--- a/src/HOL/Integ/IntDef.thy Fri Mar 19 11:06:53 2004 +0100
+++ b/src/HOL/Integ/IntDef.thy Wed Mar 24 10:50:29 2004 +0100
@@ -8,13 +8,15 @@
header{*The Integers as Equivalence Classes over Pairs of Natural Numbers*}
theory IntDef = Equiv + NatArith:
+
constdefs
intrel :: "((nat * nat) * (nat * nat)) set"
- "intrel == {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
+ --{*the equivalence relation underlying the integers*}
+ "intrel == {p. \<exists>x y u v. p = ((x,y),(u,v)) & x+v = u+y}"
typedef (Integ)
int = "UNIV//intrel"
- by (auto simp add: quotient_def)
+ by (auto simp add: quotient_def)
instance int :: ord ..
instance int :: zero ..
@@ -27,49 +29,57 @@
int :: "nat => int"
"int m == Abs_Integ(intrel `` {(m,0)})"
-
+
+
defs (overloaded)
-
- zminus_def: "- Z == Abs_Integ(\<Union>(x,y) \<in> Rep_Integ(Z). intrel``{(y,x)})"
Zero_int_def: "0 == int 0"
One_int_def: "1 == int 1"
- zadd_def:
- "z + w ==
- Abs_Integ(\<Union>(x1,y1) \<in> Rep_Integ(z). \<Union>(x2,y2) \<in> Rep_Integ(w).
- intrel``{(x1+x2, y1+y2)})"
+ minus_int_def:
+ "- z == contents (\<Union>(x,y) \<in> Rep_Integ(z). { Abs_Integ(intrel``{(y,x)}) })"
+
+ add_int_def:
+ "z + w ==
+ contents (\<Union>(x,y) \<in> Rep_Integ(z). \<Union>(u,v) \<in> Rep_Integ(w).
+ { Abs_Integ(intrel``{(x+u, y+v)}) })"
+
+ diff_int_def: "z - (w::int) == z + (-w)"
- zdiff_def: "z - (w::int) == z + (-w)"
- zmult_def:
- "z * w ==
- Abs_Integ(\<Union>(x1,y1) \<in> Rep_Integ(z). \<Union>(x2,y2) \<in> Rep_Integ(w).
- intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
+ mult_int_def:
+ "z * w ==
+ contents (\<Union>(x,y) \<in> Rep_Integ(z). \<Union>(u,v) \<in> Rep_Integ(w).
+ { Abs_Integ(intrel``{(x*u + y*v, x*v + y*u)}) })"
+
+ le_int_def:
+ "z \<le> (w::int) ==
+ \<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Integ z & (u,v) \<in> Rep_Integ w"
- zless_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
+ less_int_def: "(z < (w::int)) == (z \<le> w & z \<noteq> w)"
+
+
+subsection{*Construction of the Integers*}
- zle_def:
- "z \<le> (w::int) == \<exists>x1 y1 x2 y2. x1 + y2 \<le> x2 + y1 &
- (x1,y1) \<in> Rep_Integ z & (x2,y2) \<in> Rep_Integ w"
+subsubsection{*Preliminary Lemmas about the Equivalence Relation*}
-lemma intrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> intrel) = (x1+y2 = x2+y1)"
-by (unfold intrel_def, blast)
+lemma intrel_iff [simp]: "(((x,y),(u,v)) \<in> intrel) = (x+v = u+y)"
+by (simp add: intrel_def)
lemma equiv_intrel: "equiv UNIV intrel"
-by (unfold intrel_def equiv_def refl_def sym_def trans_def, auto)
+by (simp add: intrel_def equiv_def refl_def sym_def trans_def)
lemmas equiv_intrel_iff =
eq_equiv_class_iff [OF equiv_intrel UNIV_I UNIV_I, simp]
-lemma intrel_in_integ [simp]: "intrel``{(x,y)}:Integ"
-by (unfold Integ_def intrel_def quotient_def, fast)
+lemma intrel_in_integ [simp]: "intrel``{(x,y)} \<in> Integ"
+by (simp add: Integ_def intrel_def quotient_def, fast)
lemma inj_on_Abs_Integ: "inj_on Abs_Integ Integ"
apply (rule inj_on_inverseI)
apply (erule Abs_Integ_inverse)
done
-declare inj_on_Abs_Integ [THEN inj_on_iff, simp]
+declare inj_on_Abs_Integ [THEN inj_on_iff, simp]
