haftmann@25919: (*  Title:      Int.thy
haftmann@25919:     ID:         $Id$
haftmann@25919:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
haftmann@25919:                 Tobias Nipkow, Florian Haftmann, TU Muenchen
haftmann@25919:     Copyright   1994  University of Cambridge
haftmann@25919: 
haftmann@25919: *)
haftmann@25919: 
haftmann@25919: header {* The Integers as Equivalence Classes over Pairs of Natural Numbers *} 
haftmann@25919: 
haftmann@25919: theory Int
krauss@26748: imports Equiv_Relations Nat Wellfounded
haftmann@25919: uses
haftmann@25919:   ("Tools/numeral.ML")
haftmann@25919:   ("Tools/numeral_syntax.ML")
haftmann@25919:   ("~~/src/Provers/Arith/assoc_fold.ML")
haftmann@25919:   "~~/src/Provers/Arith/cancel_numerals.ML"
haftmann@25919:   "~~/src/Provers/Arith/combine_numerals.ML"
haftmann@28952:   ("Tools/int_arith.ML")
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: subsection {* The equivalence relation underlying the integers *}
haftmann@25919: 
haftmann@28661: definition intrel :: "((nat \<times> nat) \<times> (nat \<times> nat)) set" where
haftmann@28562:   [code del]: "intrel = {((x, y), (u, v)) | x y u v. x + v = u +y }"
haftmann@25919: 
haftmann@25919: typedef (Integ)
haftmann@25919:   int = "UNIV//intrel"
haftmann@25919:   by (auto simp add: quotient_def)
haftmann@25919: 
haftmann@25919: instantiation int :: "{zero, one, plus, minus, uminus, times, ord, abs, sgn}"
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   Zero_int_def [code del]: "0 = Abs_Integ (intrel `` {(0, 0)})"
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   One_int_def [code del]: "1 = Abs_Integ (intrel `` {(1, 0)})"
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   add_int_def [code del]: "z + w = Abs_Integ
haftmann@25919:     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u, v) \<in> Rep_Integ w.
haftmann@25919:       intrel `` {(x + u, y + v)})"
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   minus_int_def [code del]:
haftmann@25919:     "- z = Abs_Integ (\<Union>(x, y) \<in> Rep_Integ z. intrel `` {(y, x)})"
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   diff_int_def [code del]:  "z - w = z + (-w \<Colon> int)"
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   mult_int_def [code del]: "z * w = Abs_Integ
haftmann@25919:     (\<Union>(x, y) \<in> Rep_Integ z. \<Union>(u,v ) \<in> Rep_Integ w.
haftmann@25919:       intrel `` {(x*u + y*v, x*v + y*u)})"
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   le_int_def [code del]:
haftmann@25919:    "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)"
haftmann@25919: 
haftmann@25919: definition
haftmann@28562:   less_int_def [code del]: "(z\<Colon>int) < w \<longleftrightarrow> z \<le> w \<and> z \<noteq> w"
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   zabs_def: "\<bar>i\<Colon>int\<bar> = (if i < 0 then - i else i)"
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   zsgn_def: "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
haftmann@25919: 
haftmann@25919: instance ..
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*Construction of the Integers*}
haftmann@25919: 
haftmann@25919: lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
haftmann@25919: by (simp add: intrel_def)
haftmann@25919: 
haftmann@25919: lemma equiv_intrel: "equiv UNIV intrel"
haftmann@25919: by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
haftmann@25919: 
haftmann@25919: text{*Reduces equality of equivalence classes to the @{term intrel} relation:
haftmann@25919:   @{term "(intrel `` {x} = intrel `` {y}) = ((x,y) \<in> intrel)"} *}
haftmann@25919: lemmas equiv_intrel_iff [simp] = eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I]
haftmann@25919: 
haftmann@25919: text{*All equivalence classes belong to set of representatives*}
haftmann@25919: lemma [simp]: "intrel``{(x,y)} \<in> Integ"
haftmann@25919: by (auto simp add: Integ_def intrel_def quotient_def)
haftmann@25919: 
haftmann@25919: text{*Reduces equality on abstractions to equality on representatives:
haftmann@25919:   @{prop "\<lbrakk>x \<in> Integ; y \<in> Integ\<rbrakk> \<Longrightarrow> (Abs_Integ x = Abs_Integ y) = (x=y)"} *}
haftmann@25919: declare Abs_Integ_inject [simp,noatp]  Abs_Integ_inverse [simp,noatp]
haftmann@25919: 
haftmann@25919: text{*Case analysis on the representation of an integer as an equivalence
haftmann@25919:       class of pairs of naturals.*}
haftmann@25919: lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
haftmann@25919:      "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
haftmann@25919: apply (rule Abs_Integ_cases [of z]) 
haftmann@25919: apply (auto simp add: Integ_def quotient_def) 
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Arithmetic Operations *}
haftmann@25919: 
haftmann@25919: lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
haftmann@25919: proof -
haftmann@25919:   have "(\<lambda>(x,y). intrel``{(y,x)}) respects intrel"
haftmann@25919:     by (simp add: congruent_def) 
haftmann@25919:   thus ?thesis
haftmann@25919:     by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma add:
haftmann@25919:      "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
haftmann@25919:       Abs_Integ (intrel``{(x+u, y+v)})"
haftmann@25919: proof -
haftmann@25919:   have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). intrel `` {(x+u, y+v)}) w) z) 
haftmann@25919:         respects2 intrel"
haftmann@25919:     by (simp add: congruent2_def)
haftmann@25919:   thus ?thesis
haftmann@25919:     by (simp add: add_int_def UN_UN_split_split_eq
haftmann@25919:                   UN_equiv_class2 [OF equiv_intrel equiv_intrel])
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: text{*Congruence property for multiplication*}
haftmann@25919: lemma mult_congruent2:
haftmann@25919:      "(%p1 p2. (%(x,y). (%(u,v). intrel``{(x*u + y*v, x*v + y*u)}) p2) p1)
haftmann@25919:       respects2 intrel"
haftmann@25919: apply (rule equiv_intrel [THEN congruent2_commuteI])
haftmann@25919:  apply (force simp add: mult_ac, clarify) 
haftmann@25919: apply (simp add: congruent_def mult_ac)  
haftmann@25919: apply (rename_tac u v w x y z)
haftmann@25919: apply (subgoal_tac "u*y + x*y = w*y + v*y  &  u*z + x*z = w*z + v*z")
haftmann@25919: apply (simp add: mult_ac)
haftmann@25919: apply (simp add: add_mult_distrib [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma mult:
haftmann@25919:      "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
haftmann@25919:       Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
haftmann@25919: by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
haftmann@25919:               UN_equiv_class2 [OF equiv_intrel equiv_intrel])
haftmann@25919: 
haftmann@25919: text{*The integers form a @{text comm_ring_1}*}
haftmann@25919: instance int :: comm_ring_1
haftmann@25919: proof
haftmann@25919:   fix i j k :: int
haftmann@25919:   show "(i + j) + k = i + (j + k)"
haftmann@25919:     by (cases i, cases j, cases k) (simp add: add add_assoc)
haftmann@25919:   show "i + j = j + i" 
haftmann@25919:     by (cases i, cases j) (simp add: add_ac add)
haftmann@25919:   show "0 + i = i"
haftmann@25919:     by (cases i) (simp add: Zero_int_def add)
haftmann@25919:   show "- i + i = 0"
haftmann@25919:     by (cases i) (simp add: Zero_int_def minus add)
haftmann@25919:   show "i - j = i + - j"
haftmann@25919:     by (simp add: diff_int_def)
haftmann@25919:   show "(i * j) * k = i * (j * k)"
haftmann@25919:     by (cases i, cases j, cases k) (simp add: mult ring_simps)
haftmann@25919:   show "i * j = j * i"
haftmann@25919:     by (cases i, cases j) (simp add: mult ring_simps)
haftmann@25919:   show "1 * i = i"
haftmann@25919:     by (cases i) (simp add: One_int_def mult)
haftmann@25919:   show "(i + j) * k = i * k + j * k"
haftmann@25919:     by (cases i, cases j, cases k) (simp add: add mult ring_simps)
haftmann@25919:   show "0 \<noteq> (1::int)"
haftmann@25919:     by (simp add: Zero_int_def One_int_def)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma int_def: "of_nat m = Abs_Integ (intrel `` {(m, 0)})"
haftmann@25919: by (induct m, simp_all add: Zero_int_def One_int_def add)
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* The @{text "\<le>"} Ordering *}
haftmann@25919: 
haftmann@25919: lemma le:
haftmann@25919:   "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
haftmann@25919: by (force simp add: le_int_def)
haftmann@25919: 
haftmann@25919: lemma less:
haftmann@25919:   "(Abs_Integ(intrel``{(x,y)}) < Abs_Integ(intrel``{(u,v)})) = (x+v < u+y)"
haftmann@25919: by (simp add: less_int_def le order_less_le)
haftmann@25919: 
haftmann@25919: instance int :: linorder
haftmann@25919: proof
haftmann@25919:   fix i j k :: int
haftmann@27682:   show antisym: "i \<le> j \<Longrightarrow> j \<le> i \<Longrightarrow> i = j"
haftmann@27682:     by (cases i, cases j) (simp add: le)
haftmann@27682:   show "(i < j) = (i \<le> j \<and> \<not> j \<le> i)"
haftmann@27682:     by (auto simp add: less_int_def dest: antisym) 
haftmann@25919:   show "i \<le> i"
haftmann@25919:     by (cases i) (simp add: le)
haftmann@25919:   show "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
haftmann@25919:     by (cases i, cases j, cases k) (simp add: le)
haftmann@25919:   show "i \<le> j \<or> j \<le> i"
haftmann@25919:     by (cases i, cases j) (simp add: le linorder_linear)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: instantiation int :: distrib_lattice
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   "(inf \<Colon> int \<Rightarrow> int \<Rightarrow> int) = min"
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   "(sup \<Colon> int \<Rightarrow> int \<Rightarrow> int) = max"
haftmann@25919: 
haftmann@25919: instance
haftmann@25919:   by intro_classes
haftmann@25919:     (auto simp add: inf_int_def sup_int_def min_max.sup_inf_distrib1)
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: instance int :: pordered_cancel_ab_semigroup_add
haftmann@25919: proof
haftmann@25919:   fix i j k :: int
haftmann@25919:   show "i \<le> j \<Longrightarrow> k + i \<le> k + j"
haftmann@25919:     by (cases i, cases j, cases k) (simp add: le add)
haftmann@25919: qed
haftmann@25919: 
haftmann@25961: 
haftmann@25919: text{*Strict Monotonicity of Multiplication*}
haftmann@25919: 
haftmann@25919: text{*strict, in 1st argument; proof is by induction on k>0*}
haftmann@25919: lemma zmult_zless_mono2_lemma:
haftmann@25919:      "(i::int)<j ==> 0<k ==> of_nat k * i < of_nat k * j"
haftmann@25919: apply (induct "k", simp)
haftmann@25919: apply (simp add: left_distrib)
haftmann@25919: apply (case_tac "k=0")
haftmann@25919: apply (simp_all add: add_strict_mono)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = of_nat n"
haftmann@25919: apply (cases k)
haftmann@25919: apply (auto simp add: le add int_def Zero_int_def)
haftmann@25919: apply (rule_tac x="x-y" in exI, simp)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = of_nat n"
haftmann@25919: apply (cases k)
haftmann@25919: apply (simp add: less int_def Zero_int_def)
haftmann@25919: apply (rule_tac x="x-y" in exI, simp)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
haftmann@25919: apply (drule zero_less_imp_eq_int)
haftmann@25919: apply (auto simp add: zmult_zless_mono2_lemma)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text{*The integers form an ordered integral domain*}
haftmann@25919: instance int :: ordered_idom
haftmann@25919: proof
haftmann@25919:   fix i j k :: int
haftmann@25919:   show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
haftmann@25919:     by (rule zmult_zless_mono2)
haftmann@25919:   show "\<bar>i\<bar> = (if i < 0 then -i else i)"
haftmann@25919:     by (simp only: zabs_def)
haftmann@25919:   show "sgn (i\<Colon>int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
haftmann@25919:     by (simp only: zsgn_def)
haftmann@25919: qed
haftmann@25919: 
haftmann@25961: instance int :: lordered_ring
haftmann@25961: proof  
haftmann@25961:   fix k :: int
haftmann@25961:   show "abs k = sup k (- k)"
haftmann@25961:     by (auto simp add: sup_int_def zabs_def max_def less_minus_self_iff [symmetric])
haftmann@25961: qed
haftmann@25961: 
haftmann@25919: lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1\<Colon>int) \<le> z"
haftmann@25919: apply (cases w, cases z) 
haftmann@25919: apply (simp add: less le add One_int_def)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma zless_iff_Suc_zadd:
haftmann@25919:   "(w \<Colon> int) < z \<longleftrightarrow> (\<exists>n. z = w + of_nat (Suc n))"
haftmann@25919: apply (cases z, cases w)
haftmann@25919: apply (auto simp add: less add int_def)
haftmann@25919: apply (rename_tac a b c d) 
haftmann@25919: apply (rule_tac x="a+d - Suc(c+b)" in exI) 
haftmann@25919: apply arith
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemmas int_distrib =
haftmann@25919:   left_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919:   right_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919:   left_diff_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919:   right_diff_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Embedding of the Integers into any @{text ring_1}: @{text of_int}*}
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   of_int :: "int \<Rightarrow> 'a"
haftmann@25919: where
haftmann@28562:   [code del]: "of_int z = contents (\<Union>(i, j) \<in> Rep_Integ z. { of_nat i - of_nat j })"
haftmann@25919: 
haftmann@25919: lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
haftmann@25919: proof -
haftmann@25919:   have "(\<lambda>(i,j). { of_nat i - (of_nat j :: 'a) }) respects intrel"
haftmann@25919:     by (simp add: congruent_def compare_rls of_nat_add [symmetric]
haftmann@25919:             del: of_nat_add) 
haftmann@25919:   thus ?thesis
haftmann@25919:     by (simp add: of_int_def UN_equiv_class [OF equiv_intrel])
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma of_int_0 [simp]: "of_int 0 = 0"
haftmann@25919:   by (simp add: of_int Zero_int_def)
haftmann@25919: 
haftmann@25919: lemma of_int_1 [simp]: "of_int 1 = 1"
haftmann@25919:   by (simp add: of_int One_int_def)
haftmann@25919: 
haftmann@25919: lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
haftmann@25919:   by (cases w, cases z, simp add: compare_rls of_int OrderedGroup.compare_rls add)
haftmann@25919: 
haftmann@25919: lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
haftmann@25919:   by (cases z, simp add: compare_rls of_int minus)
haftmann@25919: 
haftmann@25919: lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
haftmann@25919:   by (simp add: OrderedGroup.diff_minus diff_minus)
haftmann@25919: 
haftmann@25919: lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
haftmann@25919: apply (cases w, cases z)
haftmann@25919: apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
haftmann@25919:                  mult add_ac of_nat_mult)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text{*Collapse nested embeddings*}
haftmann@25919: lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
haftmann@25919:   by (induct n) auto
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: context ordered_idom
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: lemma of_int_le_iff [simp]:
haftmann@25919:   "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
haftmann@25919:   by (cases w, cases z, simp add: of_int le minus compare_rls of_nat_add [symmetric] del: of_nat_add)
haftmann@25919: 
haftmann@25919: text{*Special cases where either operand is zero*}
haftmann@25919: lemmas of_int_0_le_iff [simp] = of_int_le_iff [of 0, simplified]
haftmann@25919: lemmas of_int_le_0_iff [simp] = of_int_le_iff [of _ 0, simplified]
haftmann@25919: 
haftmann@25919: lemma of_int_less_iff [simp]:
haftmann@25919:   "of_int w < of_int z \<longleftrightarrow> w < z"
haftmann@25919:   by (simp add: not_le [symmetric] linorder_not_le [symmetric])
haftmann@25919: 
haftmann@25919: text{*Special cases where either operand is zero*}
haftmann@25919: lemmas of_int_0_less_iff [simp] = of_int_less_iff [of 0, simplified]
haftmann@25919: lemmas of_int_less_0_iff [simp] = of_int_less_iff [of _ 0, simplified]
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: text{*Class for unital rings with characteristic zero.
