ _ => true | _ => false)
huffman@31024
  1531
*}
huffman@31024
  1532
wenzelm@33523
  1533
simproc_setup reorient_numeral ("number_of w = x") = Reorient_Proc.proc
huffman@31024
  1534
haftmann@25919
  1535
haftmann@25919
  1536
subsection{*Lemmas About Small Numerals*}
haftmann@25919
  1537
haftmann@25919
  1538
lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
haftmann@25919
  1539
proof -
haftmann@25919
  1540
  have "(of_int -1 :: 'a) = of_int (- 1)" by simp
haftmann@25919
  1541
  also have "... = - of_int 1" by (simp only: of_int_minus)
haftmann@25919
  1542
  also have "... = -1" by simp
haftmann@25919
  1543
  finally show ?thesis .
haftmann@25919
  1544
qed
haftmann@25919
  1545
haftmann@35028
  1546
lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{linordered_idom,number_ring})"
haftmann@25919
  1547
by (simp add: abs_if)
haftmann@25919
  1548
haftmann@25919
  1549
lemma abs_power_minus_one [simp]:
haftmann@35028
  1550
  "abs(-1 ^ n) = (1::'a::{linordered_idom,number_ring})"
haftmann@25919
  1551
by (simp add: power_abs)
haftmann@25919
  1552
huffman@30000
  1553
lemma of_int_number_of_eq [simp]:
haftmann@25919
  1554
     "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
haftmann@25919
  1555
by (simp add: number_of_eq) 
haftmann@25919
  1556
haftmann@25919
  1557
text{*Lemmas for specialist use, NOT as default simprules*}
haftmann@25919
  1558
lemma mult_2: "2 * z = (z+z::'a::number_ring)"
haftmann@33296
  1559
unfolding one_add_one_is_two [symmetric] left_distrib by simp
haftmann@25919
  1560
haftmann@25919
  1561
lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
haftmann@25919
  1562
by (subst mult_commute, rule mult_2)
haftmann@25919
  1563
haftmann@25919
  1564
haftmann@25919
  1565
subsection{*More Inequality Reasoning*}
haftmann@25919
  1566
haftmann@25919
  1567
lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919
  1568
by arith
haftmann@25919
  1569
haftmann@25919
  1570
lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919
  1571
by arith
haftmann@25919
  1572
haftmann@25919
  1573
lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919
  1574
by arith
haftmann@25919
  1575
haftmann@25919
  1576
lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919
  1577
by arith
haftmann@25919
  1578
haftmann@25919
  1579
lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919
  1580
by arith
haftmann@25919
  1581
haftmann@25919
  1582
huffman@28958
  1583
subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919
  1584
haftmann@25919
  1585
text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
haftmann@25919
  1586
  @{term "w + - z"}*}
haftmann@25919
  1587
declare Zero_int_def [symmetric, simp]
haftmann@25919
  1588
declare One_int_def [symmetric, simp]
haftmann@25919
  1589
haftmann@25919
  1590
lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
haftmann@25919
  1591
huffman@35216
  1592
(* FIXME: duplicates nat_zero *)
haftmann@25919
  1593
lemma nat_0: "nat 0 = 0"
haftmann@25919
  1594
by (simp add: nat_eq_iff)
haftmann@25919
  1595
haftmann@25919
  1596
lemma nat_1: "nat 1 = Suc 0"
haftmann@25919
  1597
by (subst nat_eq_iff, simp)
haftmann@25919
  1598
haftmann@25919
  1599
lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@25919
  1600
by (subst nat_eq_iff, simp)
haftmann@25919
  1601
haftmann@25919
  1602
lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919
  1603
apply (insert zless_nat_conj [of 1 z])
haftmann@25919
  1604
apply (auto simp add: nat_1)
haftmann@25919
  1605
done
haftmann@25919
  1606
haftmann@25919
  1607
text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919
  1608
      z is an integer literal.*}
haftmann@25919
  1609
lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
haftmann@25919
  1610
haftmann@25919
  1611
lemma split_nat [arith_split]:
haftmann@25919
  1612
  "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919
  1613
  (is "?P = (?L & ?R)")
haftmann@25919
  1614
proof (cases "i < 0")
haftmann@25919
  1615
  case True thus ?thesis by auto
haftmann@25919
  1616
next
haftmann@25919
  1617
  case False
haftmann@25919
  1618
  have "?P = ?L"
haftmann@25919
  1619
  proof
haftmann@25919
  1620
    assume ?P thus ?L using False by clarsimp
haftmann@25919
  1621
  next
haftmann@25919
  1622
    assume ?L thus ?P using False by simp
haftmann@25919
  1623
  qed
haftmann@25919
  1624
  with False show ?thesis by simp
haftmann@25919
  1625
qed
haftmann@25919
  1626
haftmann@25919
  1627
context ring_1
haftmann@25919
  1628
begin
haftmann@25919
  1629
blanchet@33056
  1630
lemma of_int_of_nat [nitpick_simp]:
haftmann@25919
  1631
  "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919
  1632
proof (cases "k < 0")
haftmann@25919
  1633
  case True then have "0 \<le> - k" by simp
haftmann@25919
  1634
  then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919
  1635
  with True show ?thesis by simp
haftmann@25919
  1636
next
haftmann@25919
  1637
  case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919
  1638
qed
haftmann@25919
  1639
haftmann@25919
  1640
end
haftmann@25919
  1641
haftmann@25919
  1642
lemma nat_mult_distrib:
haftmann@25919
  1643
  fixes z z' :: int
haftmann@25919
  1644
  assumes "0 \<le> z"
haftmann@25919
  1645
  shows "nat (z * z') = nat z * nat z'"
haftmann@25919
  1646
proof (cases "0 \<le> z'")
haftmann@25919
  1647
  case False with assms have "z * z' \<le> 0"
haftmann@25919
  1648
    by (simp add: not_le mult_le_0_iff)
haftmann@25919
  1649
  then have "nat (z * z') = 0" by simp
haftmann@25919
  1650
  moreover from False have "nat z' = 0" by simp
haftmann@25919
  1651
  ultimately show ?thesis by simp
haftmann@25919
  1652
next
haftmann@25919
  1653
  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919
  1654
  show ?thesis
haftmann@25919
  1655
    by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919
  1656
      (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919
  1657
         of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919
  1658
qed
haftmann@25919
  1659