Abs_Integ_inverse [simp]
lemma inj_Rep_Integ: "inj(Rep_Integ)"
@@ -77,85 +87,64 @@
apply (rule Rep_Integ_inverse)
done
+text{*Case analysis on the representation of an integer as an equivalence
+ class.*}
+lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
+ "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
+apply (rule Rep_Integ [of z, unfolded Integ_def, THEN quotientE])
+apply (drule arg_cong [where f=Abs_Integ])
+apply (auto simp add: Rep_Integ_inverse)
+done
-(** int: the injection from "nat" to "int" **)
+
+subsubsection{*@{term int}: Embedding the Naturals into the Integers*}
lemma inj_int: "inj int"
-apply (rule inj_onI)
-apply (unfold int_def)
-apply (drule inj_on_Abs_Integ [THEN inj_onD])
-apply (rule intrel_in_integ)+
-apply (drule eq_equiv_class)
-apply (rule equiv_intrel, fast)
-apply (simp add: intrel_def)
-done
+by (simp add: inj_on_def int_def)
lemma int_int_eq [iff]: "(int m = int n) = (m = n)"
by (fast elim!: inj_int [THEN injD])
-
-subsection{*zminus: unary negation on Integ*}
-
-lemma zminus_congruent: "congruent intrel (%(x,y). intrel``{(y,x)})"
-apply (unfold congruent_def intrel_def)
-apply (auto simp add: add_ac)
-done
-
-lemma zminus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
-by (simp add: zminus_def equiv_intrel [THEN UN_equiv_class] zminus_congruent)
+subsubsection{*Integer Unary Negation*}
-(*Every integer can be written in the form Abs_Integ(...) *)
-lemma eq_Abs_Integ:
- "(!!x y. z = Abs_Integ(intrel``{(x,y)}) ==> P) ==> P"
-apply (rule_tac x1=z in Rep_Integ [unfolded Integ_def, THEN quotientE])
-apply (drule_tac f = Abs_Integ in arg_cong)
-apply (rule_tac p = x in PairE)
-apply (simp add: Rep_Integ_inverse)
-done
+lemma minus: "- Abs_Integ(intrel``{(x,y)}) = Abs_Integ(intrel `` {(y,x)})"
+proof -
+ have "congruent intrel (\<lambda>(x,y). {Abs_Integ (intrel``{(y,x)})})"
+ by (simp add: congruent_def)
+ thus ?thesis
+ by (simp add: minus_int_def UN_equiv_class [OF equiv_intrel])
+qed
-lemma zminus_zminus [simp]: "- (- z) = (z::int)"
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zminus)
-done
+lemma zminus_zminus: "- (- z) = (z::int)"
+by (cases z, simp add: minus)
-lemma inj_zminus: "inj(%z::int. -z)"
-apply (rule inj_onI)
-apply (drule_tac f = uminus in arg_cong, simp)
-done
-
-lemma zminus_0 [simp]: "- 0 = (0::int)"
-by (simp add: int_def Zero_int_def zminus)
+lemma zminus_0: "- 0 = (0::int)"
+by (simp add: int_def Zero_int_def minus)
-subsection{*zadd: addition on Integ*}
+subsection{*Integer Addition*}
-lemma zadd:
- "Abs_Integ(intrel``{(x1,y1)}) + Abs_Integ(intrel``{(x2,y2)}) =
- Abs_Integ(intrel``{(x1+x2, y1+y2)})"
-apply (simp add: zadd_def UN_UN_split_split_eq)
-apply (subst equiv_intrel [THEN UN_equiv_class2])
-apply (auto simp add: congruent2_def)
-done
+lemma add:
+ "Abs_Integ (intrel``{(x,y)}) + Abs_Integ (intrel``{(u,v)}) =
+ Abs_Integ (intrel``{(x+u, y+v)})"
+proof -
+ have "congruent2 intrel
+ (\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Integ (intrel `` {(x+u, y+v)})}) w) z)"
+ by (simp add: congruent2_def)
+ thus ?thesis
+ by (simp add: add_int_def UN_UN_split_split_eq