haftmann@25919:  Includes non-ordered rings like the complex numbers.*}
haftmann@25919: class ring_char_0 = ring_1 + semiring_char_0
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: lemma of_int_eq_iff [simp]:
haftmann@25919:    "of_int w = of_int z \<longleftrightarrow> w = z"
haftmann@25919: apply (cases w, cases z, simp add: of_int)
haftmann@25919: apply (simp only: diff_eq_eq diff_add_eq eq_diff_eq)
haftmann@25919: apply (simp only: of_nat_add [symmetric] of_nat_eq_iff)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text{*Special cases where either operand is zero*}
haftmann@25919: lemmas of_int_0_eq_iff [simp] = of_int_eq_iff [of 0, simplified]
haftmann@25919: lemmas of_int_eq_0_iff [simp] = of_int_eq_iff [of _ 0, simplified]
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: text{*Every @{text ordered_idom} has characteristic zero.*}
haftmann@25919: subclass (in ordered_idom) ring_char_0 by intro_locales
haftmann@25919: 
haftmann@25919: lemma of_int_eq_id [simp]: "of_int = id"
haftmann@25919: proof
haftmann@25919:   fix z show "of_int z = id z"
haftmann@25919:     by (cases z) (simp add: of_int add minus int_def diff_minus)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Magnitude of an Integer, as a Natural Number: @{text nat} *}
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   nat :: "int \<Rightarrow> nat"
haftmann@25919: where
haftmann@28562:   [code del]: "nat z = contents (\<Union>(x, y) \<in> Rep_Integ z. {x-y})"
haftmann@25919: 
haftmann@25919: lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
haftmann@25919: proof -
haftmann@25919:   have "(\<lambda>(x,y). {x-y}) respects intrel"
haftmann@25919:     by (simp add: congruent_def) arith
haftmann@25919:   thus ?thesis
haftmann@25919:     by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma nat_int [simp]: "nat (of_nat n) = n"
haftmann@25919: by (simp add: nat int_def)
haftmann@25919: 
haftmann@25919: lemma nat_zero [simp]: "nat 0 = 0"
haftmann@25919: by (simp add: Zero_int_def nat)
haftmann@25919: 
haftmann@25919: lemma int_nat_eq [simp]: "of_nat (nat z) = (if 0 \<le> z then z else 0)"
haftmann@25919: by (cases z, simp add: nat le int_def Zero_int_def)
haftmann@25919: 
haftmann@25919: corollary nat_0_le: "0 \<le> z ==> of_nat (nat z) = z"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
haftmann@25919: by (cases z, simp add: nat le Zero_int_def)
haftmann@25919: 
haftmann@25919: lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
haftmann@25919: apply (cases w, cases z) 
haftmann@25919: apply (simp add: nat le linorder_not_le [symmetric] Zero_int_def, arith)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text{*An alternative condition is @{term "0 \<le> w"} *}
haftmann@25919: corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
haftmann@25919: by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919: 
haftmann@25919: corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
haftmann@25919: by (simp add: nat_le_eq_zle linorder_not_le [symmetric]) 
haftmann@25919: 
haftmann@25919: lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
haftmann@25919: apply (cases w, cases z) 
haftmann@25919: apply (simp add: nat le Zero_int_def linorder_not_le [symmetric], arith)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma nonneg_eq_int:
haftmann@25919:   fixes z :: int
haftmann@25919:   assumes "0 \<le> z" and "\<And>m. z = of_nat m \<Longrightarrow> P"
haftmann@25919:   shows P
haftmann@25919:   using assms by (blast dest: nat_0_le sym)
haftmann@25919: 
haftmann@25919: lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = of_nat m else m=0)"
haftmann@25919: by (cases w, simp add: nat le int_def Zero_int_def, arith)
haftmann@25919: 
haftmann@25919: corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = of_nat m else m=0)"
haftmann@25919: by (simp only: eq_commute [of m] nat_eq_iff)
haftmann@25919: 
haftmann@25919: lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
haftmann@25919: apply (cases w)
haftmann@25919: apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
haftmann@25919: by (auto simp add: nat_eq_iff2)
haftmann@25919: 
haftmann@25919: lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
haftmann@25919: by (insert zless_nat_conj [of 0], auto)
haftmann@25919: 
haftmann@25919: lemma nat_add_distrib:
haftmann@25919:      "[| (0::int) \<le> z;  0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
haftmann@25919: by (cases z, cases z', simp add: nat add le Zero_int_def)
haftmann@25919: 
haftmann@25919: lemma nat_diff_distrib:
haftmann@25919:      "[| (0::int) \<le> z';  z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
haftmann@25919: by (cases z, cases z', 
haftmann@25919:     simp add: nat add minus diff_minus le Zero_int_def)
haftmann@25919: 
haftmann@25919: lemma nat_zminus_int [simp]: "nat (- (of_nat n)) = 0"
haftmann@25919: by (simp add: int_def minus nat Zero_int_def) 
haftmann@25919: 
haftmann@25919: lemma zless_nat_eq_int_zless: "(m < nat z) = (of_nat m < z)"
haftmann@25919: by (cases z, simp add: nat less int_def, arith)
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: lemma of_nat_nat: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
haftmann@25919:   by (cases z rule: eq_Abs_Integ)
haftmann@25919:    (simp add: nat le of_int Zero_int_def of_nat_diff)
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*Lemmas about the Function @{term of_nat} and Orderings*}
haftmann@25919: 
haftmann@25919: lemma negative_zless_0: "- (of_nat (Suc n)) < (0 \<Colon> int)"
haftmann@25919: by (simp add: order_less_le del: of_nat_Suc)
haftmann@25919: 
haftmann@25919: lemma negative_zless [iff]: "- (of_nat (Suc n)) < (of_nat m \<Colon> int)"
haftmann@25919: by (rule negative_zless_0 [THEN order_less_le_trans], simp)
haftmann@25919: 
haftmann@25919: lemma negative_zle_0: "- of_nat n \<le> (0 \<Colon> int)"
haftmann@25919: by (simp add: minus_le_iff)
haftmann@25919: 
haftmann@25919: lemma negative_zle [iff]: "- of_nat n \<le> (of_nat m \<Colon> int)"
haftmann@25919: by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
haftmann@25919: 
haftmann@25919: lemma not_zle_0_negative [simp]: "~ (0 \<le> - (of_nat (Suc n) \<Colon> int))"
haftmann@25919: by (subst le_minus_iff, simp del: of_nat_Suc)
haftmann@25919: 
haftmann@25919: lemma int_zle_neg: "((of_nat n \<Colon> int) \<le> - of_nat m) = (n = 0 & m = 0)"
haftmann@25919: by (simp add: int_def le minus Zero_int_def)
haftmann@25919: 
haftmann@25919: lemma not_int_zless_negative [simp]: "~ ((of_nat n \<Colon> int) < - of_nat m)"
haftmann@25919: by (simp add: linorder_not_less)
haftmann@25919: 
haftmann@25919: lemma negative_eq_positive [simp]: "((- of_nat n \<Colon> int) = of_nat m) = (n = 0 & m = 0)"
haftmann@25919: by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
haftmann@25919: 
haftmann@25919: lemma zle_iff_zadd: "(w\<Colon>int) \<le> z \<longleftrightarrow> (\<exists>n. z = w + of_nat n)"
haftmann@25919: proof -
haftmann@25919:   have "(w \<le> z) = (0 \<le> z - w)"
haftmann@25919:     by (simp only: le_diff_eq add_0_left)
haftmann@25919:   also have "\<dots> = (\<exists>n. z - w = of_nat n)"
haftmann@25919:     by (auto elim: zero_le_imp_eq_int)
haftmann@25919:   also have "\<dots> = (\<exists>n. z = w + of_nat n)"
haftmann@25919:     by (simp only: group_simps)
haftmann@25919:   finally show ?thesis .
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma zadd_int_left: "of_nat m + (of_nat n + z) = of_nat (m + n) + (z\<Colon>int)"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: lemma int_Suc0_eq_1: "of_nat (Suc 0) = (1\<Colon>int)"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: text{*This version is proved for all ordered rings, not just integers!
haftmann@25919:       It is proved here because attribute @{text arith_split} is not available
haftmann@25919:       in theory @{text Ring_and_Field}.
haftmann@25919:       But is it really better than just rewriting with @{text abs_if}?*}
haftmann@25919: lemma abs_split [arith_split,noatp]:
haftmann@25919:      "P(abs(a::'a::ordered_idom)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
haftmann@25919: by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
haftmann@25919: 
haftmann@25919: lemma negD: "(x \<Colon> int) < 0 \<Longrightarrow> \<exists>n. x = - (of_nat (Suc n))"
haftmann@25919: apply (cases x)
haftmann@25919: apply (auto simp add: le minus Zero_int_def int_def order_less_le)
haftmann@25919: apply (rule_tac x="y - Suc x" in exI, arith)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Cases and induction *}
haftmann@25919: 
haftmann@25919: text{*Now we replace the case analysis rule by a more conventional one:
haftmann@25919: whether an integer is negative or not.*}
haftmann@25919: 
haftmann@25919: theorem int_cases [cases type: int, case_names nonneg neg]:
haftmann@25919:   "[|!! n. (z \<Colon> int) = of_nat n ==> P;  !! n. z =  - (of_nat (Suc n)) ==> P |] ==> P"
haftmann@25919: apply (cases "z < 0", blast dest!: negD)
haftmann@25919: apply (simp add: linorder_not_less del: of_nat_Suc)
haftmann@25919: apply auto
haftmann@25919: apply (blast dest: nat_0_le [THEN sym])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: theorem int_induct [induct type: int, case_names nonneg neg]:
haftmann@25919:      "[|!! n. P (of_nat n \<Colon> int);  !!n. P (- (of_nat (Suc n))) |] ==> P z"
haftmann@25919:   by (cases z rule: int_cases) auto
haftmann@25919: 
haftmann@25919: text{*Contributed by Brian Huffman*}
haftmann@25919: theorem int_diff_cases:
haftmann@25919:   obtains (diff) m n where "(z\<Colon>int) = of_nat m - of_nat n"
haftmann@25919: apply (cases z rule: eq_Abs_Integ)
haftmann@25919: apply (rule_tac m=x and n=y in diff)
haftmann@25919: apply (simp add: int_def diff_def minus add)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Binary representation *}
haftmann@25919: 
haftmann@25919: text {*
haftmann@25919:   This formalization defines binary arithmetic in terms of the integers
haftmann@25919:   rather than using a datatype. This avoids multiple representations (leading
haftmann@25919:   zeroes, etc.)  See @{text "ZF/Tools/twos-compl.ML"}, function @{text
haftmann@25919:   int_of_binary}, for the numerical interpretation.