haftmann@25919
  1660
lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919
  1661
apply (rule trans)
haftmann@25919
  1662
apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919
  1663
done
haftmann@25919
  1664
haftmann@25919
  1665
lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919
  1666
apply (cases "z=0 | w=0")
haftmann@25919
  1667
apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919
  1668
                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919
  1669
done
haftmann@25919
  1670
haftmann@25919
  1671
haftmann@25919
  1672
subsection "Induction principles for int"
haftmann@25919
  1673
haftmann@25919
  1674
text{*Well-founded segments of the integers*}
haftmann@25919
  1675
haftmann@25919
  1676
definition
haftmann@25919
  1677
  int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919
  1678
where
haftmann@25919
  1679
  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919
  1680
haftmann@25919
  1681
theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919
  1682
proof -
haftmann@25919
  1683
  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919
  1684
    by (auto simp add: int_ge_less_than_def)
haftmann@25919
  1685
  thus ?thesis 
haftmann@25919
  1686
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1687
qed
haftmann@25919
  1688
haftmann@25919
  1689
text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919
  1690
by RankFinder.*}
haftmann@25919
  1691
haftmann@25919
  1692
definition
haftmann@25919
  1693
  int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919
  1694
where
haftmann@25919
  1695
  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919
  1696
haftmann@25919
  1697
theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919
  1698
proof -
haftmann@25919
  1699
  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919
  1700
    by (auto simp add: int_ge_less_than2_def)
haftmann@25919
  1701
  thus ?thesis 
haftmann@25919
  1702
    by (rule wf_subset [OF wf_measure]) 
haftmann@25919
  1703
qed
haftmann@25919
  1704
haftmann@25919
  1705
abbreviation
haftmann@25919
  1706
  int :: "nat \<Rightarrow> int"
haftmann@25919
  1707
where
haftmann@25919
  1708
  "int \<equiv> of_nat"
haftmann@25919
  1709
haftmann@25919
  1710
(* `set:int': dummy construction *)
haftmann@25919
  1711
theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919
  1712
  fixes i :: int
haftmann@25919
  1713
  assumes ge: "k \<le> i" and
haftmann@25919
  1714
    base: "P k" and
haftmann@25919
  1715
    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919
  1716
  shows "P i"
haftmann@25919
  1717
proof -
wenzelm@42676
  1718
  { fix n
wenzelm@42676
  1719
    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919
  1720
    proof (induct n)
haftmann@25919
  1721
      case 0
haftmann@25919
  1722
      hence "i = k" by arith
haftmann@25919
  1723
      thus "P i" using base by simp
haftmann@25919
  1724
    next
haftmann@25919
  1725
      case (Suc n)
haftmann@25919
  1726
      then have "n = nat((i - 1) - k)" by arith
haftmann@25919
  1727
      moreover
haftmann@25919
  1728
      have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919
  1729
      ultimately
wenzelm@42676
  1730
      have "P (i - 1)" by (rule Suc.hyps)
wenzelm@42676
  1731
      from step [OF ki1 this] show ?case by simp
haftmann@25919
  1732
    qed
haftmann@25919
  1733
  }
haftmann@25919
  1734
  with ge show ?thesis by fast
haftmann@25919
  1735
qed
haftmann@25919
  1736
haftmann@25928
  1737
(* `set:int': dummy construction *)
haftmann@25928
  1738
theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919
  1739
  assumes gr: "k < (i::int)" and
haftmann@25919
  1740
        base: "P(k+1)" and
haftmann@25919
  1741
        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919
  1742
  shows "P i"
haftmann@25919
  1743
apply(rule int_ge_induct[of "k + 1"])
haftmann@25919
  1744
  using gr apply arith
haftmann@25919
  1745
 apply(rule base)
haftmann@25919
  1746
apply (rule step, simp+)
haftmann@25919
  1747
done
haftmann@25919
  1748
wenzelm@42676
  1749
theorem int_le_induct [consumes 1, case_names base step]:
haftmann@25919
  1750
  assumes le: "i \<le> (k::int)" and
haftmann@25919
  1751
        base: "P(k)" and
haftmann@25919
  1752
        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1753
  shows "P i"
haftmann@25919
  1754
proof -
wenzelm@42676
  1755
  { fix n
wenzelm@42676
  1756
    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919
  1757
    proof (induct n)
haftmann@25919
  1758
      case 0
haftmann@25919
  1759
      hence "i = k" by arith
haftmann@25919
  1760
      thus "P i" using base by simp
haftmann@25919
  1761
    next
haftmann@25919
  1762
      case (Suc n)
wenzelm@42676
  1763
      hence "n = nat (k - (i + 1))" by arith
haftmann@25919
  1764
      moreover
haftmann@25919
  1765
      have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919
  1766
      ultimately
wenzelm@42676
  1767
      have "P (i + 1)" by(rule Suc.hyps)
haftmann@25919
  1768
      from step[OF ki1 this] show ?case by simp
haftmann@25919
  1769
    qed
haftmann@25919
  1770
  }
haftmann@25919
  1771
  with le show ?thesis by fast
haftmann@25919
  1772
qed
haftmann@25919
  1773
wenzelm@42676
  1774
theorem int_less_induct [consumes 1, case_names base step]:
haftmann@25919
  1775
  assumes less: "(i::int) < k" and
haftmann@25919
  1776
        base: "P(k - 1)" and
haftmann@25919
  1777
        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919
  1778
  shows "P i"
haftmann@25919
  1779
apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919
  1780
  using less apply arith
haftmann@25919
  1781
 apply(rule base)
haftmann@25919
  1782
apply (rule step, simp+)
haftmann@25919
  1783
done
haftmann@25919
  1784
haftmann@36811
  1785
theorem int_induct [case_names base step1 step2]:
haftmann@36801
  1786
  fixes k :: int
haftmann@36801
  1787
  assumes base: "P k"
haftmann@36801
  1788
    and step1: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@36801
  1789