+ UN_equiv_class2 [OF equiv_intrel])
+qed
-lemma zminus_zadd_distrib [simp]: "- (z + w) = (- z) + (- w::int)"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zminus zadd)
-done
+lemma zminus_zadd_distrib: "- (z + w) = (- z) + (- w::int)"
+by (cases z, cases w, simp add: minus add)
lemma zadd_commute: "(z::int) + w = w + z"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: add_ac zadd)
-done
+by (cases z, cases w, simp add: add_ac add)
lemma zadd_assoc: "((z1::int) + z2) + z3 = z1 + (z2 + z3)"
-apply (rule eq_Abs_Integ [of z1])
-apply (rule eq_Abs_Integ [of z2])
-apply (rule eq_Abs_Integ [of z3])
-apply (simp add: zadd add_assoc)
-done
+by (cases z1, cases z2, cases z3, simp add: add add_assoc)
(*For AC rewriting*)
lemma zadd_left_commute: "x + (y + z) = y + ((x + z) ::int)"
@@ -164,13 +153,12 @@
apply (rule zadd_commute)
done
-(*Integer addition is an AC operator*)
lemmas zadd_ac = zadd_assoc zadd_commute zadd_left_commute
lemmas zmult_ac = Ring_and_Field.mult_ac
lemma zadd_int: "(int m) + (int n) = int (m + n)"
-by (simp add: int_def zadd)
+by (simp add: int_def add)
lemma zadd_int_left: "(int m) + (int n + z) = int (m + n) + z"
by (simp add: zadd_int zadd_assoc [symmetric])
@@ -179,104 +167,56 @@
by (simp add: One_int_def zadd_int)
(*also for the instance declaration int :: plus_ac0*)
-lemma zadd_0 [simp]: "(0::int) + z = z"
-apply (unfold Zero_int_def int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zadd)
-done
-
-lemma zadd_0_right [simp]: "z + (0::int) = z"
-by (rule trans [OF zadd_commute zadd_0])
-
-lemma zadd_zminus_inverse [simp]: "z + (- z) = (0::int)"
-apply (unfold int_def Zero_int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zminus zadd add_commute)
-done
-
-lemma zadd_zminus_inverse2 [simp]: "(- z) + z = (0::int)"
-apply (rule zadd_commute [THEN trans])
-apply (rule zadd_zminus_inverse)
+lemma zadd_0: "(0::int) + z = z"
+apply (simp add: Zero_int_def int_def)
+apply (cases z, simp add: add)
done
-lemma zadd_zminus_cancel [simp]: "z + (- z + w) = (w::int)"
-by (simp add: zadd_assoc [symmetric] zadd_0)
-
-lemma zminus_zadd_cancel [simp]: "(-z) + (z + w) = (w::int)"
-by (simp add: zadd_assoc [symmetric] zadd_0)
+lemma zadd_0_right: "z + (0::int) = z"
+by (rule trans [OF zadd_commute zadd_0])
-lemma zdiff0 [simp]: "(0::int) - x = -x"
-by (simp add: zdiff_def)
-
-lemma zdiff0_right [simp]: "x - (0::int) = x"
-by (simp add: zdiff_def)
-
-lemma zdiff_self [simp]: "x - x = (0::int)"
-by (simp add: zdiff_def Zero_int_def)
+lemma zadd_zminus_inverse2: "(- z) + z = (0::int)"
+by (cases z, simp add: int_def Zero_int_def minus add)
-(** Lemmas **)
-
-lemma zadd_assoc_cong: "(z::int) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
-by (simp add: zadd_assoc [symmetric])
-
-
-subsection{*zmult: multiplication on Integ*}
+subsection{*Integer Multiplication*}
text{*Congruence property for multiplication*}
-lemma zmult_congruent2: "congruent2 intrel
- (%p1 p2. (%(x1,y1). (%(x2,y2).
- intrel``{(x1*x2 + y1*y2, x1*y2 + y1*x2)}) p2) p1)"
+lemma mult_congruent2:
+ "congruent2 intrel
+ (%p1 p2. (%(x,y). (%(u,v).