haftmann@25919: 
haftmann@25919:   The representation expects that @{text "(m mod 2)"} is 0 or 1,
haftmann@25919:   even if m is negative;
haftmann@25919:   For instance, @{text "-5 div 2 = -3"} and @{text "-5 mod 2 = 1"}; thus
haftmann@25919:   @{text "-5 = (-3)*2 + 1"}.
haftmann@25919:   
haftmann@25919:   This two's complement binary representation derives from the paper 
haftmann@25919:   "An Efficient Representation of Arithmetic for Term Rewriting" by
haftmann@25919:   Dave Cohen and Phil Watson, Rewriting Techniques and Applications,
haftmann@25919:   Springer LNCS 488 (240-251), 1991.
haftmann@25919: *}
haftmann@25919: 
huffman@28958: subsubsection {* The constructors @{term Bit0}, @{term Bit1}, @{term Pls} and @{term Min} *}
huffman@28958: 
haftmann@25919: definition
haftmann@25919:   Pls :: int where
haftmann@28562:   [code del]: "Pls = 0"
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   Min :: int where
haftmann@28562:   [code del]: "Min = - 1"
haftmann@25919: 
haftmann@25919: definition
huffman@26086:   Bit0 :: "int \<Rightarrow> int" where
haftmann@28562:   [code del]: "Bit0 k = k + k"
huffman@26086: 
huffman@26086: definition
huffman@26086:   Bit1 :: "int \<Rightarrow> int" where
haftmann@28562:   [code del]: "Bit1 k = 1 + k + k"
haftmann@25919: 
haftmann@25919: class number = type + -- {* for numeric types: nat, int, real, \dots *}
haftmann@25919:   fixes number_of :: "int \<Rightarrow> 'a"
haftmann@25919: 
haftmann@25919: use "Tools/numeral.ML"
haftmann@25919: 
haftmann@25919: syntax
haftmann@25919:   "_Numeral" :: "num_const \<Rightarrow> 'a"    ("_")
haftmann@25919: 
haftmann@25919: use "Tools/numeral_syntax.ML"
haftmann@25919: setup NumeralSyntax.setup
haftmann@25919: 
haftmann@25919: abbreviation
haftmann@25919:   "Numeral0 \<equiv> number_of Pls"
haftmann@25919: 
haftmann@25919: abbreviation
huffman@26086:   "Numeral1 \<equiv> number_of (Bit1 Pls)"
haftmann@25919: 
haftmann@25919: lemma Let_number_of [simp]: "Let (number_of v) f = f (number_of v)"
haftmann@25919:   -- {* Unfold all @{text let}s involving constants *}
haftmann@25919:   unfolding Let_def ..
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   succ :: "int \<Rightarrow> int" where
haftmann@28562:   [code del]: "succ k = k + 1"
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   pred :: "int \<Rightarrow> int" where
haftmann@28562:   [code del]: "pred k = k - 1"
haftmann@25919: 
haftmann@25919: lemmas
haftmann@25919:   max_number_of [simp] = max_def
haftmann@25919:     [of "number_of u" "number_of v", standard, simp]
haftmann@25919: and
haftmann@25919:   min_number_of [simp] = min_def 
haftmann@25919:     [of "number_of u" "number_of v", standard, simp]
haftmann@25919:   -- {* unfolding @{text minx} and @{text max} on numerals *}
haftmann@25919: 
haftmann@25919: lemmas numeral_simps = 
huffman@26086:   succ_def pred_def Pls_def Min_def Bit0_def Bit1_def
haftmann@25919: 
haftmann@25919: text {* Removal of leading zeroes *}
haftmann@25919: 
huffman@26086: lemma Bit0_Pls [simp, code post]:
huffman@26086:   "Bit0 Pls = Pls"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@26086: lemma Bit1_Min [simp, code post]:
huffman@26086:   "Bit1 Min = Min"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@26075: lemmas normalize_bin_simps =
huffman@26086:   Bit0_Pls Bit1_Min
huffman@26075: 
haftmann@25919: 
huffman@28958: subsubsection {* Successor and predecessor functions *}
huffman@28958: 
huffman@28958: text {* Successor *}
huffman@28958: 
huffman@28958: lemma succ_Pls:
huffman@26086:   "succ Pls = Bit1 Pls"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma succ_Min:
haftmann@25919:   "succ Min = Pls"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma succ_Bit0:
huffman@26086:   "succ (Bit0 k) = Bit1 k"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma succ_Bit1:
huffman@26086:   "succ (Bit1 k) = Bit0 (succ k)"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemmas succ_bin_simps [simp] =
huffman@26086:   succ_Pls succ_Min succ_Bit0 succ_Bit1
huffman@26075: 
huffman@28958: text {* Predecessor *}
huffman@28958: 
huffman@28958: lemma pred_Pls:
haftmann@25919:   "pred Pls = Min"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma pred_Min:
huffman@26086:   "pred Min = Bit0 Min"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma pred_Bit0:
huffman@26086:   "pred (Bit0 k) = Bit1 (pred k)"
haftmann@25919:   unfolding numeral_simps by simp 
haftmann@25919: 
huffman@28958: lemma pred_Bit1:
huffman@26086:   "pred (Bit1 k) = Bit0 k"
huffman@26086:   unfolding numeral_simps by simp
huffman@26086: 
huffman@28958: lemmas pred_bin_simps [simp] =
huffman@26086:   pred_Pls pred_Min pred_Bit0 pred_Bit1
huffman@26075: 
huffman@28958: 
huffman@28958: subsubsection {* Binary arithmetic *}
huffman@28958: 
huffman@28958: text {* Addition *}
huffman@28958: 
huffman@28958: lemma add_Pls:
huffman@28958:   "Pls + k = k"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma add_Min:
huffman@28958:   "Min + k = pred k"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma add_Bit0_Bit0:
huffman@28958:   "(Bit0 k) + (Bit0 l) = Bit0 (k + l)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma add_Bit0_Bit1:
huffman@28958:   "(Bit0 k) + (Bit1 l) = Bit1 (k + l)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma add_Bit1_Bit0:
huffman@28958:   "(Bit1 k) + (Bit0 l) = Bit1 (k + l)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma add_Bit1_Bit1:
huffman@28958:   "(Bit1 k) + (Bit1 l) = Bit0 (k + succ l)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma add_Pls_right:
huffman@28958:   "k + Pls = k"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma add_Min_right:
huffman@28958:   "k + Min = pred k"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemmas add_bin_simps [simp] =
huffman@28958:   add_Pls add_Min add_Pls_right add_Min_right
huffman@28958:   add_Bit0_Bit0 add_Bit0_Bit1 add_Bit1_Bit0 add_Bit1_Bit1
huffman@28958: 
huffman@28958: text {* Negation *}
huffman@28958: 
huffman@28958: lemma minus_Pls:
haftmann@25919:   "- Pls = Pls"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma minus_Min:
huffman@26086:   "- Min = Bit1 Pls"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma minus_Bit0:
huffman@26086:   "- (Bit0 k) = Bit0 (- k)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma minus_Bit1:
huffman@26086:   "- (Bit1 k) = Bit1 (pred (- k))"
huffman@26086:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemmas minus_bin_simps [simp] =
huffman@26086:   minus_Pls minus_Min minus_Bit0 minus_Bit1
huffman@26075: 
huffman@28958: text {* Subtraction *}
huffman@28958: 
huffman@28958: lemma diff_Pls:
huffman@28958:   "Pls - k = - k"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma diff_Min:
huffman@28958:   "Min - k = pred (- k)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma diff_Bit0_Bit0:
huffman@28958:   "(Bit0 k) - (Bit0 l) = Bit0 (k - l)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma diff_Bit0_Bit1:
huffman@28958:   "(Bit0 k) - (Bit1 l) = Bit1 (pred k - l)"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma diff_Bit1_Bit0:
huffman@28958:   "(Bit1 k) - (Bit0 l) = Bit1 (k - l)"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma diff_Bit1_Bit1:
huffman@28958:   "(Bit1 k) - (Bit1 l) = Bit0 (k - l)"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma diff_Pls_right:
huffman@28958:   "k - Pls = k"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: lemma diff_Min_right:
huffman@28958:   "k - Min = succ k"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemmas diff_bin_simps [simp] =
huffman@28958:   diff_Pls diff_Min diff_Pls_right diff_Min_right
huffman@28958:   diff_Bit0_Bit0 diff_Bit0_Bit1 diff_Bit1_Bit0 diff_Bit1_Bit1
huffman@28958: 
huffman@28958: text {* Multiplication *}
huffman@28958: 
huffman@28958: lemma mult_Pls:
huffman@28958:   "Pls * w = Pls"
huffman@26086:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma mult_Min:
haftmann@25919:   "Min * k = - k"
haftmann@25919:   unfolding numeral_simps by simp
haftmann@25919: 
huffman@28958: lemma mult_Bit0:
huffman@26086:   "(Bit0 k) * l = Bit0 (k * l)"
huffman@26086:   unfolding numeral_simps int_distrib by simp
haftmann@25919: 
huffman@28958: lemma mult_Bit1:
huffman@26086:   "(Bit1 k) * l = (Bit0 (k * l)) + l"
huffman@28958:   unfolding numeral_simps int_distrib by simp
huffman@28958: 
huffman@28958: lemmas mult_bin_simps [simp] =
huffman@26086:   mult_Pls mult_Min mult_Bit0 mult_Bit1
huffman@26075: 
haftmann@25919: 
huffman@28958: subsubsection {* Binary comparisons *}
huffman@28958: 
huffman@28958: text {* Preliminaries *}
huffman@28958: 
huffman@28958: lemma even_less_0_iff:
huffman@28958:   "a + a < 0 \<longleftrightarrow> a < (0::'a::ordered_idom)"
huffman@28958: proof -
huffman@28958:   have "a + a < 0 \<longleftrightarrow> (1+1)*a < 0" by (simp add: left_distrib)
huffman@28958:   also have "(1+1)*a < 0 \<longleftrightarrow> a < 0"
huffman@28958:     by (simp add: mult_less_0_iff zero_less_two 
huffman@28958:                   order_less_not_sym [OF zero_less_two])
huffman@28958:   finally show ?thesis .
huffman@28958: qed
huffman@28958: 
huffman@28958: lemma le_imp_0_less: 
huffman@28958:   assumes le: "0 \<le> z"
huffman@28958:   shows "(0::int) < 1 + z"
huffman@28958: proof -
huffman@28958:   have "0 \<le> z" by fact
huffman@28958:   also have "... < z + 1" by (rule less_add_one) 
huffman@28958:   also have "... = 1 + z" by (simp add: add_ac)
huffman@28958:   finally show "0 < 1 + z" .
huffman@28958: qed
huffman@28958: 
huffman@28958: lemma odd_less_0_iff:
huffman@28958:   "(1 + z + z < 0) = (z < (0::int))"
huffman@28958: proof (cases z rule: int_cases)
huffman@28958:   case (nonneg n)
huffman@28958:   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
huffman@28958:                              le_imp_0_less [THEN order_less_imp_le])  
huffman@28958: next
huffman@28958:   case (neg n)
huffman@28958:   thus ?thesis by (simp del: of_nat_Suc of_nat_add
huffman@28958:     add: compare_rls of_nat_1 [symmetric] of_nat_add [symmetric])
huffman@28958: qed
huffman@28958: 
huffman@28958: lemma neg_simps:
huffman@28958:   "Pls < 0 \<longleftrightarrow> False"
huffman@28958:   "Min < 0 \<longleftrightarrow> True"
huffman@28958:   "Bit0 w < 0 \<longleftrightarrow> w < 0"
huffman@28958:   "Bit1 w < 0 \<longleftrightarrow> w < 0"
huffman@28958:   unfolding numeral_simps
huffman@28958:   by (simp_all add: even_less_0_iff odd_less_0_iff)
huffman@28958: 
huffman@28958: lemma less_bin_lemma: "k < l \<longleftrightarrow> k - l < (0::int)"
huffman@28958:   by simp
huffman@28958: 
huffman@28958: lemma le_iff_pred_less: "k \<le> l \<longleftrightarrow> pred k < l"
huffman@28958:   unfolding numeral_simps
huffman@28958:   proof
huffman@28958:     have "k - 1 < k" by simp
huffman@28958:     also assume "k \<le> l"
huffman@28958:     finally show "k - 1 < l" .