    and step2: "\<And>i. k \<ge> i \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
haftmann@36801
  1790
  shows "P i"
haftmann@36801
  1791
proof -
haftmann@36801
  1792
  have "i \<le> k \<or> i \<ge> k" by arith
wenzelm@42676
  1793
  then show ?thesis
wenzelm@42676
  1794
  proof
wenzelm@42676
  1795
    assume "i \<ge> k"
wenzelm@42676
  1796
    then show ?thesis using base
haftmann@36801
  1797
      by (rule int_ge_induct) (fact step1)
haftmann@36801
  1798
  next
wenzelm@42676
  1799
    assume "i \<le> k"
wenzelm@42676
  1800
    then show ?thesis using base
haftmann@36801
  1801
      by (rule int_le_induct) (fact step2)
haftmann@36801
  1802
  qed
haftmann@36801
  1803
qed
haftmann@36801
  1804
haftmann@25919
  1805
subsection{*Intermediate value theorems*}
haftmann@25919
  1806
haftmann@25919
  1807
lemma int_val_lemma:
haftmann@25919
  1808
     "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919
  1809
      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
huffman@30079
  1810
unfolding One_nat_def
wenzelm@42676
  1811
apply (induct n)
wenzelm@42676
  1812
apply simp
haftmann@25919
  1813
apply (intro strip)
haftmann@25919
  1814
apply (erule impE, simp)
haftmann@25919
  1815
apply (erule_tac x = n in allE, simp)
huffman@30079
  1816
apply (case_tac "k = f (Suc n)")
haftmann@27106
  1817
apply force
haftmann@25919
  1818
apply (erule impE)
haftmann@25919
  1819
 apply (simp add: abs_if split add: split_if_asm)
haftmann@25919
  1820
apply (blast intro: le_SucI)
haftmann@25919
  1821
done
haftmann@25919
  1822
haftmann@25919
  1823
lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919
  1824
haftmann@25919
  1825
lemma nat_intermed_int_val:
haftmann@25919
  1826
     "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919
  1827
         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919
  1828
apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919
  1829
       in int_val_lemma)
huffman@30079
  1830
unfolding One_nat_def
haftmann@25919
  1831
apply simp
haftmann@25919
  1832
apply (erule exE)
haftmann@25919
  1833
apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919
  1834
done
haftmann@25919
  1835
haftmann@25919
  1836
haftmann@25919
  1837
subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919
  1838
haftmann@25919
  1839
lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919
  1840
by arith
haftmann@25919
  1841
paulson@34055
  1842
lemma abs_zmult_eq_1:
paulson@34055
  1843
  assumes mn: "\<bar>m * n\<bar> = 1"
paulson@34055
  1844
  shows "\<bar>m\<bar> = (1::int)"
paulson@34055
  1845
proof -
paulson@34055
  1846
  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
paulson@34055
  1847
    by auto
paulson@34055
  1848
  have "~ (2 \<le> \<bar>m\<bar>)"
paulson@34055
  1849
  proof
paulson@34055
  1850
    assume "2 \<le> \<bar>m\<bar>"
paulson@34055
  1851
    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
paulson@34055
  1852
      by (simp add: mult_mono 0) 
paulson@34055
  1853
    also have "... = \<bar>m*n\<bar>" 
paulson@34055
  1854
      by (simp add: abs_mult)
paulson@34055
  1855
    also have "... = 1"
paulson@34055
  1856
      by (simp add: mn)
paulson@34055
  1857
    finally have "2*\<bar>n\<bar> \<le> 1" .
paulson@34055
  1858
    thus "False" using 0
paulson@34055
  1859
      by auto
paulson@34055
  1860
  qed
paulson@34055
  1861
  thus ?thesis using 0
paulson@34055
  1862
    by auto
paulson@34055
  1863
qed
haftmann@25919
  1864
haftmann@25919
  1865
lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919
  1866
by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919
  1867
boehmes@35815
  1868
lemma pos_zmult_eq_1_iff:
boehmes@35815
  1869
  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
boehmes@35815
  1870
proof -
boehmes@35815
  1871
  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1872
  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
boehmes@35815
  1873
qed
haftmann@25919
  1874
haftmann@25919
  1875
lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919
  1876
apply (rule iffI) 
haftmann@25919
  1877
 apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919
  1878
 apply (simp add: mult_commute [of m]) 
haftmann@25919
  1879
 apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919
  1880
done
haftmann@25919
  1881
haftmann@33296
  1882
lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
haftmann@25919
  1883
proof
haftmann@33296
  1884
  assume "finite (UNIV::int set)"
haftmann@33296
  1885
  moreover have "inj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1886
    by (rule injI) simp
haftmann@33296
  1887
  ultimately have "surj (\<lambda>i\<Colon>int. 2 * i)"
haftmann@33296
  1888
    by (rule finite_UNIV_inj_surj)
haftmann@33296
  1889
  then obtain i :: int where "1 = 2 * i" by (rule surjE)
haftmann@33296
  1890
  then show False by (simp add: pos_zmult_eq_1_iff)
haftmann@25919
  1891
qed
haftmann@25919
  1892
haftmann@25919
  1893
haftmann@30652
  1894
subsection {* Further theorems on numerals *}
haftmann@30652
  1895
haftmann@30652
  1896
subsubsection{*Special Simplification for Constants*}
haftmann@30652
  1897
haftmann@30652
  1898
text{*These distributive laws move literals inside sums and differences.*}
haftmann@30652
  1899
haftmann@30652
  1900
lemmas left_distrib_number_of [simp] =
haftmann@30652
  1901
  left_distrib [of _ _ "number_of v", standard]
haftmann@30652
  1902
haftmann@30652
  1903
lemmas right_distrib_number_of [simp] =
haftmann@30652
  1904
  right_distrib [of "number_of v", standard]
haftmann@30652
  1905
haftmann@30652
  1906
lemmas left_diff_distrib_number_of [simp] =
haftmann@30652
  1907
  left_diff_distrib [of _ _ "number_of v", standard]
haftmann@30652
  1908
haftmann@30652
  1909
lemmas right_diff_distrib_number_of [simp] =
haftmann@30652
  1910
  right_diff_distrib [of "number_of v", standard]
haftmann@30652
  1911
haftmann@30652
  1912
text{*These are actually for fields, like real: but where else to put them?*}
haftmann@30652
  1913
blanchet@35828