+ { Abs_Integ(intrel``{(x*u + y*v, x*v + y*u)}) }) p2) p1)"
apply (rule equiv_intrel [THEN congruent2_commuteI])
- apply (force simp add: add_ac mult_ac)
-apply (clarify, simp del: equiv_intrel_iff add: add_ac mult_ac)
-apply (rename_tac x1 x2 y1 y2 z1 z2)
-apply (rule equiv_class_eq [OF equiv_intrel intrel_iff [THEN iffD2]])
-apply (subgoal_tac "x1*z1 + y2*z1 = y1*z1 + x2*z1 & x1*z2 + y2*z2 = y1*z2 + x2*z2")
-apply (simp add: mult_ac, arith)
+apply (auto simp add: congruent_def mult_ac)
+apply (rename_tac u v w x y z)
+apply (subgoal_tac "u*y + x*y = w*y + v*y & u*z + x*z = w*z + v*z")
+apply (simp add: mult_ac, arith)
apply (simp add: add_mult_distrib [symmetric])
done
-lemma zmult:
- "Abs_Integ((intrel``{(x1,y1)})) * Abs_Integ((intrel``{(x2,y2)})) =
- Abs_Integ(intrel `` {(x1*x2 + y1*y2, x1*y2 + y1*x2)})"
-by (simp add: zmult_def UN_UN_split_split_eq zmult_congruent2
+lemma mult:
+ "Abs_Integ((intrel``{(x,y)})) * Abs_Integ((intrel``{(u,v)})) =
+ Abs_Integ(intrel `` {(x*u + y*v, x*v + y*u)})"
+by (simp add: mult_int_def UN_UN_split_split_eq mult_congruent2
equiv_intrel [THEN UN_equiv_class2])
lemma zmult_zminus: "(- z) * w = - (z * (w::int))"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zminus zmult add_ac)
-done
+by (cases z, cases w, simp add: minus mult add_ac)
lemma zmult_commute: "(z::int) * w = w * z"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zmult add_ac mult_ac)
-done
+by (cases z, cases w, simp add: mult add_ac mult_ac)
lemma zmult_assoc: "((z1::int) * z2) * z3 = z1 * (z2 * z3)"
-apply (rule eq_Abs_Integ [of z1])
-apply (rule eq_Abs_Integ [of z2])
-apply (rule eq_Abs_Integ [of z3])
-apply (simp add: add_mult_distrib2 zmult add_ac mult_ac)
-done
+by (cases z1, cases z2, cases z3, simp add: mult add_mult_distrib2 mult_ac)
lemma zadd_zmult_distrib: "((z1::int) + z2) * w = (z1 * w) + (z2 * w)"
-apply (rule eq_Abs_Integ [of z1])
-apply (rule eq_Abs_Integ [of z2])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: add_mult_distrib2 zadd zmult add_ac mult_ac)
-done
-
-lemma zmult_zminus_right: "w * (- z) = - (w * (z::int))"
-by (simp add: zmult_commute [of w] zmult_zminus)
+by (cases z1, cases z2, cases w, simp add: add mult add_mult_distrib2 mult_ac)
lemma zadd_zmult_distrib2: "(w::int) * (z1 + z2) = (w * z1) + (w * z2)"
by (simp add: zmult_commute [of w] zadd_zmult_distrib)
lemma zdiff_zmult_distrib: "((z1::int) - z2) * w = (z1 * w) - (z2 * w)"
-apply (unfold zdiff_def)
+apply (simp add: diff_int_def)
apply (subst zadd_zmult_distrib)
apply (simp add: zmult_zminus)
done
@@ -285,28 +225,16 @@
by (simp add: zmult_commute [of w] zdiff_zmult_distrib)
lemmas int_distrib =
- zadd_zmult_distrib zadd_zmult_distrib2
+ zadd_zmult_distrib zadd_zmult_distrib2
zdiff_zmult_distrib zdiff_zmult_distrib2
lemma zmult_int: "(int m) * (int n) = int (m * n)"
-by (simp add: int_def zmult)
-
-lemma zmult_0 [simp]: "0 * z = (0::int)"
-apply (unfold Zero_int_def int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zmult)
-done
+by (simp add: int_def mult)
-lemma zmult_1 [simp]: "(1::int) * z = z"
-apply (unfold One_int_def int_def)
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zmult)
-done
+lemma zmult_1: "(1::int) * z = z"
+by (cases z, simp add: One_int_def int_def mult)
-lemma zmult_0_right [simp]: "z * 0 = (0::int)"
-by (rule trans [OF zmult_commute zmult_0])
-
-lemma zmult_1_right [simp]: "z * (1::int) = z"