huffman@28958:   next
huffman@28958:     assume "k - 1 < l"
huffman@28958:     hence "(k - 1) + 1 \<le> l" by (rule zless_imp_add1_zle)
huffman@28958:     thus "k \<le> l" by simp
huffman@28958:   qed
huffman@28958: 
huffman@28958: lemma succ_pred: "succ (pred x) = x"
huffman@28958:   unfolding numeral_simps by simp
huffman@28958: 
huffman@28958: text {* Less-than *}
huffman@28958: 
huffman@28958: lemma less_bin_simps [simp]:
huffman@28958:   "Pls < Pls \<longleftrightarrow> False"
huffman@28958:   "Pls < Min \<longleftrightarrow> False"
huffman@28958:   "Pls < Bit0 k \<longleftrightarrow> Pls < k"
huffman@28958:   "Pls < Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958:   "Min < Pls \<longleftrightarrow> True"
huffman@28958:   "Min < Min \<longleftrightarrow> False"
huffman@28958:   "Min < Bit0 k \<longleftrightarrow> Min < k"
huffman@28958:   "Min < Bit1 k \<longleftrightarrow> Min < k"
huffman@28958:   "Bit0 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958:   "Bit0 k < Min \<longleftrightarrow> k \<le> Min"
huffman@28958:   "Bit1 k < Pls \<longleftrightarrow> k < Pls"
huffman@28958:   "Bit1 k < Min \<longleftrightarrow> k < Min"
huffman@28958:   "Bit0 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958:   "Bit0 k < Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958:   "Bit1 k < Bit0 l \<longleftrightarrow> k < l"
huffman@28958:   "Bit1 k < Bit1 l \<longleftrightarrow> k < l"
huffman@28958:   unfolding le_iff_pred_less
huffman@28958:     less_bin_lemma [of Pls]
huffman@28958:     less_bin_lemma [of Min]
huffman@28958:     less_bin_lemma [of "k"]
huffman@28958:     less_bin_lemma [of "Bit0 k"]
huffman@28958:     less_bin_lemma [of "Bit1 k"]
huffman@28958:     less_bin_lemma [of "pred Pls"]
huffman@28958:     less_bin_lemma [of "pred k"]
huffman@28958:   by (simp_all add: neg_simps succ_pred)
huffman@28958: 
huffman@28958: text {* Less-than-or-equal *}
huffman@28958: 
huffman@28958: lemma le_bin_simps [simp]:
huffman@28958:   "Pls \<le> Pls \<longleftrightarrow> True"
huffman@28958:   "Pls \<le> Min \<longleftrightarrow> False"
huffman@28958:   "Pls \<le> Bit0 k \<longleftrightarrow> Pls \<le> k"
huffman@28958:   "Pls \<le> Bit1 k \<longleftrightarrow> Pls \<le> k"
huffman@28958:   "Min \<le> Pls \<longleftrightarrow> True"
huffman@28958:   "Min \<le> Min \<longleftrightarrow> True"
huffman@28958:   "Min \<le> Bit0 k \<longleftrightarrow> Min < k"
huffman@28958:   "Min \<le> Bit1 k \<longleftrightarrow> Min \<le> k"
huffman@28958:   "Bit0 k \<le> Pls \<longleftrightarrow> k \<le> Pls"
huffman@28958:   "Bit0 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958:   "Bit1 k \<le> Pls \<longleftrightarrow> k < Pls"
huffman@28958:   "Bit1 k \<le> Min \<longleftrightarrow> k \<le> Min"
huffman@28958:   "Bit0 k \<le> Bit0 l \<longleftrightarrow> k \<le> l"
huffman@28958:   "Bit0 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958:   "Bit1 k \<le> Bit0 l \<longleftrightarrow> k < l"
huffman@28958:   "Bit1 k \<le> Bit1 l \<longleftrightarrow> k \<le> l"
huffman@28958:   unfolding not_less [symmetric]
huffman@28958:   by (simp_all add: not_le)
huffman@28958: 
huffman@28958: text {* Equality *}
huffman@28958: 
huffman@28958: lemma eq_bin_simps [simp]:
huffman@28958:   "Pls = Pls \<longleftrightarrow> True"
huffman@28958:   "Pls = Min \<longleftrightarrow> False"
huffman@28958:   "Pls = Bit0 l \<longleftrightarrow> Pls = l"
huffman@28958:   "Pls = Bit1 l \<longleftrightarrow> False"
huffman@28958:   "Min = Pls \<longleftrightarrow> False"
huffman@28958:   "Min = Min \<longleftrightarrow> True"
huffman@28958:   "Min = Bit0 l \<longleftrightarrow> False"
huffman@28958:   "Min = Bit1 l \<longleftrightarrow> Min = l"
huffman@28958:   "Bit0 k = Pls \<longleftrightarrow> k = Pls"
huffman@28958:   "Bit0 k = Min \<longleftrightarrow> False"
huffman@28958:   "Bit1 k = Pls \<longleftrightarrow> False"
huffman@28958:   "Bit1 k = Min \<longleftrightarrow> k = Min"
huffman@28958:   "Bit0 k = Bit0 l \<longleftrightarrow> k = l"
huffman@28958:   "Bit0 k = Bit1 l \<longleftrightarrow> False"
huffman@28958:   "Bit1 k = Bit0 l \<longleftrightarrow> False"
huffman@28958:   "Bit1 k = Bit1 l \<longleftrightarrow> k = l"
huffman@28958:   unfolding order_eq_iff [where 'a=int]
huffman@28958:   by (simp_all add: not_less)
huffman@28958: 
huffman@28958: 
haftmann@25919: subsection {* Converting Numerals to Rings: @{term number_of} *}
haftmann@25919: 
haftmann@25919: class number_ring = number + comm_ring_1 +
haftmann@25919:   assumes number_of_eq: "number_of k = of_int k"
haftmann@25919: 
haftmann@25919: text {* self-embedding of the integers *}
haftmann@25919: 
haftmann@25919: instantiation int :: number_ring
haftmann@25919: begin
haftmann@25919: 
haftmann@28724: definition int_number_of_def [code del]:
haftmann@28724:   "number_of w = (of_int w \<Colon> int)"
haftmann@25919: 
haftmann@28724: instance proof
haftmann@28724: qed (simp only: int_number_of_def)
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: lemma number_of_is_id:
haftmann@25919:   "number_of (k::int) = k"
haftmann@25919:   unfolding int_number_of_def by simp
haftmann@25919: 
haftmann@25919: lemma number_of_succ:
haftmann@25919:   "number_of (succ k) = (1 + number_of k ::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: lemma number_of_pred:
haftmann@25919:   "number_of (pred w) = (- 1 + number_of w ::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: lemma number_of_minus:
haftmann@25919:   "number_of (uminus w) = (- (number_of w)::'a::number_ring)"
huffman@28958:   unfolding number_of_eq by (rule of_int_minus)
haftmann@25919: 
haftmann@25919: lemma number_of_add:
haftmann@25919:   "number_of (v + w) = (number_of v + number_of w::'a::number_ring)"
huffman@28958:   unfolding number_of_eq by (rule of_int_add)
huffman@28958: 
huffman@28958: lemma number_of_diff:
huffman@28958:   "number_of (v - w) = (number_of v - number_of w::'a::number_ring)"
huffman@28958:   unfolding number_of_eq by (rule of_int_diff)
haftmann@25919: 
haftmann@25919: lemma number_of_mult:
haftmann@25919:   "number_of (v * w) = (number_of v * number_of w::'a::number_ring)"
huffman@28958:   unfolding number_of_eq by (rule of_int_mult)
haftmann@25919: 
haftmann@25919: text {*
haftmann@25919:   The correctness of shifting.
haftmann@25919:   But it doesn't seem to give a measurable speed-up.
haftmann@25919: *}
haftmann@25919: 
huffman@26086: lemma double_number_of_Bit0:
huffman@26086:   "(1 + 1) * number_of w = (number_of (Bit0 w) ::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps left_distrib by simp
haftmann@25919: 
haftmann@25919: text {*
haftmann@25919:   Converting numerals 0 and 1 to their abstract versions.
haftmann@25919: *}
haftmann@25919: 
haftmann@25919: lemma numeral_0_eq_0 [simp]:
haftmann@25919:   "Numeral0 = (0::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: lemma numeral_1_eq_1 [simp]:
haftmann@25919:   "Numeral1 = (1::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: text {*
haftmann@25919:   Special-case simplification for small constants.
haftmann@25919: *}
haftmann@25919: 
haftmann@25919: text{*
haftmann@25919:   Unary minus for the abstract constant 1. Cannot be inserted
haftmann@25919:   as a simprule until later: it is @{text number_of_Min} re-oriented!
haftmann@25919: *}
haftmann@25919: 
haftmann@25919: lemma numeral_m1_eq_minus_1:
haftmann@25919:   "(-1::'a::number_ring) = - 1"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: lemma mult_minus1 [simp]:
haftmann@25919:   "-1 * z = -(z::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: lemma mult_minus1_right [simp]:
haftmann@25919:   "z * -1 = -(z::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: (*Negation of a coefficient*)
haftmann@25919: lemma minus_number_of_mult [simp]:
haftmann@25919:    "- (number_of w) * z = number_of (uminus w) * (z::'a::number_ring)"
haftmann@25919:    unfolding number_of_eq by simp
haftmann@25919: 
haftmann@25919: text {* Subtraction *}
haftmann@25919: 
haftmann@25919: lemma diff_number_of_eq:
haftmann@25919:   "number_of v - number_of w =
haftmann@25919:     (number_of (v + uminus w)::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq by simp
haftmann@25919: 
haftmann@25919: lemma number_of_Pls:
haftmann@25919:   "number_of Pls = (0::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: lemma number_of_Min:
haftmann@25919:   "number_of Min = (- 1::'a::number_ring)"
haftmann@25919:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
huffman@26086: lemma number_of_Bit0:
huffman@26086:   "number_of (Bit0 w) = (0::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086:   unfolding number_of_eq numeral_simps by simp
huffman@26086: 
huffman@26086: lemma number_of_Bit1:
huffman@26086:   "number_of (Bit1 w) = (1::'a::number_ring) + (number_of w) + (number_of w)"
huffman@26086:   unfolding number_of_eq numeral_simps by simp
haftmann@25919: 
haftmann@25919: 
huffman@28958: subsubsection {* Equality of Binary Numbers *}
haftmann@25919: 
haftmann@25919: text {* First version by Norbert Voelker *}
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   neg  :: "'a\<Colon>ordered_idom \<Rightarrow> bool"
haftmann@25919: where
haftmann@25919:   "neg Z \<longleftrightarrow> Z < 0"
haftmann@25919: 
haftmann@25919: definition (*for simplifying equalities*)
haftmann@25919:   iszero :: "'a\<Colon>semiring_1 \<Rightarrow> bool"
haftmann@25919: where
haftmann@25919:   "iszero z \<longleftrightarrow> z = 0"
haftmann@25919: 
haftmann@25919: lemma not_neg_int [simp]: "~ neg (of_nat n)"
haftmann@25919: by (simp add: neg_def)
haftmann@25919: 
haftmann@25919: lemma neg_zminus_int [simp]: "neg (- (of_nat (Suc n)))"
haftmann@25919: by (simp add: neg_def neg_less_0_iff_less del: of_nat_Suc)
haftmann@25919: 
haftmann@25919: lemmas neg_eq_less_0 = neg_def
haftmann@25919: 
haftmann@25919: lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
haftmann@25919: by (simp add: neg_def linorder_not_less)
haftmann@25919: 
haftmann@25919: text{*To simplify inequalities when Numeral1 can get simplified to 1*}
haftmann@25919: 
haftmann@25919: lemma not_neg_0: "~ neg 0"
haftmann@25919: by (simp add: One_int_def neg_def)
haftmann@25919: 
haftmann@25919: lemma not_neg_1: "~ neg 1"
haftmann@25919: by (simp add: neg_def linorder_not_less zero_le_one)
haftmann@25919: 
haftmann@25919: lemma iszero_0: "iszero 0"
haftmann@25919: by (simp add: iszero_def)
haftmann@25919: 
haftmann@25919: lemma not_iszero_1: "~ iszero 1"
haftmann@25919: by (simp add: iszero_def eq_commute)
haftmann@25919: 
haftmann@25919: lemma neg_nat: "neg z ==> nat z = 0"
haftmann@25919: by (simp add: neg_def order_less_imp_le) 
haftmann@25919: 
haftmann@25919: lemma not_neg_nat: "~ neg z ==> of_nat (nat z) = z"
haftmann@25919: by (simp add: linorder_not_less neg_def)
haftmann@25919: 
haftmann@25919: lemma eq_number_of_eq:
haftmann@25919:   "((number_of x::'a::number_ring) = number_of y) =
haftmann@25919:    iszero (number_of (x + uminus y) :: 'a)"
haftmann@25919:   unfolding iszero_def number_of_add number_of_minus
haftmann@25919:   by (simp add: compare_rls)
haftmann@25919: 
haftmann@25919: lemma iszero_number_of_Pls:
haftmann@25919:   "iszero ((number_of Pls)::'a::number_ring)"
haftmann@25919:   unfolding iszero_def numeral_0_eq_0 ..
haftmann@25919: 
haftmann@25919: lemma nonzero_number_of_Min:
haftmann@25919:   "~ iszero ((number_of Min)::'a::number_ring)"
haftmann@25919:   unfolding iszero_def numeral_m1_eq_minus_1 by simp
haftmann@25919: 
haftmann@25919: 
huffman@28958: subsubsection {* Comparisons, for Ordered Rings *}
haftmann@25919: 
haftmann@25919: lemmas double_eq_0_iff = double_zero
haftmann@25919: 
haftmann@25919: lemma odd_nonzero:
haftmann@25919:   "1 + z + z \<noteq> (0::int)";
haftmann@25919: proof (cases z rule: int_cases)
haftmann@25919:   case (nonneg n)
haftmann@25919:   have le: "0 \<le> z+z" by (simp add: nonneg add_increasing) 
haftmann@25919:   thus ?thesis using  le_imp_0_less [OF le]
haftmann@25919:     by (auto simp add: add_assoc) 
haftmann@25919: next
haftmann@25919:   case (neg n)
haftmann@25919:   show ?thesis
haftmann@25919:   proof
haftmann@25919:     assume eq: "1 + z + z = 0"
haftmann@25919:     have "(0::int) < 1 + (of_nat n + of_nat n)"
haftmann@25919:       by (simp add: le_imp_0_less add_increasing) 
haftmann@25919:     also have "... = - (1 + z + z)" 
haftmann@25919:       by (simp add: neg add_assoc [symmetric]) 
haftmann@25919:     also have "... = 0" by (simp add: eq) 
haftmann@25919:     finally have "0<0" ..