  1914
lemmas zero_less_divide_iff_number_of [simp, no_atp] =
haftmann@30652
  1915
  zero_less_divide_iff [of "number_of w", standard]
haftmann@30652
  1916
blanchet@35828
  1917
lemmas divide_less_0_iff_number_of [simp, no_atp] =
haftmann@30652
  1918
  divide_less_0_iff [of "number_of w", standard]
haftmann@30652
  1919
blanchet@35828
  1920
lemmas zero_le_divide_iff_number_of [simp, no_atp] =
haftmann@30652
  1921
  zero_le_divide_iff [of "number_of w", standard]
haftmann@30652
  1922
blanchet@35828
  1923
lemmas divide_le_0_iff_number_of [simp, no_atp] =
haftmann@30652
  1924
  divide_le_0_iff [of "number_of w", standard]
haftmann@30652
  1925
haftmann@30652
  1926
haftmann@30652
  1927
text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}.  It looks
haftmann@30652
  1928
  strange, but then other simprocs simplify the quotient.*}
haftmann@30652
  1929
haftmann@30652
  1930
lemmas inverse_eq_divide_number_of [simp] =
haftmann@30652
  1931
  inverse_eq_divide [of "number_of w", standard]
haftmann@30652
  1932
haftmann@30652
  1933
text {*These laws simplify inequalities, moving unary minus from a term
haftmann@30652
  1934
into the literal.*}
haftmann@30652
  1935
blanchet@35828
  1936
lemmas less_minus_iff_number_of [simp, no_atp] =
haftmann@30652
  1937
  less_minus_iff [of "number_of v", standard]
haftmann@30652
  1938
blanchet@35828
  1939
lemmas le_minus_iff_number_of [simp, no_atp] =
haftmann@30652
  1940
  le_minus_iff [of "number_of v", standard]
haftmann@30652
  1941
blanchet@35828
  1942
lemmas equation_minus_iff_number_of [simp, no_atp] =
haftmann@30652
  1943
  equation_minus_iff [of "number_of v", standard]
haftmann@30652
  1944
blanchet@35828
  1945
lemmas minus_less_iff_number_of [simp, no_atp] =
haftmann@30652
  1946
  minus_less_iff [of _ "number_of v", standard]
haftmann@30652
  1947
blanchet@35828
  1948
lemmas minus_le_iff_number_of [simp, no_atp] =
haftmann@30652
  1949
  minus_le_iff [of _ "number_of v", standard]
haftmann@30652
  1950
blanchet@35828
  1951
lemmas minus_equation_iff_number_of [simp, no_atp] =
haftmann@30652
  1952
  minus_equation_iff [of _ "number_of v", standard]
haftmann@30652
  1953
haftmann@30652
  1954
haftmann@30652
  1955
text{*To Simplify Inequalities Where One Side is the Constant 1*}
haftmann@30652
  1956
blanchet@35828
  1957
lemma less_minus_iff_1 [simp,no_atp]:
haftmann@35028
  1958
  fixes b::"'b::{linordered_idom,number_ring}"
haftmann@30652
  1959
  shows "(1 < - b) = (b < -1)"
haftmann@30652
  1960
by auto
haftmann@30652
  1961
blanchet@35828
  1962
lemma le_minus_iff_1 [simp,no_atp]:
haftmann@35028
  1963
  fixes b::"'b::{linordered_idom,number_ring}"
haftmann@30652
  1964
  shows "(1 \<le> - b) = (b \<le> -1)"
haftmann@30652
  1965
by auto
haftmann@30652
  1966
blanchet@35828
  1967
lemma equation_minus_iff_1 [simp,no_atp]:
haftmann@30652
  1968
  fixes b::"'b::number_ring"
haftmann@30652
  1969
  shows "(1 = - b) = (b = -1)"
haftmann@30652
  1970
by (subst equation_minus_iff, auto)
haftmann@30652
  1971
blanchet@35828
  1972
lemma minus_less_iff_1 [simp,no_atp]:
haftmann@35028
  1973
  fixes a::"'b::{linordered_idom,number_ring}"
haftmann@30652
  1974
  shows "(- a < 1) = (-1 < a)"
haftmann@30652
  1975
by auto
haftmann@30652
  1976
blanchet@35828
  1977
lemma minus_le_iff_1 [simp,no_atp]:
haftmann@35028
  1978
  fixes a::"'b::{linordered_idom,number_ring}"
haftmann@30652
  1979
  shows "(- a \<le> 1) = (-1 \<le> a)"
haftmann@30652
  1980
by auto
haftmann@30652
  1981
blanchet@35828
  1982
lemma minus_equation_iff_1 [simp,no_atp]:
haftmann@30652
  1983
  fixes a::"'b::number_ring"
haftmann@30652
  1984
  shows "(- a = 1) = (a = -1)"
haftmann@30652
  1985
by (subst minus_equation_iff, auto)
haftmann@30652
  1986
haftmann@30652
  1987
haftmann@30652
  1988
text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
haftmann@30652
  1989
blanchet@35828
  1990
lemmas mult_less_cancel_left_number_of [simp, no_atp] =
haftmann@30652
  1991
  mult_less_cancel_left [of "number_of v", standard]
haftmann@30652
  1992
blanchet@35828
  1993
lemmas mult_less_cancel_right_number_of [simp, no_atp] =
haftmann@30652
  1994
  mult_less_cancel_right [of _ "number_of v", standard]
haftmann@30652
  1995
blanchet@35828
  1996
lemmas mult_le_cancel_left_number_of [simp, no_atp] =
haftmann@30652
  1997
  mult_le_cancel_left [of "number_of v", standard]
haftmann@30652
  1998
blanchet@35828
  1999
lemmas mult_le_cancel_right_number_of [simp, no_atp] =
haftmann@30652
  2000
  mult_le_cancel_right [of _ "number_of v", standard]
haftmann@30652
  2001
haftmann@30652
  2002
haftmann@30652
  2003
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *}
haftmann@30652
  2004
haftmann@30652
  2005
lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard]
haftmann@30652
  2006
lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard]
haftmann@30652
  2007
lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard]
haftmann@30652
  2008
lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard]
haftmann@30652
  2009
lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard]
haftmann@30652
  2010
lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard]
haftmann@30652
  2011
haftmann@30652
  2012
haftmann@30652
  2013
subsubsection{*Optional Simplification Rules Involving Constants*}
haftmann@30652
  2014
haftmann@30652
  2015
text{*Simplify quotients that are compared with a literal constant.*}
haftmann@30652
  2016
haftmann@30652
  2017
lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard]
haftmann@30652
  2018
lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard]
haftmann@30652
  2019
lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard]
haftmann@30652
  2020
lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard]
haftmann@30652
  2021
lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard]
haftmann@30652
  2022
lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard]
haftmann@30652
  2023
haftmann@30652
  2024
haftmann@30652
  2025
text{*Not good as automatic simprules because they cause case splits.*}