+lemma zmult_1_right: "z * (1::int) = z"
by (rule trans [OF zmult_commute zmult_1])
@@ -316,46 +244,36 @@
fix i j k :: int
show "(i + j) + k = i + (j + k)" by (simp add: zadd_assoc)
show "i + j = j + i" by (simp add: zadd_commute)
- show "0 + i = i" by simp
- show "- i + i = 0" by simp
- show "i - j = i + (-j)" by (simp add: zdiff_def)
+ show "0 + i = i" by (rule zadd_0)
+ show "- i + i = 0" by (rule zadd_zminus_inverse2)
+ show "i - j = i + (-j)" by (simp add: diff_int_def)
show "(i * j) * k = i * (j * k)" by (rule zmult_assoc)
show "i * j = j * i" by (rule zmult_commute)
- show "1 * i = i" by simp
+ show "1 * i = i" by (rule zmult_1)
show "(i + j) * k = i * k + j * k" by (simp add: int_distrib)
- show "0 \<noteq> (1::int)"
+ show "0 \<noteq> (1::int)"
by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
qed
subsection{*The @{text "\<le>"} Ordering*}
-lemma zle:
- "(Abs_Integ(intrel``{(x1,y1)}) \<le> Abs_Integ(intrel``{(x2,y2)})) =
- (x1 + y2 \<le> x2 + y1)"
-by (force simp add: zle_def)
+lemma le:
+ "(Abs_Integ(intrel``{(x,y)}) \<le> Abs_Integ(intrel``{(u,v)})) = (x+v \<le> u+y)"
+by (force simp add: le_int_def)
lemma zle_refl: "w \<le> (w::int)"
-apply (rule eq_Abs_Integ [of w])
-apply (force simp add: zle)
-done
+by (cases w, simp add: le)
lemma zle_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::int)"
-apply (rule eq_Abs_Integ [of i])
-apply (rule eq_Abs_Integ [of j])
-apply (rule eq_Abs_Integ [of k])
-apply (simp add: zle)
-done
+by (cases i, cases j, cases k, simp add: le)
lemma zle_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::int)"
-apply (rule eq_Abs_Integ [of w])
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: zle)
-done
+by (cases w, cases z, simp add: le)
(* Axiom 'order_less_le' of class 'order': *)
lemma zless_le: "((w::int) < z) = (w \<le> z & w \<noteq> z)"
-by (simp add: zless_def)
+by (simp add: less_int_def)
instance int :: order
proof qed
@@ -364,10 +282,7 @@
(* Axiom 'linorder_linear' of class 'linorder': *)
lemma zle_linear: "(z::int) \<le> w | w \<le> z"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: zle linorder_linear)
-done
+by (cases z, cases w, simp add: le linorder_linear)
instance int :: linorder
proof qed (rule zle_linear)
@@ -379,21 +294,8 @@
lemma int_eq_0_conv [simp]: "(int n = 0) = (n = 0)"
by (simp add: Zero_int_def)
-(*This lemma allows direct proofs of other <-properties*)
-lemma zless_iff_Suc_zadd:
- "(w < z) = (\<exists>n. z = w + int(Suc n))"
-apply (rule eq_Abs_Integ [of z])
-apply (rule eq_Abs_Integ [of w])
-apply (simp add: linorder_not_le [where 'a = int, symmetric]
- linorder_not_le [where 'a = nat]
- zle int_def zdiff_def zadd zminus)
-apply (safe dest!: less_imp_Suc_add)
-apply (rule_tac x = k in exI)
-apply (simp_all add: add_ac)
-done
-
lemma zless_int [simp]: "(int m < int n) = (m<n)"
-by (simp add: less_iff_Suc_add zless_iff_Suc_zadd zadd_int)
+by (simp add: le add int_def linorder_not_le [symmetric])
lemma int_less_0_conv [simp]: "~ (int k < 0)"
by (simp add: Zero_int_def)
@@ -425,25 +327,20 @@
lemma int_Suc0_eq_1: "int (Suc 0) = 1"
by (simp add: One_int_def One_nat_def)
+
subsection{*Monotonicity results*}
-lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
-apply (rule eq_Abs_Integ [of i])
-apply (rule eq_Abs_Integ [of j])
-apply (rule eq_Abs_Integ [of k])
-apply (simp add: zle zadd)
-done