haftmann@25919:     thus False by blast
haftmann@25919:   qed
haftmann@25919: qed
haftmann@25919: 
huffman@26086: lemma iszero_number_of_Bit0:
huffman@26086:   "iszero (number_of (Bit0 w)::'a) = 
huffman@26086:    iszero (number_of w::'a::{ring_char_0,number_ring})"
haftmann@25919: proof -
haftmann@25919:   have "(of_int w + of_int w = (0::'a)) \<Longrightarrow> (w = 0)"
haftmann@25919:   proof -
haftmann@25919:     assume eq: "of_int w + of_int w = (0::'a)"
haftmann@25919:     then have "of_int (w + w) = (of_int 0 :: 'a)" by simp
haftmann@25919:     then have "w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919:     then show "w = 0" by (simp only: double_eq_0_iff)
haftmann@25919:   qed
huffman@26086:   thus ?thesis
huffman@26086:     by (auto simp add: iszero_def number_of_eq numeral_simps)
huffman@26086: qed
huffman@26086: 
huffman@26086: lemma iszero_number_of_Bit1:
huffman@26086:   "~ iszero (number_of (Bit1 w)::'a::{ring_char_0,number_ring})"
huffman@26086: proof -
huffman@26086:   have "1 + of_int w + of_int w \<noteq> (0::'a)"
haftmann@25919:   proof
haftmann@25919:     assume eq: "1 + of_int w + of_int w = (0::'a)"
haftmann@25919:     hence "of_int (1 + w + w) = (of_int 0 :: 'a)" by simp 
haftmann@25919:     hence "1 + w + w = 0" by (simp only: of_int_eq_iff)
haftmann@25919:     with odd_nonzero show False by blast
haftmann@25919:   qed
huffman@26086:   thus ?thesis
huffman@26086:     by (auto simp add: iszero_def number_of_eq numeral_simps)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
huffman@28958: subsubsection {* The Less-Than Relation *}
haftmann@25919: 
haftmann@25919: lemma less_number_of_eq_neg:
haftmann@25919:   "((number_of x::'a::{ordered_idom,number_ring}) < number_of y)
haftmann@25919:   = neg (number_of (x + uminus y) :: 'a)"
haftmann@25919: apply (subst less_iff_diff_less_0) 
haftmann@25919: apply (simp add: neg_def diff_minus number_of_add number_of_minus)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text {*
haftmann@25919:   If @{term Numeral0} is rewritten to 0 then this rule can't be applied:
haftmann@25919:   @{term Numeral0} IS @{term "number_of Pls"}
haftmann@25919: *}
haftmann@25919: 
haftmann@25919: lemma not_neg_number_of_Pls:
haftmann@25919:   "~ neg (number_of Pls ::'a::{ordered_idom,number_ring})"
haftmann@25919:   by (simp add: neg_def numeral_0_eq_0)
haftmann@25919: 
haftmann@25919: lemma neg_number_of_Min:
haftmann@25919:   "neg (number_of Min ::'a::{ordered_idom,number_ring})"
haftmann@25919:   by (simp add: neg_def zero_less_one numeral_m1_eq_minus_1)
haftmann@25919: 
haftmann@25919: lemma double_less_0_iff:
haftmann@25919:   "(a + a < 0) = (a < (0::'a::ordered_idom))"
haftmann@25919: proof -
haftmann@25919:   have "(a + a < 0) = ((1+1)*a < 0)" by (simp add: left_distrib)
haftmann@25919:   also have "... = (a < 0)"
haftmann@25919:     by (simp add: mult_less_0_iff zero_less_two 
haftmann@25919:                   order_less_not_sym [OF zero_less_two]) 
haftmann@25919:   finally show ?thesis .
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma odd_less_0:
haftmann@25919:   "(1 + z + z < 0) = (z < (0::int))";
haftmann@25919: proof (cases z rule: int_cases)
haftmann@25919:   case (nonneg n)
haftmann@25919:   thus ?thesis by (simp add: linorder_not_less add_assoc add_increasing
haftmann@25919:                              le_imp_0_less [THEN order_less_imp_le])  
haftmann@25919: next
haftmann@25919:   case (neg n)
haftmann@25919:   thus ?thesis by (simp del: of_nat_Suc of_nat_add
haftmann@25919:     add: compare_rls of_nat_1 [symmetric] of_nat_add [symmetric])
haftmann@25919: qed
haftmann@25919: 
huffman@26086: lemma neg_number_of_Bit0:
huffman@26086:   "neg (number_of (Bit0 w)::'a) = 
huffman@26086:   neg (number_of w :: 'a::{ordered_idom,number_ring})"
huffman@26086: by (simp add: neg_def number_of_eq numeral_simps double_less_0_iff)
huffman@26086: 
huffman@26086: lemma neg_number_of_Bit1:
huffman@26086:   "neg (number_of (Bit1 w)::'a) = 
haftmann@25919:   neg (number_of w :: 'a::{ordered_idom,number_ring})"
haftmann@25919: proof -
haftmann@25919:   have "((1::'a) + of_int w + of_int w < 0) = (of_int (1 + w + w) < (of_int 0 :: 'a))"
haftmann@25919:     by simp
haftmann@25919:   also have "... = (w < 0)" by (simp only: of_int_less_iff odd_less_0)
haftmann@25919:   finally show ?thesis
huffman@26086:   by (simp add: neg_def number_of_eq numeral_simps)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
haftmann@25919: text {* Less-Than or Equals *}
haftmann@25919: 
haftmann@25919: text {* Reduces @{term "a\<le>b"} to @{term "~ (b<a)"} for ALL numerals. *}
haftmann@25919: 
haftmann@25919: lemmas le_number_of_eq_not_less =
haftmann@25919:   linorder_not_less [of "number_of w" "number_of v", symmetric, 
haftmann@25919:   standard]
haftmann@25919: 
haftmann@25919: lemma le_number_of_eq:
haftmann@25919:     "((number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y)
haftmann@25919:      = (~ (neg (number_of (y + uminus x) :: 'a)))"
haftmann@25919: by (simp add: le_number_of_eq_not_less less_number_of_eq_neg)
haftmann@25919: 
haftmann@25919: 
haftmann@25919: text {* Absolute value (@{term abs}) *}
haftmann@25919: 
haftmann@25919: lemma abs_number_of:
haftmann@25919:   "abs(number_of x::'a::{ordered_idom,number_ring}) =
haftmann@25919:    (if number_of x < (0::'a) then -number_of x else number_of x)"
haftmann@25919:   by (simp add: abs_if)
haftmann@25919: 
haftmann@25919: 
haftmann@25919: text {* Re-orientation of the equation nnn=x *}
haftmann@25919: 
haftmann@25919: lemma number_of_reorient:
haftmann@25919:   "(number_of w = x) = (x = number_of w)"
haftmann@25919:   by auto
haftmann@25919: 
haftmann@25919: 
huffman@28958: subsubsection {* Simplification of arithmetic operations on integer constants. *}
haftmann@25919: 
haftmann@25919: lemmas arith_extra_simps [standard, simp] =
haftmann@25919:   number_of_add [symmetric]
huffman@28958:   number_of_minus [symmetric]
huffman@28958:   numeral_m1_eq_minus_1 [symmetric]
haftmann@25919:   number_of_mult [symmetric]
haftmann@25919:   diff_number_of_eq abs_number_of 
haftmann@25919: 
haftmann@25919: text {*
haftmann@25919:   For making a minimal simpset, one must include these default simprules.
haftmann@25919:   Also include @{text simp_thms}.
haftmann@25919: *}
haftmann@25919: 
haftmann@25919: lemmas arith_simps = 
huffman@26075:   normalize_bin_simps pred_bin_simps succ_bin_simps
huffman@26075:   add_bin_simps minus_bin_simps mult_bin_simps
haftmann@25919:   abs_zero abs_one arith_extra_simps
haftmann@25919: 
haftmann@25919: text {* Simplification of relational operations *}
haftmann@25919: 
huffman@28962: lemma less_number_of [simp]:
huffman@28962:   "(number_of x::'a::{ordered_idom,number_ring}) < number_of y \<longleftrightarrow> x < y"
huffman@28962:   unfolding number_of_eq by (rule of_int_less_iff)
huffman@28962: 
huffman@28962: lemma le_number_of [simp]:
huffman@28962:   "(number_of x::'a::{ordered_idom,number_ring}) \<le> number_of y \<longleftrightarrow> x \<le> y"
huffman@28962:   unfolding number_of_eq by (rule of_int_le_iff)
huffman@28962: 
huffman@28967: lemma eq_number_of [simp]:
huffman@28967:   "(number_of x::'a::{ring_char_0,number_ring}) = number_of y \<longleftrightarrow> x = y"
huffman@28967:   unfolding number_of_eq by (rule of_int_eq_iff)
huffman@28967: 
haftmann@25919: lemmas rel_simps [simp] = 
huffman@28962:   less_number_of less_bin_simps
huffman@28962:   le_number_of le_bin_simps
huffman@28967:   eq_number_of eq_bin_simps
huffman@28967:   iszero_0 nonzero_number_of_Min
huffman@26086:   iszero_number_of_Bit0 iszero_number_of_Bit1
haftmann@25919:   not_neg_number_of_Pls not_neg_0 not_neg_1 not_iszero_1
huffman@26086:   neg_number_of_Min neg_number_of_Bit0 neg_number_of_Bit1
haftmann@25919: (* iszero_number_of_Pls would never be used
haftmann@25919:    because its lhs simplifies to "iszero 0" *)
haftmann@25919: 
haftmann@25919: 
huffman@28958: subsubsection {* Simplification of arithmetic when nested to the right. *}
haftmann@25919: 
haftmann@25919: lemma add_number_of_left [simp]:
haftmann@25919:   "number_of v + (number_of w + z) =
haftmann@25919:    (number_of(v + w) + z::'a::number_ring)"
haftmann@25919:   by (simp add: add_assoc [symmetric])
haftmann@25919: 
haftmann@25919: lemma mult_number_of_left [simp]:
haftmann@25919:   "number_of v * (number_of w * z) =
haftmann@25919:    (number_of(v * w) * z::'a::number_ring)"
haftmann@25919:   by (simp add: mult_assoc [symmetric])
haftmann@25919: 
haftmann@25919: lemma add_number_of_diff1:
haftmann@25919:   "number_of v + (number_of w - c) = 
haftmann@25919:   number_of(v + w) - (c::'a::number_ring)"
haftmann@25919:   by (simp add: diff_minus add_number_of_left)
haftmann@25919: 
haftmann@25919: lemma add_number_of_diff2 [simp]:
haftmann@25919:   "number_of v + (c - number_of w) =
haftmann@25919:    number_of (v + uminus w) + (c::'a::number_ring)"
haftmann@25919: apply (subst diff_number_of_eq [symmetric])
haftmann@25919: apply (simp only: compare_rls)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* The Set of Integers *}
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   Ints  :: "'a set"
haftmann@25919: where
haftmann@28562:   [code del]: "Ints = range of_int"
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: notation (xsymbols)
haftmann@25919:   Ints  ("\<int>")
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: lemma Ints_0 [simp]: "0 \<in> \<int>"
haftmann@25919: apply (simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_0 [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma Ints_1 [simp]: "1 \<in> \<int>"
haftmann@25919: apply (simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_1 [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
haftmann@25919: apply (auto simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_add [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
haftmann@25919: apply (auto simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_minus [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
haftmann@25919: apply (auto simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_mult [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma Ints_cases [cases set: Ints]:
haftmann@25919:   assumes "q \<in> \<int>"
haftmann@25919:   obtains (of_int) z where "q = of_int z"
haftmann@25919:   unfolding Ints_def
haftmann@25919: proof -
haftmann@25919:   from `q \<in> \<int>` have "q \<in> range of_int" unfolding Ints_def .
haftmann@25919:   then obtain z where "q = of_int z" ..
haftmann@25919:   then show thesis ..
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma Ints_induct [case_names of_int, induct set: Ints]:
haftmann@25919:   "q \<in> \<int> \<Longrightarrow> (\<And>z. P (of_int z)) \<Longrightarrow> P q"
haftmann@25919:   by (rule Ints_cases) auto
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a-b \<in> \<int>"
haftmann@25919: apply (auto simp add: Ints_def)
haftmann@25919: apply (rule range_eqI)
haftmann@25919: apply (rule of_int_diff [symmetric])
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text {* The premise involving @{term Ints} prevents @{term "a = 1/2"}. *}
haftmann@25919: 
haftmann@25919: lemma Ints_double_eq_0_iff:
haftmann@25919:   assumes in_Ints: "a \<in> Ints"
haftmann@25919:   shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
haftmann@25919: proof -
haftmann@25919:   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919:   then obtain z where a: "a = of_int z" ..
haftmann@25919:   show ?thesis
haftmann@25919:   proof
haftmann@25919:     assume "a = 0"
haftmann@25919:     thus "a + a = 0" by simp
haftmann@25919:   next
haftmann@25919:     assume eq: "a + a = 0"
haftmann@25919:     hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919:     hence "z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919:     hence "z = 0" by (simp only: double_eq_0_iff)
haftmann@25919:     thus "a = 0" by (simp add: a)
haftmann@25919:   qed
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma Ints_odd_nonzero:
haftmann@25919:   assumes in_Ints: "a \<in> Ints"
haftmann@25919:   shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
haftmann@25919: proof -
haftmann@25919:   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919:   then obtain z where a: "a = of_int z" ..
haftmann@25919:   show ?thesis
haftmann@25919:   proof
haftmann@25919:     assume eq: "1 + a + a = 0"
haftmann@25919:     hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
haftmann@25919:     hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
haftmann@25919:     with odd_nonzero show False by blast
haftmann@25919:   qed
haftmann@25919: qed 
haftmann@25919: 
haftmann@25919: lemma Ints_number_of:
haftmann@25919:   "(number_of w :: 'a::number_ring) \<in> Ints"
haftmann@25919:   unfolding number_of_eq Ints_def by simp
haftmann@25919: 
haftmann@25919: lemma Ints_odd_less_0: 
haftmann@25919:   assumes in_Ints: "a \<in> Ints"
haftmann@25919:   shows "(1 + a + a < 0) = (a < (0::'a::ordered_idom))";
haftmann@25919: proof -
haftmann@25919:   from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
haftmann@25919:   then obtain z where a: "a = of_int z" ..
haftmann@25919:   hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
haftmann@25919:     by (simp add: a)
haftmann@25919:   also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0)
haftmann@25919:   also have "... = (a < 0)" by (simp add: a)
haftmann@25919:   finally show ?thesis .