haftmann@30652
  2026
lemmas divide_const_simps =
haftmann@30652
  2027
  le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of
haftmann@30652
  2028
  divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of
haftmann@30652
  2029
  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
haftmann@30652
  2030
haftmann@30652
  2031
text{*Division By @{text "-1"}*}
haftmann@30652
  2032
haftmann@30652
  2033
lemma divide_minus1 [simp]:
haftmann@36409
  2034
     "x/-1 = -(x::'a::{field_inverse_zero, number_ring})"
haftmann@30652
  2035
by simp
haftmann@30652
  2036
haftmann@30652
  2037
lemma minus1_divide [simp]:
haftmann@36409
  2038
     "-1 / (x::'a::{field_inverse_zero, number_ring}) = - (1/x)"
huffman@35216
  2039
by (simp add: divide_inverse)
haftmann@30652
  2040
haftmann@30652
  2041
lemma half_gt_zero_iff:
haftmann@36409
  2042
     "(0 < r/2) = (0 < (r::'a::{linordered_field_inverse_zero,number_ring}))"
haftmann@30652
  2043
by auto
haftmann@30652
  2044
haftmann@30652
  2045
lemmas half_gt_zero [simp] = half_gt_zero_iff [THEN iffD2, standard]
haftmann@30652
  2046
haftmann@36719
  2047
lemma divide_Numeral1:
haftmann@36719
  2048
  "(x::'a::{field, number_ring}) / Numeral1 = x"
haftmann@36719
  2049
  by simp
haftmann@36719
  2050
haftmann@36719
  2051
lemma divide_Numeral0:
haftmann@36719
  2052
  "(x::'a::{field_inverse_zero, number_ring}) / Numeral0 = 0"
haftmann@36719
  2053
  by simp
haftmann@36719
  2054
haftmann@30652
  2055
haftmann@33320
  2056
subsection {* The divides relation *}
haftmann@33320
  2057
nipkow@33657
  2058
lemma zdvd_antisym_nonneg:
nipkow@33657
  2059
    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
haftmann@33320
  2060
  apply (simp add: dvd_def, auto)
nipkow@33657
  2061
  apply (auto simp add: mult_assoc zero_le_mult_iff zmult_eq_1_iff)
haftmann@33320
  2062
  done
haftmann@33320
  2063
nipkow@33657
  2064
lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a" 
haftmann@33320
  2065
  shows "\<bar>a\<bar> = \<bar>b\<bar>"
nipkow@33657
  2066
proof cases
nipkow@33657
  2067
  assume "a = 0" with assms show ?thesis by simp
nipkow@33657
  2068
next
nipkow@33657
  2069
  assume "a \<noteq> 0"
haftmann@33320
  2070
  from `a dvd b` obtain k where k:"b = a*k" unfolding dvd_def by blast 
haftmann@33320
  2071
  from `b dvd a` obtain k' where k':"a = b*k'" unfolding dvd_def by blast 
haftmann@33320
  2072
  from k k' have "a = a*k*k'" by simp
haftmann@33320
  2073
  with mult_cancel_left1[where c="a" and b="k*k'"]
haftmann@33320
  2074
  have kk':"k*k' = 1" using `a\<noteq>0` by (simp add: mult_assoc)
haftmann@33320
  2075
  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
haftmann@33320
  2076
  thus ?thesis using k k' by auto
haftmann@33320
  2077
qed
haftmann@33320
  2078
haftmann@33320
  2079
lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
haftmann@33320
  2080
  apply (subgoal_tac "m = n + (m - n)")
haftmann@33320
  2081
   apply (erule ssubst)
haftmann@33320
  2082
   apply (blast intro: dvd_add, simp)
haftmann@33320
  2083
  done
haftmann@33320
  2084
haftmann@33320
  2085
lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
haftmann@33320
  2086
apply (rule iffI)
haftmann@33320
  2087
 apply (erule_tac [2] dvd_add)
haftmann@33320
  2088
 apply (subgoal_tac "n = (n + k * m) - k * m")
haftmann@33320
  2089
  apply (erule ssubst)
haftmann@33320
  2090
  apply (erule dvd_diff)
haftmann@33320
  2091
  apply(simp_all)
haftmann@33320
  2092
done
haftmann@33320
  2093
haftmann@33320
  2094
lemma dvd_imp_le_int:
haftmann@33320
  2095
  fixes d i :: int
haftmann@33320
  2096
  assumes "i \<noteq> 0" and "d dvd i"
haftmann@33320
  2097
  shows "\<bar>d\<bar> \<le> \<bar>i\<bar>"
haftmann@33320
  2098
proof -
haftmann@33320
  2099
  from `d dvd i` obtain k where "i = d * k" ..
haftmann@33320
  2100
  with `i \<noteq> 0` have "k \<noteq> 0" by auto
haftmann@33320
  2101
  then have "1 \<le> \<bar>k\<bar>" and "0 \<le> \<bar>d\<bar>" by auto
haftmann@33320
  2102
  then have "\<bar>d\<bar> * 1 \<le> \<bar>d\<bar> * \<bar>k\<bar>" by (rule mult_left_mono)
haftmann@33320
  2103
  with `i = d * k` show ?thesis by (simp add: abs_mult)
haftmann@33320
  2104
qed
haftmann@33320
  2105
haftmann@33320
  2106
lemma zdvd_not_zless:
haftmann@33320
  2107
  fixes m n :: int
haftmann@33320
  2108
  assumes "0 < m" and "m < n"
haftmann@33320
  2109
  shows "\<not> n dvd m"
haftmann@33320
  2110
proof
haftmann@33320
  2111
  from assms have "0 < n" by auto
haftmann@33320
  2112
  assume "n dvd m" then obtain k where k: "m = n * k" ..
haftmann@33320
  2113
  with `0 < m` have "0 < n * k" by auto
haftmann@33320
  2114
  with `0 < n` have "0 < k" by (simp add: zero_less_mult_iff)
haftmann@33320
  2115
  with k `0 < n` `m < n` have "n * k < n * 1" by simp
haftmann@33320
  2116
  with `0 < n` `0 < k` show False unfolding mult_less_cancel_left by auto
haftmann@33320
  2117
qed
haftmann@33320
  2118
haftmann@33320
  2119
lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
haftmann@33320
  2120
  shows "m dvd n"
haftmann@33320
  2121
proof-
haftmann@33320
  2122
  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
haftmann@33320
  2123
  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
haftmann@33320
  2124
    with h have False by (simp add: mult_assoc)}
haftmann@33320
  2125
  hence "n = m * h" by blast
haftmann@33320
  2126
  thus ?thesis by simp
haftmann@33320
  2127
qed
haftmann@33320
  2128
haftmann@33320
  2129
theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
haftmann@33320
  2130
proof -
haftmann@33320
  2131
  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
haftmann@33320
  2132
  proof -
haftmann@33320
  2133
    fix k
haftmann@33320
  2134
    assume A: "int y = int x * k"
wenzelm@42676
  2135
    then show "x dvd y"
wenzelm@42676
  2136
    proof (cases k)
wenzelm@42676
  2137
      case (nonneg n)
wenzelm@42676
  2138
      with A have "y = x * n" by (simp add: of_nat_mult [symmetric])
haftmann@33320
  2139
      then show ?thesis ..