+lemma zadd_left_mono: "i \<le> j ==> k + i \<le> k + (j::int)"
+by (cases i, cases j, cases k, simp add: le add)
-lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
-apply (rule eq_Abs_Integ [of i])
-apply (rule eq_Abs_Integ [of j])
-apply (rule eq_Abs_Integ [of k])
-apply (simp add: linorder_not_le [where 'a = int, symmetric]
- linorder_not_le [where 'a = nat] zle zadd)
+lemma zadd_strict_right_mono: "i < j ==> i + k < j + (k::int)"
+apply (cases i, cases j, cases k)
+apply (simp add: linorder_not_le [where 'a = int, symmetric]
+ linorder_not_le [where 'a = nat] le add)
done
lemma zadd_zless_mono: "[| w'<w; z'\<le>z |] ==> w' + z' < w + (z::int)"
-by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
+by (rule order_less_le_trans [OF zadd_strict_right_mono zadd_left_mono])
subsection{*Strict Monotonicity of Multiplication*}
@@ -451,7 +348,7 @@
text{*strict, in 1st argument; proof is by induction on k>0*}
lemma zmult_zless_mono2_lemma [rule_format]:
"i<j ==> 0<k --> int k * i < int k * j"
-apply (induct_tac "k", simp)
+apply (induct_tac "k", simp)
apply (simp add: int_Suc)
apply (case_tac "n=0")
apply (simp_all add: zadd_zmult_distrib int_Suc0_eq_1 order_le_less)
@@ -459,14 +356,14 @@
done
lemma zero_le_imp_eq_int: "0 \<le> k ==> \<exists>n. k = int n"
-apply (rule eq_Abs_Integ [of k])
-apply (auto simp add: zle zadd int_def Zero_int_def)
-apply (rule_tac x="x-y" in exI, simp)
+apply (cases k)
+apply (auto simp add: le add int_def Zero_int_def)
+apply (rule_tac x="x-y" in exI, simp)
done
lemma zmult_zless_mono2: "[| i<j; (0::int) < k |] ==> k*i < k*j"
-apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
-apply (auto simp add: zmult_zless_mono2_lemma)
+apply (frule order_less_imp_le [THEN zero_le_imp_eq_int])
+apply (auto simp add: zmult_zless_mono2_lemma)
done
@@ -484,49 +381,100 @@
qed
+lemma zless_imp_add1_zle: "w<z ==> w + (1::int) \<le> z"
+apply (cases w, cases z)
+apply (simp add: linorder_not_le [symmetric] le int_def add One_int_def)
+done
+
subsection{*Magnitide of an Integer, as a Natural Number: @{term nat}*}
constdefs
nat :: "int => nat"
- "nat(Z) == if Z<0 then 0 else (THE m. Z = int m)"
+ "nat z == contents (\<Union>(x,y) \<in> Rep_Integ(z). { x-y })"
+
+lemma nat: "nat (Abs_Integ (intrel``{(x,y)})) = x-y"
+proof -
+ have "congruent intrel (\<lambda>(x,y). {x-y})"
+ by (simp add: congruent_def, arith)
+ thus ?thesis
+ by (simp add: nat_def UN_equiv_class [OF equiv_intrel])
+qed
lemma nat_int [simp]: "nat(int n) = n"
-by (unfold nat_def, auto)
+by (simp add: nat int_def)
lemma nat_zero [simp]: "nat 0 = 0"
-apply (unfold Zero_int_def)
-apply (rule nat_int)
-done
+by (simp only: Zero_int_def nat_int)
-lemma nat_0_le [simp]: "0 \<le> z ==> int (nat z) = z"
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: nat_def linorder_not_le [symmetric] zle int_def Zero_int_def)
-apply (subgoal_tac "(THE m. x = m + y) = x-y")
-apply (auto simp add: the_equality)
+lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
+by (cases z, simp add: nat le int_def Zero_int_def)
+
+corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
+apply simp
done
lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
-by (simp add: nat_def order_less_le eq_commute [of 0])
+by (cases z, simp add: nat le int_def Zero_int_def)
+
+lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
+apply (cases w, cases z)
+apply (simp add: nat le linorder_not_le [symmetric] int_def Zero_int_def, arith)