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* @{term setsum} and @{term setprod} *}
haftmann@25919: 
haftmann@25919: text {*By Jeremy Avigad*}
haftmann@25919: 
haftmann@25919: lemma of_nat_setsum: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemma of_int_setsum: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemma of_nat_setprod: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto simp add: of_nat_mult)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemma of_int_setprod: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
haftmann@25919:   apply (cases "finite A")
haftmann@25919:   apply (erule finite_induct, auto)
haftmann@25919:   done
haftmann@25919: 
haftmann@25919: lemma setprod_nonzero_nat:
haftmann@25919:     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::nat)) ==> setprod f A \<noteq> 0"
haftmann@25919:   by (rule setprod_nonzero, auto)
haftmann@25919: 
haftmann@25919: lemma setprod_zero_eq_nat:
haftmann@25919:     "finite A ==> (setprod f A = (0::nat)) = (\<exists>x \<in> A. f x = 0)"
haftmann@25919:   by (rule setprod_zero_eq, auto)
haftmann@25919: 
haftmann@25919: lemma setprod_nonzero_int:
haftmann@25919:     "finite A ==> (\<forall>x \<in> A. f x \<noteq> (0::int)) ==> setprod f A \<noteq> 0"
haftmann@25919:   by (rule setprod_nonzero, auto)
haftmann@25919: 
haftmann@25919: lemma setprod_zero_eq_int:
haftmann@25919:     "finite A ==> (setprod f A = (0::int)) = (\<exists>x \<in> A. f x = 0)"
haftmann@25919:   by (rule setprod_zero_eq, auto)
haftmann@25919: 
haftmann@25919: lemmas int_setsum = of_nat_setsum [where 'a=int]
haftmann@25919: lemmas int_setprod = of_nat_setprod [where 'a=int]
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*Inequality Reasoning for the Arithmetic Simproc*}
haftmann@25919: 
haftmann@25919: lemma add_numeral_0: "Numeral0 + a = (a::'a::number_ring)"
haftmann@25919: by simp 
haftmann@25919: 
haftmann@25919: lemma add_numeral_0_right: "a + Numeral0 = (a::'a::number_ring)"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: lemma mult_numeral_1: "Numeral1 * a = (a::'a::number_ring)"
haftmann@25919: by simp 
haftmann@25919: 
haftmann@25919: lemma mult_numeral_1_right: "a * Numeral1 = (a::'a::number_ring)"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: lemma divide_numeral_1: "a / Numeral1 = (a::'a::{number_ring,field})"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: lemma inverse_numeral_1:
haftmann@25919:   "inverse Numeral1 = (Numeral1::'a::{number_ring,field})"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: text{*Theorem lists for the cancellation simprocs. The use of binary numerals
haftmann@25919: for 0 and 1 reduces the number of special cases.*}
haftmann@25919: 
haftmann@25919: lemmas add_0s = add_numeral_0 add_numeral_0_right
haftmann@25919: lemmas mult_1s = mult_numeral_1 mult_numeral_1_right 
haftmann@25919:                  mult_minus1 mult_minus1_right
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*Special Arithmetic Rules for Abstract 0 and 1*}
haftmann@25919: 
haftmann@25919: text{*Arithmetic computations are defined for binary literals, which leaves 0
haftmann@25919: and 1 as special cases. Addition already has rules for 0, but not 1.
haftmann@25919: Multiplication and unary minus already have rules for both 0 and 1.*}
haftmann@25919: 
haftmann@25919: 
haftmann@25919: lemma binop_eq: "[|f x y = g x y; x = x'; y = y'|] ==> f x' y' = g x' y'"
haftmann@25919: by simp
haftmann@25919: 
haftmann@25919: 
haftmann@25919: lemmas add_number_of_eq = number_of_add [symmetric]
haftmann@25919: 
haftmann@25919: text{*Allow 1 on either or both sides*}
haftmann@25919: lemma one_add_one_is_two: "1 + 1 = (2::'a::number_ring)"
haftmann@25919: by (simp del: numeral_1_eq_1 add: numeral_1_eq_1 [symmetric] add_number_of_eq)
haftmann@25919: 
haftmann@25919: lemmas add_special =
haftmann@25919:     one_add_one_is_two
haftmann@25919:     binop_eq [of "op +", OF add_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919:     binop_eq [of "op +", OF add_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919: 
haftmann@25919: text{*Allow 1 on either or both sides (1-1 already simplifies to 0)*}
haftmann@25919: lemmas diff_special =
haftmann@25919:     binop_eq [of "op -", OF diff_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919:     binop_eq [of "op -", OF diff_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919: 
haftmann@25919: text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919: lemmas eq_special =
haftmann@25919:     binop_eq [of "op =", OF eq_number_of_eq numeral_0_eq_0 refl, standard]
haftmann@25919:     binop_eq [of "op =", OF eq_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919:     binop_eq [of "op =", OF eq_number_of_eq refl numeral_0_eq_0, standard]
haftmann@25919:     binop_eq [of "op =", OF eq_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919: 
haftmann@25919: text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919: lemmas less_special =
haftmann@25919:   binop_eq [of "op <", OF less_number_of_eq_neg numeral_0_eq_0 refl, standard]
haftmann@25919:   binop_eq [of "op <", OF less_number_of_eq_neg numeral_1_eq_1 refl, standard]
haftmann@25919:   binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_0_eq_0, standard]
haftmann@25919:   binop_eq [of "op <", OF less_number_of_eq_neg refl numeral_1_eq_1, standard]
haftmann@25919: 
haftmann@25919: text{*Allow 0 or 1 on either side with a binary numeral on the other*}
haftmann@25919: lemmas le_special =
haftmann@25919:     binop_eq [of "op \<le>", OF le_number_of_eq numeral_0_eq_0 refl, standard]
haftmann@25919:     binop_eq [of "op \<le>", OF le_number_of_eq numeral_1_eq_1 refl, standard]
haftmann@25919:     binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_0_eq_0, standard]
haftmann@25919:     binop_eq [of "op \<le>", OF le_number_of_eq refl numeral_1_eq_1, standard]
haftmann@25919: 
haftmann@25919: lemmas arith_special[simp] = 
haftmann@25919:        add_special diff_special eq_special less_special le_special
haftmann@25919: 
haftmann@25919: 
haftmann@25919: lemma min_max_01: "min (0::int) 1 = 0 & min (1::int) 0 = 0 &
haftmann@25919:                    max (0::int) 1 = 1 & max (1::int) 0 = 1"
haftmann@25919: by(simp add:min_def max_def)
haftmann@25919: 
haftmann@25919: lemmas min_max_special[simp] =
haftmann@25919:  min_max_01
haftmann@25919:  max_def[of "0::int" "number_of v", standard, simp]
haftmann@25919:  min_def[of "0::int" "number_of v", standard, simp]
haftmann@25919:  max_def[of "number_of u" "0::int", standard, simp]
haftmann@25919:  min_def[of "number_of u" "0::int", standard, simp]
haftmann@25919:  max_def[of "1::int" "number_of v", standard, simp]
haftmann@25919:  min_def[of "1::int" "number_of v", standard, simp]
haftmann@25919:  max_def[of "number_of u" "1::int", standard, simp]
haftmann@25919:  min_def[of "number_of u" "1::int", standard, simp]
haftmann@25919: 
haftmann@25919: text {* Legacy theorems *}
haftmann@25919: 
haftmann@25919: lemmas zle_int = of_nat_le_iff [where 'a=int]
haftmann@25919: lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
haftmann@25919: 
haftmann@25919: use "~~/src/Provers/Arith/assoc_fold.ML"
haftmann@28952: use "Tools/int_arith.ML"
haftmann@25919: declaration {* K int_arith_setup *}
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*Lemmas About Small Numerals*}
haftmann@25919: 
haftmann@25919: lemma of_int_m1 [simp]: "of_int -1 = (-1 :: 'a :: number_ring)"
haftmann@25919: proof -
haftmann@25919:   have "(of_int -1 :: 'a) = of_int (- 1)" by simp
haftmann@25919:   also have "... = - of_int 1" by (simp only: of_int_minus)
haftmann@25919:   also have "... = -1" by simp
haftmann@25919:   finally show ?thesis .
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma abs_minus_one [simp]: "abs (-1) = (1::'a::{ordered_idom,number_ring})"
haftmann@25919: by (simp add: abs_if)
haftmann@25919: 
haftmann@25919: lemma abs_power_minus_one [simp]:
haftmann@25919:      "abs(-1 ^ n) = (1::'a::{ordered_idom,number_ring,recpower})"
haftmann@25919: by (simp add: power_abs)
haftmann@25919: 
haftmann@25919: lemma of_int_number_of_eq:
haftmann@25919:      "of_int (number_of v) = (number_of v :: 'a :: number_ring)"
haftmann@25919: by (simp add: number_of_eq) 
haftmann@25919: 
haftmann@25919: text{*Lemmas for specialist use, NOT as default simprules*}
haftmann@25919: lemma mult_2: "2 * z = (z+z::'a::number_ring)"
haftmann@25919: proof -
haftmann@25919:   have "2*z = (1 + 1)*z" by simp
haftmann@25919:   also have "... = z+z" by (simp add: left_distrib)
haftmann@25919:   finally show ?thesis .