haftmann@33320
  2140
    next
wenzelm@42676
  2141
      case (neg n)
wenzelm@42676
  2142
      with A have "int y = int x * (- int (Suc n))" by simp
haftmann@33320
  2143
      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
haftmann@33320
  2144
      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
haftmann@33320
  2145
      finally have "- int (x * Suc n) = int y" ..
haftmann@33320
  2146
      then show ?thesis by (simp only: negative_eq_positive) auto
haftmann@33320
  2147
    qed
haftmann@33320
  2148
  qed
haftmann@33320
  2149
  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
haftmann@33320
  2150
qed
haftmann@33320
  2151
wenzelm@42676
  2152
lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
haftmann@33320
  2153
proof
haftmann@33320
  2154
  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
haftmann@33320
  2155
  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
haftmann@33320
  2156
  hence "nat \<bar>x\<bar> = 1"  by simp
wenzelm@42676
  2157
  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
haftmann@33320
  2158
next
haftmann@33320
  2159
  assume "\<bar>x\<bar>=1"
haftmann@33320
  2160
  then have "x = 1 \<or> x = -1" by auto
haftmann@33320
  2161
  then show "x dvd 1" by (auto intro: dvdI)
haftmann@33320
  2162
qed
haftmann@33320
  2163
haftmann@33320
  2164
lemma zdvd_mult_cancel1: 
haftmann@33320
  2165
  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
haftmann@33320
  2166
proof
haftmann@33320
  2167
  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m" 
wenzelm@42676
  2168
    by (cases "n >0") (auto simp add: minus_equation_iff)
haftmann@33320
  2169
next
haftmann@33320
  2170
  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
haftmann@33320
  2171
  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
haftmann@33320
  2172
qed
haftmann@33320
  2173
haftmann@33320
  2174
lemma int_dvd_iff: "(int m dvd z) = (m dvd nat (abs z))"
haftmann@33320
  2175
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  2176
haftmann@33320
  2177
lemma dvd_int_iff: "(z dvd int m) = (nat (abs z) dvd m)"
haftmann@33320
  2178
  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
haftmann@33320
  2179
haftmann@33320
  2180
lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
haftmann@33320
  2181
  by (auto simp add: dvd_int_iff)
haftmann@33320
  2182
haftmann@33341
  2183
lemma eq_nat_nat_iff:
haftmann@33341
  2184
  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
haftmann@33341
  2185
  by (auto elim!: nonneg_eq_int)
haftmann@33341
  2186
haftmann@33341
  2187
lemma nat_power_eq:
haftmann@33341
  2188
  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
haftmann@33341
  2189
  by (induct n) (simp_all add: nat_mult_distrib)
haftmann@33341
  2190
haftmann@33320
  2191
lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
wenzelm@42676
  2192
  apply (cases n)
haftmann@33320
  2193
  apply (auto simp add: dvd_int_iff)
wenzelm@42676
  2194
  apply (cases z)
haftmann@33320
  2195
  apply (auto simp add: dvd_imp_le)
haftmann@33320
  2196
  done
haftmann@33320
  2197
haftmann@36749
  2198
lemma zdvd_period:
haftmann@36749
  2199
  fixes a d :: int
haftmann@36749
  2200
  assumes "a dvd d"
haftmann@36749
  2201
  shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
haftmann@36749
  2202
proof -
haftmann@36749
  2203
  from assms obtain k where "d = a * k" by (rule dvdE)
wenzelm@42676
  2204
  show ?thesis
wenzelm@42676
  2205
  proof
haftmann@36749
  2206
    assume "a dvd (x + t)"
haftmann@36749
  2207
    then obtain l where "x + t = a * l" by (rule dvdE)
haftmann@36749
  2208
    then have "x = a * l - t" by simp
haftmann@36749
  2209
    with `d = a * k` show "a dvd x + c * d + t" by simp
haftmann@36749
  2210
  next
haftmann@36749
  2211
    assume "a dvd x + c * d + t"
haftmann@36749
  2212
    then obtain l where "x + c * d + t = a * l" by (rule dvdE)
haftmann@36749
  2213
    then have "x = a * l - c * d - t" by simp
haftmann@36749
  2214
    with `d = a * k` show "a dvd (x + t)" by simp
haftmann@36749
  2215
  qed
haftmann@36749
  2216
qed
haftmann@36749
  2217
haftmann@33320
  2218
haftmann@25919
  2219
subsection {* Configuration of the code generator *}
haftmann@25919
  2220
haftmann@26507
  2221
code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"
haftmann@26507
  2222
haftmann@28562
  2223
lemmas pred_succ_numeral_code [code] =
haftmann@26507
  2224
  pred_bin_simps succ_bin_simps
haftmann@26507
  2225
haftmann@28562
  2226
lemmas plus_numeral_code [code] =
haftmann@26507
  2227
  add_bin_simps
haftmann@26507
  2228
  arith_extra_simps(1) [where 'a = int]
haftmann@26507
  2229
haftmann@28562
  2230
lemmas minus_numeral_code [code] =
haftmann@26507
  2231
  minus_bin_simps
haftmann@26507
  2232
  arith_extra_simps(2) [where 'a = int]
haftmann@26507
  2233
  arith_extra_simps(5) [where 'a = int]
haftmann@26507
  2234
haftmann@28562
  2235
lemmas times_numeral_code [code] =
haftmann@26507
  2236
  mult_bin_simps
haftmann@26507
  2237
  arith_extra_simps(4) [where 'a = int]
haftmann@26507
  2238
haftmann@38857
  2239
instantiation int :: equal
haftmann@26507
  2240
begin
haftmann@26507
  2241
haftmann@37767
  2242
definition
haftmann@38857
  2243
  "HOL.equal k l \<longleftrightarrow> k - l = (0\<Colon>int)"
haftmann@38857
  2244
haftmann@38857
  2245
instance by default (simp add: equal_int_def)
haftmann@26507
  2246
haftmann@26507
  2247
end
haftmann@26507
  2248
haftmann@28562
  2249
lemma eq_number_of_int_code [code]:
haftmann@38857
  2250
  "HOL.equal (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> HOL.equal k l"
haftmann@38857
  2251
  unfolding equal_int_def number_of_is_id ..