+done
text{*An alternative condition is @{term "0 \<le> w"} *}
-lemma nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
-apply (subst zless_int [symmetric])
-apply (simp add: order_le_less)
-apply (case_tac "w < 0")
- apply (simp add: order_less_imp_le)
-apply (simp add: linorder_not_less)
+corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
+by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
+
+corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
+by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
+
+lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
+apply (cases w, cases z)
+apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
done
-lemma zless_nat_conj: "(nat w < nat z) = (0 < z & w < z)"
-apply (case_tac "0 < z")
-apply (auto simp add: nat_mono_iff linorder_not_less)
+lemma nonneg_eq_int: "[| 0 \<le> z; !!m. z = int m ==> P |] ==> P"
+by (blast dest: nat_0_le sym)
+
+lemma nat_eq_iff: "(nat w = m) = (if 0 \<le> w then w = int m else m=0)"
+by (cases w, simp add: nat le int_def Zero_int_def, arith)
+
+corollary nat_eq_iff2: "(m = nat w) = (if 0 \<le> w then w = int m else m=0)"
+by (simp only: eq_commute [of m] nat_eq_iff)
+
+lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < int m)"
+apply (cases w)
+apply (simp add: nat le int_def Zero_int_def linorder_not_le [symmetric], arith)
done
+lemma int_eq_iff: "(int m = z) = (m = nat z & 0 \<le> z)"
+by (auto simp add: nat_eq_iff2)
+
+lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
+by (insert zless_nat_conj [of 0], auto)
+
+
+lemma nat_add_distrib:
+ "[| (0::int) \<le> z; 0 \<le> z' |] ==> nat (z+z') = nat z + nat z'"
+by (cases z, cases z', simp add: nat add le int_def Zero_int_def)
+
+lemma nat_diff_distrib:
+ "[| (0::int) \<le> z'; z' \<le> z |] ==> nat (z-z') = nat z - nat z'"
+by (cases z, cases z',
+ simp add: nat add minus diff_minus le int_def Zero_int_def)
+
+
+lemma nat_zminus_int [simp]: "nat (- (int n)) = 0"
+by (simp add: int_def minus nat Zero_int_def)
+
+lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
+by (cases z, simp add: nat le int_def linorder_not_le [symmetric], arith)
+
subsection{*Lemmas about the Function @{term int} and Orderings*}
lemma negative_zless_0: "- (int (Suc n)) < 0"
-by (simp add: zless_def)
+by (simp add: order_less_le)
lemma negative_zless [iff]: "- (int (Suc n)) < int m"
by (rule negative_zless_0 [THEN order_less_le_trans], simp)
@@ -541,10 +489,7 @@
by (subst le_minus_iff, simp)
lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
-apply safe
-apply (drule_tac [2] le_minus_iff [THEN iffD1])
-apply (auto dest: zle_trans [OF _ negative_zle_0])
-done
+by (simp add: int_def le minus Zero_int_def)
lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
by (simp add: linorder_not_less)
@@ -553,10 +498,15 @@
by (force simp add: order_eq_iff [of "- int n"] int_zle_neg)
lemma zle_iff_zadd: "(w \<le> z) = (\<exists>n. z = w + int n)"
-by (force intro: exI [of _ "0::nat"]
- intro!: not_sym [THEN not0_implies_Suc]
- simp add: zless_iff_Suc_zadd order_le_less)
+apply (cases w, cases z)
+apply (auto simp add: le add int_def)
+apply (rename_tac a b c d)
+apply (rule_tac x="c+b - (a+d)" in exI)
+apply arith
+done
+lemma abs_int_eq [simp]: "abs (int m) = int m"
+by (simp add: zabs_def)
text{*This version is proved for all ordered rings, not just integers!