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma mult_2_right: "z * 2 = (z+z::'a::number_ring)"
haftmann@25919: by (subst mult_commute, rule mult_2)
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*More Inequality Reasoning*}
haftmann@25919: 
haftmann@25919: lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: 
huffman@28958: subsection{*The functions @{term nat} and @{term int}*}
haftmann@25919: 
haftmann@25919: text{*Simplify the terms @{term "int 0"}, @{term "int(Suc 0)"} and
haftmann@25919:   @{term "w + - z"}*}
haftmann@25919: declare Zero_int_def [symmetric, simp]
haftmann@25919: declare One_int_def [symmetric, simp]
haftmann@25919: 
haftmann@25919: lemmas diff_int_def_symmetric = diff_int_def [symmetric, simp]
haftmann@25919: 
haftmann@25919: lemma nat_0: "nat 0 = 0"
haftmann@25919: by (simp add: nat_eq_iff)
haftmann@25919: 
haftmann@25919: lemma nat_1: "nat 1 = Suc 0"
haftmann@25919: by (subst nat_eq_iff, simp)
haftmann@25919: 
haftmann@25919: lemma nat_2: "nat 2 = Suc (Suc 0)"
haftmann@25919: by (subst nat_eq_iff, simp)
haftmann@25919: 
haftmann@25919: lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
haftmann@25919: apply (insert zless_nat_conj [of 1 z])
haftmann@25919: apply (auto simp add: nat_1)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: text{*This simplifies expressions of the form @{term "int n = z"} where
haftmann@25919:       z is an integer literal.*}
haftmann@25919: lemmas int_eq_iff_number_of [simp] = int_eq_iff [of _ "number_of v", standard]
haftmann@25919: 
haftmann@25919: lemma split_nat [arith_split]:
haftmann@25919:   "P(nat(i::int)) = ((\<forall>n. i = of_nat n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
haftmann@25919:   (is "?P = (?L & ?R)")
haftmann@25919: proof (cases "i < 0")
haftmann@25919:   case True thus ?thesis by auto
haftmann@25919: next
haftmann@25919:   case False
haftmann@25919:   have "?P = ?L"
haftmann@25919:   proof
haftmann@25919:     assume ?P thus ?L using False by clarsimp
haftmann@25919:   next
haftmann@25919:     assume ?L thus ?P using False by simp
haftmann@25919:   qed
haftmann@25919:   with False show ?thesis by simp
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: context ring_1
haftmann@25919: begin
haftmann@25919: 
haftmann@25919: lemma of_int_of_nat:
haftmann@25919:   "of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
haftmann@25919: proof (cases "k < 0")
haftmann@25919:   case True then have "0 \<le> - k" by simp
haftmann@25919:   then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
haftmann@25919:   with True show ?thesis by simp
haftmann@25919: next
haftmann@25919:   case False then show ?thesis by (simp add: not_less of_nat_nat)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: end
haftmann@25919: 
haftmann@25919: lemma nat_mult_distrib:
haftmann@25919:   fixes z z' :: int
haftmann@25919:   assumes "0 \<le> z"
haftmann@25919:   shows "nat (z * z') = nat z * nat z'"
haftmann@25919: proof (cases "0 \<le> z'")
haftmann@25919:   case False with assms have "z * z' \<le> 0"
haftmann@25919:     by (simp add: not_le mult_le_0_iff)
haftmann@25919:   then have "nat (z * z') = 0" by simp
haftmann@25919:   moreover from False have "nat z' = 0" by simp
haftmann@25919:   ultimately show ?thesis by simp
haftmann@25919: next
haftmann@25919:   case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
haftmann@25919:   show ?thesis
haftmann@25919:     by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
haftmann@25919:       (simp only: of_nat_mult of_nat_nat [OF True]
haftmann@25919:          of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
haftmann@25919: apply (rule trans)
haftmann@25919: apply (rule_tac [2] nat_mult_distrib, auto)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma nat_abs_mult_distrib: "nat (abs (w * z)) = nat (abs w) * nat (abs z)"
haftmann@25919: apply (cases "z=0 | w=0")
haftmann@25919: apply (auto simp add: abs_if nat_mult_distrib [symmetric] 
haftmann@25919:                       nat_mult_distrib_neg [symmetric] mult_less_0_iff)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection "Induction principles for int"
haftmann@25919: 
haftmann@25919: text{*Well-founded segments of the integers*}
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   int_ge_less_than  ::  "int => (int * int) set"
haftmann@25919: where
haftmann@25919:   "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
haftmann@25919: 
haftmann@25919: theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
haftmann@25919: proof -
haftmann@25919:   have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
haftmann@25919:     by (auto simp add: int_ge_less_than_def)
haftmann@25919:   thus ?thesis 
haftmann@25919:     by (rule wf_subset [OF wf_measure]) 
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: text{*This variant looks odd, but is typical of the relations suggested
haftmann@25919: by RankFinder.*}
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   int_ge_less_than2 ::  "int => (int * int) set"
haftmann@25919: where
haftmann@25919:   "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
haftmann@25919: 
haftmann@25919: theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
haftmann@25919: proof -
haftmann@25919:   have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))" 
haftmann@25919:     by (auto simp add: int_ge_less_than2_def)
haftmann@25919:   thus ?thesis 
haftmann@25919:     by (rule wf_subset [OF wf_measure]) 
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: abbreviation
haftmann@25919:   int :: "nat \<Rightarrow> int"
haftmann@25919: where
haftmann@25919:   "int \<equiv> of_nat"
haftmann@25919: 
haftmann@25919: (* `set:int': dummy construction *)
haftmann@25919: theorem int_ge_induct [case_names base step, induct set: int]:
haftmann@25919:   fixes i :: int
haftmann@25919:   assumes ge: "k \<le> i" and
haftmann@25919:     base: "P k" and
haftmann@25919:     step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
haftmann@25919:   shows "P i"
haftmann@25919: proof -
haftmann@25919:   { fix n have "\<And>i::int. n = nat(i-k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
haftmann@25919:     proof (induct n)
haftmann@25919:       case 0
haftmann@25919:       hence "i = k" by arith
haftmann@25919:       thus "P i" using base by simp
haftmann@25919:     next
haftmann@25919:       case (Suc n)
haftmann@25919:       then have "n = nat((i - 1) - k)" by arith
haftmann@25919:       moreover
haftmann@25919:       have ki1: "k \<le> i - 1" using Suc.prems by arith
haftmann@25919:       ultimately
haftmann@25919:       have "P(i - 1)" by(rule Suc.hyps)
haftmann@25919:       from step[OF ki1 this] show ?case by simp
haftmann@25919:     qed
haftmann@25919:   }
haftmann@25919:   with ge show ?thesis by fast
haftmann@25919: qed
haftmann@25919: 
haftmann@25928: (* `set:int': dummy construction *)
haftmann@25928: theorem int_gr_induct [case_names base step, induct set: int]:
haftmann@25919:   assumes gr: "k < (i::int)" and
haftmann@25919:         base: "P(k+1)" and
haftmann@25919:         step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
haftmann@25919:   shows "P i"
haftmann@25919: apply(rule int_ge_induct[of "k + 1"])
haftmann@25919:   using gr apply arith
haftmann@25919:  apply(rule base)
haftmann@25919: apply (rule step, simp+)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: theorem int_le_induct[consumes 1,case_names base step]:
haftmann@25919:   assumes le: "i \<le> (k::int)" and
haftmann@25919:         base: "P(k)" and
haftmann@25919:         step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919:   shows "P i"
haftmann@25919: proof -
haftmann@25919:   { fix n have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
haftmann@25919:     proof (induct n)
haftmann@25919:       case 0
haftmann@25919:       hence "i = k" by arith
haftmann@25919:       thus "P i" using base by simp
haftmann@25919:     next
haftmann@25919:       case (Suc n)
haftmann@25919:       hence "n = nat(k - (i+1))" by arith
haftmann@25919:       moreover
haftmann@25919:       have ki1: "i + 1 \<le> k" using Suc.prems by arith
haftmann@25919:       ultimately
haftmann@25919:       have "P(i+1)" by(rule Suc.hyps)
haftmann@25919:       from step[OF ki1 this] show ?case by simp
haftmann@25919:     qed
haftmann@25919:   }
haftmann@25919:   with le show ?thesis by fast
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: theorem int_less_induct [consumes 1,case_names base step]:
haftmann@25919:   assumes less: "(i::int) < k" and
haftmann@25919:         base: "P(k - 1)" and
haftmann@25919:         step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
haftmann@25919:   shows "P i"
haftmann@25919: apply(rule int_le_induct[of _ "k - 1"])
haftmann@25919:   using less apply arith
haftmann@25919:  apply(rule base)
haftmann@25919: apply (rule step, simp+)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: subsection{*Intermediate value theorems*}
haftmann@25919: 
haftmann@25919: lemma int_val_lemma:
haftmann@25919:      "(\<forall>i<n::nat. abs(f(i+1) - f i) \<le> 1) -->  
haftmann@25919:       f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
haftmann@27106: apply (induct n, simp)
haftmann@25919: apply (intro strip)
haftmann@25919: apply (erule impE, simp)
haftmann@25919: apply (erule_tac x = n in allE, simp)
haftmann@25919: apply (case_tac "k = f (n+1) ")
haftmann@27106: apply force
haftmann@25919: apply (erule impE)
haftmann@25919:  apply (simp add: abs_if split add: split_if_asm)
haftmann@25919: apply (blast intro: le_SucI)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]
haftmann@25919: 
haftmann@25919: lemma nat_intermed_int_val:
haftmann@25919:      "[| \<forall>i. m \<le> i & i < n --> abs(f(i + 1::nat) - f i) \<le> 1; m < n;  
haftmann@25919:          f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
haftmann@25919: apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k 
haftmann@25919:        in int_val_lemma)
haftmann@25919: apply simp
haftmann@25919: apply (erule exE)
haftmann@25919: apply (rule_tac x = "i+m" in exI, arith)
haftmann@25919: done
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection{*Products and 1, by T. M. Rasmussen*}
haftmann@25919: 
haftmann@25919: lemma zabs_less_one_iff [simp]: "(\<bar>z\<bar> < 1) = (z = (0::int))"
haftmann@25919: by arith
haftmann@25919: 
haftmann@25919: lemma abs_zmult_eq_1: "(\<bar>m * n\<bar> = 1) ==> \<bar>m\<bar> = (1::int)"
haftmann@25919: apply (cases "\<bar>n\<bar>=1") 
haftmann@25919: apply (simp add: abs_mult) 
haftmann@25919: apply (rule ccontr) 
haftmann@25919: apply (auto simp add: linorder_neq_iff abs_mult) 
haftmann@25919: apply (subgoal_tac "2 \<le> \<bar>m\<bar> & 2 \<le> \<bar>n\<bar>")
haftmann@25919:  prefer 2 apply arith 
haftmann@25919: apply (subgoal_tac "2*2 \<le> \<bar>m\<bar> * \<bar>n\<bar>", simp) 
haftmann@25919: apply (rule mult_mono, auto) 
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
haftmann@25919: by (insert abs_zmult_eq_1 [of m n], arith)
haftmann@25919: 
haftmann@25919: lemma pos_zmult_eq_1_iff: "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)"
haftmann@25919: apply (auto dest: pos_zmult_eq_1_iff_lemma) 
haftmann@25919: apply (simp add: mult_commute [of m]) 
haftmann@25919: apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919: done
haftmann@25919: 
haftmann@25919: lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
haftmann@25919: apply (rule iffI) 
haftmann@25919:  apply (frule pos_zmult_eq_1_iff_lemma)
haftmann@25919:  apply (simp add: mult_commute [of m]) 
haftmann@25919:  apply (frule pos_zmult_eq_1_iff_lemma, auto) 
haftmann@25919: done
haftmann@25919: 
haftmann@25919: (* Could be simplified but Presburger only becomes available too late *)
haftmann@25919: lemma infinite_UNIV_int: "~finite(UNIV::int set)"
haftmann@25919: proof
haftmann@25919:   assume "finite(UNIV::int set)"
haftmann@25919:   moreover have "~(EX i::int. 2*i = 1)"
haftmann@25919:     by (auto simp: pos_zmult_eq_1_iff)
haftmann@25919:   ultimately show False using finite_UNIV_inj_surj[of "%n::int. n+n"]
haftmann@25919:     by (simp add:inj_on_def surj_def) (blast intro:sym)
haftmann@25919: qed
haftmann@25919: 
haftmann@25919: 
haftmann@25961: subsection{*Integer Powers*} 
haftmann@25961: 
haftmann@25961: instantiation int :: recpower
haftmann@25961: begin
haftmann@25961: 
haftmann@25961: primrec power_int where
haftmann@25961:   "p ^ 0 = (1\<Colon>int)"
haftmann@25961:   | "p ^ (Suc n) = (p\<Colon>int) * (p ^ n)"
haftmann@25961: 
haftmann@25961: instance proof
haftmann@25961:   fix z :: int
haftmann@25961:   fix n :: nat
haftmann@25961:   show "z ^ 0 = 1" by simp
haftmann@25961:   show "z ^ Suc n = z * (z ^ n)" by simp
haftmann@25961: qed
haftmann@25961: 
haftmann@25961: end
haftmann@25961: 
haftmann@25961: lemma zpower_zadd_distrib: "x ^ (y + z) = ((x ^ y) * (x ^ z)::int)"
haftmann@25961:   by (rule Power.power_add)
haftmann@25961: 
haftmann@25961: lemma zpower_zpower: "(x ^ y) ^ z = (x ^ (y * z)::int)"
haftmann@25961:   by (rule Power.power_mult [symmetric])
haftmann@25961: 
haftmann@25961: lemma zero_less_zpower_abs_iff [simp]:
haftmann@25961:   "(0 < abs x ^ n) \<longleftrightarrow> (x \<noteq> (0::int) | n = 0)"
haftmann@25961:   by (induct n) (auto simp add: zero_less_mult_iff)
haftmann@25961: 
haftmann@25961: lemma zero_le_zpower_abs [simp]: "(0::int) \<le> abs x ^ n"
haftmann@25961:   by (induct n) (auto simp add: zero_le_mult_iff)
haftmann@25961: 
haftmann@25961: lemma of_int_power:
haftmann@25961:   "of_int (z ^ n) = (of_int z ^ n :: 'a::{recpower, ring_1})"
haftmann@25961:   by (induct n) (simp_all add: power_Suc)
haftmann@25961: 
haftmann@25961: lemma int_power: "int (m^n) = (int m) ^ n"
haftmann@25961:   by (rule of_nat_power)
haftmann@25961: 
haftmann@25961: lemmas zpower_int = int_power [symmetric]
haftmann@25961: 
haftmann@25919: subsection {* Configuration of the code generator *}
haftmann@25919: 
haftmann@26507: code_datatype Pls Min Bit0 Bit1 "number_of \<Colon> int \<Rightarrow> int"
haftmann@26507: 
haftmann@28562: lemmas pred_succ_numeral_code [code] =
haftmann@26507:   pred_bin_simps succ_bin_simps
haftmann@26507: 
haftmann@28562: lemmas plus_numeral_code [code] =
haftmann@26507:   add_bin_simps
haftmann@26507:   arith_extra_simps(1) [where 'a = int]
haftmann@26507: 
haftmann@28562: lemmas minus_numeral_code [code] =
haftmann@26507:   minus_bin_simps
haftmann@26507:   arith_extra_simps(2) [where 'a = int]
haftmann@26507:   arith_extra_simps(5) [where 'a = int]
haftmann@26507: 
haftmann@28562: lemmas times_numeral_code [code] =
haftmann@26507:   mult_bin_simps
haftmann@26507:   arith_extra_simps(4) [where 'a = int]
haftmann@26507: 
haftmann@26507: instantiation int :: eq
haftmann@26507: begin
haftmann@26507: 
haftmann@28562: definition [code del]: "eq_class.eq k l \<longleftrightarrow> k - l = (0\<Colon>int)"
haftmann@26507: 
haftmann@26507: instance by default (simp add: eq_int_def)
haftmann@26507: 
haftmann@26507: end
haftmann@26507: 
haftmann@28562: lemma eq_number_of_int_code [code]:
haftmann@26732:   "eq_class.eq (number_of k \<Colon> int) (number_of l) \<longleftrightarrow> eq_class.eq k l"
haftmann@26507:   unfolding eq_int_def number_of_is_id ..