haftmann@26507
  2252
haftmann@28562
  2253
lemma eq_int_code [code]:
haftmann@38857
  2254
  "HOL.equal Int.Pls Int.Pls \<longleftrightarrow> True"
haftmann@38857
  2255
  "HOL.equal Int.Pls Int.Min \<longleftrightarrow> False"
haftmann@38857
  2256
  "HOL.equal Int.Pls (Int.Bit0 k2) \<longleftrightarrow> HOL.equal Int.Pls k2"
haftmann@38857
  2257
  "HOL.equal Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@38857
  2258
  "HOL.equal Int.Min Int.Pls \<longleftrightarrow> False"
haftmann@38857
  2259
  "HOL.equal Int.Min Int.Min \<longleftrightarrow> True"
haftmann@38857
  2260
  "HOL.equal Int.Min (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@38857
  2261
  "HOL.equal Int.Min (Int.Bit1 k2) \<longleftrightarrow> HOL.equal Int.Min k2"
haftmann@38857
  2262
  "HOL.equal (Int.Bit0 k1) Int.Pls \<longleftrightarrow> HOL.equal k1 Int.Pls"
haftmann@38857
  2263
  "HOL.equal (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False"
haftmann@38857
  2264
  "HOL.equal (Int.Bit0 k1) Int.Min \<longleftrightarrow> False"
haftmann@38857
  2265
  "HOL.equal (Int.Bit1 k1) Int.Min \<longleftrightarrow> HOL.equal k1 Int.Min"
haftmann@38857
  2266
  "HOL.equal (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> HOL.equal k1 k2"
haftmann@38857
  2267
  "HOL.equal (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@38857
  2268
  "HOL.equal (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@38857
  2269
  "HOL.equal (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> HOL.equal k1 k2"
haftmann@38857
  2270
  unfolding equal_eq by simp_all
haftmann@25919
  2271
haftmann@28351
  2272
lemma eq_int_refl [code nbe]:
haftmann@38857
  2273
  "HOL.equal (k::int) k \<longleftrightarrow> True"
haftmann@38857
  2274
  by (rule equal_refl)
haftmann@28351
  2275
haftmann@28562
  2276
lemma less_eq_number_of_int_code [code]:
haftmann@26507
  2277
  "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
haftmann@26507
  2278
  unfolding number_of_is_id ..
haftmann@26507
  2279
haftmann@28562
  2280
lemma less_eq_int_code [code]:
haftmann@26507
  2281
  "Int.Pls \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507
  2282
  "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
haftmann@26507
  2283
  "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  2284
  "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  2285
  "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507
  2286
  "Int.Min \<le> Int.Min \<longleftrightarrow> True"
haftmann@26507
  2287
  "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  2288
  "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"
haftmann@26507
  2289
  "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
haftmann@26507
  2290
  "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  2291
  "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  2292
  "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  2293
  "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  2294
  "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  2295
  "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  2296
  "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
huffman@28958
  2297
  by simp_all
haftmann@26507
  2298
haftmann@28562
  2299
lemma less_number_of_int_code [code]:
haftmann@26507
  2300
  "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
haftmann@26507
  2301
  unfolding number_of_is_id ..
haftmann@26507
  2302
haftmann@28562
  2303
lemma less_int_code [code]:
haftmann@26507
  2304
  "Int.Pls < Int.Pls \<longleftrightarrow> False"
haftmann@26507
  2305
  "Int.Pls < Int.Min \<longleftrightarrow> False"
haftmann@26507
  2306
  "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"
haftmann@26507
  2307
  "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507
  2308
  "Int.Min < Int.Pls \<longleftrightarrow> True"
haftmann@26507
  2309
  "Int.Min < Int.Min \<longleftrightarrow> False"
haftmann@26507
  2310
  "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  2311
  "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"
haftmann@26507
  2312
  "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  2313
  "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507
  2314
  "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507
  2315
  "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"
haftmann@26507
  2316
  "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  2317
  "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507
  2318
  "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507
  2319
  "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"
huffman@28958
  2320
  by simp_all
haftmann@25919
  2321
haftmann@25919
  2322
definition
haftmann@25919
  2323
  nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
haftmann@25919
  2324
  "nat_aux i n = nat i + n"
haftmann@25919
  2325
haftmann@25919
  2326
lemma [code]:
haftmann@25919
  2327
  "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
haftmann@25919
  2328
  by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
haftmann@25919
  2329
    dest: zless_imp_add1_zle)
haftmann@25919
  2330
haftmann@25919
  2331
lemma [code]: "nat i = nat_aux i 0"
haftmann@25919
  2332
  by (simp add: nat_aux_def)
haftmann@25919
  2333
wenzelm@36176
  2334
hide_const (open) nat_aux
haftmann@25928
  2335
haftmann@32069
  2336
lemma zero_is_num_zero [code, code_unfold_post]:
haftmann@25919
  2337
  "(0\<Colon>int) = Numeral0" 
haftmann@25919
  2338
  by simp
haftmann@25919
  2339
haftmann@32069
  2340
lemma one_is_num_one [code, code_unfold_post]:
haftmann@25919
  2341
  "(1\<Colon>int) = Numeral1" 
haftmann@25961
  2342
  by simp
haftmann@25919
  2343
haftmann@25919
  2344
code_modulename SML
haftmann@33364
  2345
  Int Arith
haftmann@25919
  2346
haftmann@25919
  2347
code_modulename OCaml
haftmann@33364
  2348
  Int Arith
haftmann@25919
  2349
haftmann@25919
  2350
code_modulename Haskell
haftmann@33364
  2351
  Int Arith
haftmann@25919
  2352
haftmann@25919
  2353
types_code
haftmann@25919
  2354
  "int" ("int")
haftmann@25919
  2355
attach (term_of) {*
haftmann@25919
  2356
val term_of_int = HOLogic.mk_number HOLogic.intT;
haftmann@25919
  2357
*}
haftmann@25919
  2358
attach (test) {*
haftmann@25919
  2359
fun gen_int i =
haftmann@25919
  2360
  let val j = one_of [~1, 1] * random_range 0 i
haftmann@25919
  2361
  in (j, fn () => term_of_int j) end;
haftmann@25919
  2362
*}
haftmann@25919
  2363
haftmann@25919
  2364
setup {*
haftmann@25919
  2365
let
haftmann@25919
  2366
haftmann@25919
  2367
fun strip_number_of (@{term "Int.number_of :: int => int"} $ t) = t
haftmann@25919
  2368
  | strip_number_of t = t;
haftmann@25919
  2369
wenzelm@42411
  2370
fun numeral_codegen thy mode defs dep module b t gr =
haftmann@25919
  2371
  let val i = HOLogic.dest_numeral (strip_number_of t)
haftmann@25919
  2372
  in
haftmann@28537
  2373
    SOME (Codegen.str (string_of_int i),
wenzelm@42411
  2374
      snd (Codegen.invoke_tycodegen thy mode defs dep module false HOLogic.intT gr))
haftmann@25919
  2375
  end handle TERM _ => NONE;
haftmann@25919
  2376
haftmann@25919
  2377
in
haftmann@25919
  2378
haftmann@25919
  2379
Codegen.add_codegen "numeral_codegen" numeral_codegen
haftmann@25919
  2380
haftmann@25919
  2381
end
haftmann@25919
  2382
*}
haftmann@25919
  2383
haftmann@25919
  2384
consts_code
haftmann@25919
  2385
  "number_of :: int \<Rightarrow> int"    ("(_)")
haftmann@25919
  2386
  "0 :: int"                   ("0")
haftmann@25919
  2387
  "1 :: int"                   ("1")
haftmann@25919
  2388
  "uminus :: int => int"       ("~")
haftmann@25919
  2389
  "op + :: int => int => int"  ("(_ +/ _)")
haftmann@25919
  2390
  "op * :: int => int => int"  ("(_ */ _)")
haftmann@25919
  2391
  "op \<le> :: int => int => bool" ("(_ <=/ _)")
haftmann@25919
  2392
  "op < :: int => int => bool" ("(_ </ _)")
haftmann@25919
  2393
haftmann@25919
  2394
quickcheck_params [default_type = int]
haftmann@25919
  2395
wenzelm@36176
  2396
hide_const (open) Pls Min Bit0 Bit1 succ pred
haftmann@25919
  2397
haftmann@25919
  2398
haftmann@25919
  2399
subsection {* Legacy theorems *}
haftmann@25919
  2400
haftmann@25919
  2401
lemmas zminus_zminus = minus_minus [of "z::int", standard]
haftmann@25919
  2402
lemmas zminus_0 = minus_zero [where 'a=int]
haftmann@25919
  2403
lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
haftmann@25919
  2404
lemmas zadd_commute = add_commute [of "z::int" "w", standard]
haftmann@25919
  2405
lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919
  2406
lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]
haftmann@25919
  2407
lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
haftmann@35050
  2408
lemmas zmult_ac = mult_ac
haftmann@35050
  2409
lemmas zadd_0 = add_0_left [of "z::int", standard]
haftmann@35050
  2410
lemmas zadd_0_right = add_0_right [of "z::int", standard]
haftmann@25919
  2411
lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]
haftmann@25919
  2412
lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]
haftmann@25919
  2413
lemmas zmult_commute = mult_commute [of "z::int" "w", standard]
haftmann@25919
  2414
lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919
  2415
lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919
  2416
lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919
  2417
lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919
  2418
lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919
  2419
haftmann@25919
  2420
lemmas zmult_1 = mult_1_left [of "z::int", standard]
haftmann@25919
  2421
lemmas zmult_1_right = mult_1_right [of "z::int", standard]
haftmann@25919
  2422
haftmann@25919
  2423
lemmas zle_refl = order_refl [of "w::int", standard]
haftmann@25919
  2424
lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]
nipkow@33657
  2425
lemmas zle_antisym = order_antisym [of "z::int" "w", standard]
haftmann@25919
  2426
lemmas zle_linear = linorder_linear [of "z::int" "w", standard]
haftmann@25919
  2427
lemmas zless_linear = linorder_less_linear [where 'a = int]
haftmann@25919
  2428
haftmann@25919
  2429
lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]
haftmann@25919
  2430
lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]
haftmann@25919
  2431
lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]
haftmann@25919
  2432
haftmann@25919
  2433
lemmas int_0_less_1 = zero_less_one [where 'a=int]
haftmann@25919
  2434
lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
haftmann@25919
  2435
haftmann@25919
  2436
lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919
  2437
lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919
  2438
lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919
  2439
lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
haftmann@25919
  2440
lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  2441
lemmas zless_int = of_nat_less_iff [where 'a=int]
haftmann@25919
  2442
lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
haftmann@25919
  2443
lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919
  2444
lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
haftmann@25919
  2445
lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
haftmann@25919
  2446
lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919
  2447
lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919
  2448
lemmas int_Suc = of_nat_Suc [where 'a=int]
haftmann@25919
  2449
lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
haftmann@25919
  2450
lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919
  2451
lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
haftmann@25919
  2452
lemmas zless_le = less_int_def
haftmann@25919
  2453
lemmas int_eq_of_nat = TrueI
haftmann@25919
  2454
haftmann@30960
  2455
lemma zpower_zadd_distrib:
haftmann@30960
  2456
  "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
haftmann@30960
  2457
  by (rule power_add)
haftmann@30960
  2458
haftmann@30960
  2459
lemma zero_less_zpower_abs_iff:
haftmann@30960
  2460
  "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
haftmann@30960
  2461
  by (rule zero_less_power_abs_iff)
haftmann@30960
  2462
haftmann@30960
  2463
lemma zero_le_zpower_abs: "(0::int) \<le> abs x ^ n"
haftmann@30960
  2464
  by (rule zero_le_power_abs)
haftmann@30960
  2465
haftmann@31015
  2466
lemma zpower_zpower:
haftmann@31015
  2467
  "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@31015
  2468
  by (rule power_mult [symmetric])
haftmann@31015
  2469
haftmann@31015
  2470
lemma int_power:
haftmann@31015
  2471
  "int (m ^ n) = int m ^ n"
haftmann@31015
  2472
  by (rule of_nat_power)
haftmann@31015
  2473
haftmann@31015
  2474
lemmas zpower_int = int_power [symmetric]
haftmann@31015
  2475
haftmann@25919
  2476
end