haftmann@26507: 
haftmann@28562: lemma eq_int_code [code]:
haftmann@26732:   "eq_class.eq Int.Pls Int.Pls \<longleftrightarrow> True"
haftmann@26732:   "eq_class.eq Int.Pls Int.Min \<longleftrightarrow> False"
haftmann@26732:   "eq_class.eq Int.Pls (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq Int.Pls k2"
haftmann@26732:   "eq_class.eq Int.Pls (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@26732:   "eq_class.eq Int.Min Int.Pls \<longleftrightarrow> False"
haftmann@26732:   "eq_class.eq Int.Min Int.Min \<longleftrightarrow> True"
haftmann@26732:   "eq_class.eq Int.Min (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@26732:   "eq_class.eq Int.Min (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq Int.Min k2"
huffman@28958:   "eq_class.eq (Int.Bit0 k1) Int.Pls \<longleftrightarrow> eq_class.eq k1 Int.Pls"
haftmann@26732:   "eq_class.eq (Int.Bit1 k1) Int.Pls \<longleftrightarrow> False"
haftmann@26732:   "eq_class.eq (Int.Bit0 k1) Int.Min \<longleftrightarrow> False"
huffman@28958:   "eq_class.eq (Int.Bit1 k1) Int.Min \<longleftrightarrow> eq_class.eq k1 Int.Min"
haftmann@26732:   "eq_class.eq (Int.Bit0 k1) (Int.Bit0 k2) \<longleftrightarrow> eq_class.eq k1 k2"
haftmann@26732:   "eq_class.eq (Int.Bit0 k1) (Int.Bit1 k2) \<longleftrightarrow> False"
haftmann@26732:   "eq_class.eq (Int.Bit1 k1) (Int.Bit0 k2) \<longleftrightarrow> False"
haftmann@26732:   "eq_class.eq (Int.Bit1 k1) (Int.Bit1 k2) \<longleftrightarrow> eq_class.eq k1 k2"
huffman@28958:   unfolding eq_equals by simp_all
haftmann@25919: 
haftmann@28351: lemma eq_int_refl [code nbe]:
haftmann@28351:   "eq_class.eq (k::int) k \<longleftrightarrow> True"
haftmann@28351:   by (rule HOL.eq_refl)
haftmann@28351: 
haftmann@28562: lemma less_eq_number_of_int_code [code]:
haftmann@26507:   "(number_of k \<Colon> int) \<le> number_of l \<longleftrightarrow> k \<le> l"
haftmann@26507:   unfolding number_of_is_id ..
haftmann@26507: 
haftmann@28562: lemma less_eq_int_code [code]:
haftmann@26507:   "Int.Pls \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507:   "Int.Pls \<le> Int.Min \<longleftrightarrow> False"
haftmann@26507:   "Int.Pls \<le> Int.Bit0 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507:   "Int.Pls \<le> Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507:   "Int.Min \<le> Int.Pls \<longleftrightarrow> True"
haftmann@26507:   "Int.Min \<le> Int.Min \<longleftrightarrow> True"
haftmann@26507:   "Int.Min \<le> Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507:   "Int.Min \<le> Int.Bit1 k \<longleftrightarrow> Int.Min \<le> k"
haftmann@26507:   "Int.Bit0 k \<le> Int.Pls \<longleftrightarrow> k \<le> Int.Pls"
haftmann@26507:   "Int.Bit1 k \<le> Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507:   "Int.Bit0 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507:   "Int.Bit1 k \<le> Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507:   "Int.Bit0 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507:   "Int.Bit0 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507:   "Int.Bit1 k1 \<le> Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507:   "Int.Bit1 k1 \<le> Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
huffman@28958:   by simp_all
haftmann@26507: 
haftmann@28562: lemma less_number_of_int_code [code]:
haftmann@26507:   "(number_of k \<Colon> int) < number_of l \<longleftrightarrow> k < l"
haftmann@26507:   unfolding number_of_is_id ..
haftmann@26507: 
haftmann@28562: lemma less_int_code [code]:
haftmann@26507:   "Int.Pls < Int.Pls \<longleftrightarrow> False"
haftmann@26507:   "Int.Pls < Int.Min \<longleftrightarrow> False"
haftmann@26507:   "Int.Pls < Int.Bit0 k \<longleftrightarrow> Int.Pls < k"
haftmann@26507:   "Int.Pls < Int.Bit1 k \<longleftrightarrow> Int.Pls \<le> k"
haftmann@26507:   "Int.Min < Int.Pls \<longleftrightarrow> True"
haftmann@26507:   "Int.Min < Int.Min \<longleftrightarrow> False"
haftmann@26507:   "Int.Min < Int.Bit0 k \<longleftrightarrow> Int.Min < k"
haftmann@26507:   "Int.Min < Int.Bit1 k \<longleftrightarrow> Int.Min < k"
haftmann@26507:   "Int.Bit0 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507:   "Int.Bit1 k < Int.Pls \<longleftrightarrow> k < Int.Pls"
haftmann@26507:   "Int.Bit0 k < Int.Min \<longleftrightarrow> k \<le> Int.Min"
haftmann@26507:   "Int.Bit1 k < Int.Min \<longleftrightarrow> k < Int.Min"
haftmann@26507:   "Int.Bit0 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507:   "Int.Bit0 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 \<le> k2"
haftmann@26507:   "Int.Bit1 k1 < Int.Bit0 k2 \<longleftrightarrow> k1 < k2"
haftmann@26507:   "Int.Bit1 k1 < Int.Bit1 k2 \<longleftrightarrow> k1 < k2"
huffman@28958:   by simp_all
haftmann@25919: 
haftmann@25919: definition
haftmann@25919:   nat_aux :: "int \<Rightarrow> nat \<Rightarrow> nat" where
haftmann@25919:   "nat_aux i n = nat i + n"
haftmann@25919: 
haftmann@25919: lemma [code]:
haftmann@25919:   "nat_aux i n = (if i \<le> 0 then n else nat_aux (i - 1) (Suc n))"  -- {* tail recursive *}
haftmann@25919:   by (auto simp add: nat_aux_def nat_eq_iff linorder_not_le order_less_imp_le
haftmann@25919:     dest: zless_imp_add1_zle)
haftmann@25919: 
haftmann@25919: lemma [code]: "nat i = nat_aux i 0"
haftmann@25919:   by (simp add: nat_aux_def)
haftmann@25919: 
haftmann@28514: hide (open) const nat_aux
haftmann@25928: 
haftmann@28562: lemma zero_is_num_zero [code, code inline, symmetric, code post]:
haftmann@25919:   "(0\<Colon>int) = Numeral0" 
haftmann@25919:   by simp
haftmann@25919: 
haftmann@28562: lemma one_is_num_one [code, code inline, symmetric, code post]:
haftmann@25919:   "(1\<Colon>int) = Numeral1" 
haftmann@25961:   by simp
haftmann@25919: 
haftmann@25919: code_modulename SML
haftmann@25928:   Int Integer
haftmann@25919: 
haftmann@25919: code_modulename OCaml
haftmann@25928:   Int Integer
haftmann@25919: 
haftmann@25919: code_modulename Haskell
haftmann@25928:   Int Integer
haftmann@25919: 
haftmann@25919: types_code
haftmann@25919:   "int" ("int")
haftmann@25919: attach (term_of) {*
haftmann@25919: val term_of_int = HOLogic.mk_number HOLogic.intT;
haftmann@25919: *}
haftmann@25919: attach (test) {*
haftmann@25919: fun gen_int i =
haftmann@25919:   let val j = one_of [~1, 1] * random_range 0 i
haftmann@25919:   in (j, fn () => term_of_int j) end;
haftmann@25919: *}
haftmann@25919: 
haftmann@25919: setup {*
haftmann@25919: let
haftmann@25919: 
haftmann@25919: fun strip_number_of (@{term "Int.number_of :: int => int"} $ t) = t
haftmann@25919:   | strip_number_of t = t;
haftmann@25919: 
haftmann@28537: fun numeral_codegen thy defs dep module b t gr =
haftmann@25919:   let val i = HOLogic.dest_numeral (strip_number_of t)
haftmann@25919:   in
haftmann@28537:     SOME (Codegen.str (string_of_int i),
haftmann@28537:       snd (Codegen.invoke_tycodegen thy defs dep module false HOLogic.intT gr))
haftmann@25919:   end handle TERM _ => NONE;
haftmann@25919: 
haftmann@25919: in
haftmann@25919: 
haftmann@25919: Codegen.add_codegen "numeral_codegen" numeral_codegen
haftmann@25919: 
haftmann@25919: end
haftmann@25919: *}
haftmann@25919: 
haftmann@25919: consts_code
haftmann@25919:   "number_of :: int \<Rightarrow> int"    ("(_)")
haftmann@25919:   "0 :: int"                   ("0")
haftmann@25919:   "1 :: int"                   ("1")
haftmann@25919:   "uminus :: int => int"       ("~")
haftmann@25919:   "op + :: int => int => int"  ("(_ +/ _)")
haftmann@25919:   "op * :: int => int => int"  ("(_ */ _)")
haftmann@25919:   "op \<le> :: int => int => bool" ("(_ <=/ _)")
haftmann@25919:   "op < :: int => int => bool" ("(_ </ _)")
haftmann@25919: 
haftmann@25919: quickcheck_params [default_type = int]
haftmann@25919: 
huffman@26086: hide (open) const Pls Min Bit0 Bit1 succ pred
haftmann@25919: 
haftmann@25919: 
haftmann@25919: subsection {* Legacy theorems *}
haftmann@25919: 
haftmann@25919: lemmas zminus_zminus = minus_minus [of "z::int", standard]
haftmann@25919: lemmas zminus_0 = minus_zero [where 'a=int]
haftmann@25919: lemmas zminus_zadd_distrib = minus_add_distrib [of "z::int" "w", standard]
haftmann@25919: lemmas zadd_commute = add_commute [of "z::int" "w", standard]
haftmann@25919: lemmas zadd_assoc = add_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919: lemmas zadd_left_commute = add_left_commute [of "x::int" "y" "z", standard]
haftmann@25919: lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
haftmann@25919: lemmas zmult_ac = OrderedGroup.mult_ac
haftmann@25919: lemmas zadd_0 = OrderedGroup.add_0_left [of "z::int", standard]
haftmann@25919: lemmas zadd_0_right = OrderedGroup.add_0_left [of "z::int", standard]
haftmann@25919: lemmas zadd_zminus_inverse2 = left_minus [of "z::int", standard]
haftmann@25919: lemmas zmult_zminus = mult_minus_left [of "z::int" "w", standard]
haftmann@25919: lemmas zmult_commute = mult_commute [of "z::int" "w", standard]
haftmann@25919: lemmas zmult_assoc = mult_assoc [of "z1::int" "z2" "z3", standard]
haftmann@25919: lemmas zadd_zmult_distrib = left_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919: lemmas zadd_zmult_distrib2 = right_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919: lemmas zdiff_zmult_distrib = left_diff_distrib [of "z1::int" "z2" "w", standard]
haftmann@25919: lemmas zdiff_zmult_distrib2 = right_diff_distrib [of "w::int" "z1" "z2", standard]
haftmann@25919: 
haftmann@25919: lemmas zmult_1 = mult_1_left [of "z::int", standard]
haftmann@25919: lemmas zmult_1_right = mult_1_right [of "z::int", standard]
haftmann@25919: 
haftmann@25919: lemmas zle_refl = order_refl [of "w::int", standard]
haftmann@25919: lemmas zle_trans = order_trans [where 'a=int and x="i" and y="j" and z="k", standard]
haftmann@25919: lemmas zle_anti_sym = order_antisym [of "z::int" "w", standard]
haftmann@25919: lemmas zle_linear = linorder_linear [of "z::int" "w", standard]
haftmann@25919: lemmas zless_linear = linorder_less_linear [where 'a = int]
haftmann@25919: 
haftmann@25919: lemmas zadd_left_mono = add_left_mono [of "i::int" "j" "k", standard]
haftmann@25919: lemmas zadd_strict_right_mono = add_strict_right_mono [of "i::int" "j" "k", standard]
haftmann@25919: lemmas zadd_zless_mono = add_less_le_mono [of "w'::int" "w" "z'" "z", standard]
haftmann@25919: 
haftmann@25919: lemmas int_0_less_1 = zero_less_one [where 'a=int]
haftmann@25919: lemmas int_0_neq_1 = zero_neq_one [where 'a=int]
haftmann@25919: 
haftmann@25919: lemmas inj_int = inj_of_nat [where 'a=int]
haftmann@25919: lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
haftmann@25919: lemmas int_mult = of_nat_mult [where 'a=int]
haftmann@25919: lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
haftmann@25919: lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n", standard]
haftmann@25919: lemmas zless_int = of_nat_less_iff [where 'a=int]
haftmann@25919: lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k", standard]
haftmann@25919: lemmas zero_less_int_conv = of_nat_0_less_iff [where 'a=int]
haftmann@25919: lemmas zero_zle_int = of_nat_0_le_iff [where 'a=int]
haftmann@25919: lemmas int_le_0_conv = of_nat_le_0_iff [where 'a=int and m="n", standard]
haftmann@25919: lemmas int_0 = of_nat_0 [where 'a=int]
haftmann@25919: lemmas int_1 = of_nat_1 [where 'a=int]
haftmann@25919: lemmas int_Suc = of_nat_Suc [where 'a=int]
haftmann@25919: lemmas abs_int_eq = abs_of_nat [where 'a=int and n="m", standard]
haftmann@25919: lemmas of_int_int_eq = of_int_of_nat_eq [where 'a=int]
haftmann@25919: lemmas zdiff_int = of_nat_diff [where 'a=int, symmetric]
haftmann@25919: lemmas zless_le = less_int_def
haftmann@25919: lemmas int_eq_of_nat = TrueI
haftmann@25919: 
haftmann@25919: end