It is proved here because attribute @{text arith_split} is not available
@@ -566,38 +516,6 @@
"P(abs(a::'a::ordered_ring)) = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
-lemma abs_int_eq [simp]: "abs (int m) = int m"
-by (simp add: zabs_def)
-
-
-subsection{*Misc Results*}
-
-lemma nat_zminus_int [simp]: "nat(- (int n)) = 0"
-by (auto simp add: nat_def zero_reorient minus_less_iff)
-
-lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
-apply (case_tac "0 \<le> z")
-apply (erule nat_0_le [THEN subst], simp)
-apply (simp add: linorder_not_le)
-apply (auto dest: order_less_trans simp add: order_less_imp_le)
-done
-
-text{*A case theorem distinguishing non-negative and negative int*}
-
-lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
-by (auto simp add: zless_iff_Suc_zadd
- diff_eq_eq [symmetric] zdiff_def)
-
-lemma int_cases [cases type: int, case_names nonneg neg]:
- "[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P"
-apply (case_tac "z < 0", blast dest!: negD)
-apply (simp add: linorder_not_less)
-apply (blast dest: nat_0_le [THEN sym])
-done
-
-lemma int_induct [induct type: int, case_names nonneg neg]:
- "[|!! n. P (int n); !!n. P (- (int (Suc n))) |] ==> P z"
- by (cases z) auto
subsection{*The Constants @{term neg} and @{term iszero}*}
@@ -610,7 +528,7 @@
(*For simplifying equalities*)
iszero :: "'a::semiring => bool"
"iszero z == z = (0)"
-
+
lemma not_neg_int [simp]: "~ neg(int n)"
by (simp add: neg_def)
@@ -623,22 +541,23 @@
lemma not_neg_eq_ge_0: "(~neg x) = (0 \<le> x)"
by (simp add: neg_def linorder_not_less)
+
subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
lemma not_neg_0: "~ neg 0"
by (simp add: One_int_def neg_def)
lemma not_neg_1: "~ neg 1"
-by (simp add: neg_def linorder_not_less zero_le_one)
+by (simp add: neg_def linorder_not_less zero_le_one)
lemma iszero_0: "iszero 0"
by (simp add: iszero_def)
lemma not_iszero_1: "~ iszero 1"
-by (simp add: iszero_def eq_commute)
+by (simp add: iszero_def eq_commute)
lemma neg_nat: "neg z ==> nat z = 0"
-by (simp add: nat_def neg_def)
+by (simp add: neg_def order_less_imp_le)
lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
by (simp add: linorder_not_less neg_def)
@@ -657,36 +576,36 @@
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n"
apply (induct m)
-apply (simp_all add: add_ac)
+apply (simp_all add: add_ac)
done
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n"
-apply (induct m)
-apply (simp_all add: mult_ac add_ac right_distrib)
+apply (induct m)
+apply (simp_all add: mult_ac add_ac right_distrib)
done
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semiring)"
-apply (induct m, simp_all)
-apply (erule order_trans)
-apply (rule less_add_one [THEN order_less_imp_le])
+apply (induct m, simp_all)
+apply (erule order_trans)
+apply (rule less_add_one [THEN order_less_imp_le])
done
lemma less_imp_of_nat_less:
"m < n ==> of_nat m < (of_nat n::'a::ordered_semiring)"
-apply (induct m n rule: diff_induct, simp_all)
-apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
+apply (induct m n rule: diff_induct, simp_all)
+apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force)
done
lemma of_nat_less_imp_less:
"of_nat m < (of_nat n::'a::ordered_semiring) ==> m < n"
-apply (induct m n rule: diff_induct, simp_all)
-apply (insert zero_le_imp_of_nat)
-apply (force simp add: linorder_not_less [symmetric])
+apply (induct m n rule: diff_induct, simp_all)
+apply (insert zero_le_imp_of_nat)
+apply (force simp add: linorder_not_less [symmetric])
done
lemma of_nat_less_iff [simp]:
"(of_nat m < (of_nat n::'a::ordered_semiring)) = (m<n)"
-by (blast intro: of_nat_less_imp_less less_imp_of_nat_less )
+by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
text{*Special cases where either operand is zero*}
declare of_nat_less_iff [of 0, simplified, simp]
@@ -694,7 +613,7 @@
lemma of_nat_le_iff [simp]:
"(of_nat m \<le> (of_nat n::'a::ordered_semiring)) = (m \<le> n)"
-by (simp add: linorder_not_less [symmetric])
+by (simp add: linorder_not_less [symmetric])
text{*Special cases where either operand is zero*}
declare of_nat_le_iff [of 0, simplified, simp]
@@ -704,7 +623,7 @@
a finite field, which indeed wraps back to zero.*}
lemma of_nat_eq_iff [simp]:
"(of_nat m = (of_nat n::'a::ordered_semiring)) = (m = n)"
-by (simp add: order_eq_iff)
+by (simp add: order_eq_iff)
text{*Special cases where either operand is zero*}
declare of_nat_eq_iff [of 0, simplified, simp]
@@ -713,7 +632,7 @@
lemma of_nat_diff [simp]:
"n \<le> m ==> of_nat (m - n) = of_nat m - (of_nat n :: 'a::ring)"
by (simp del: of_nat_add
- add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
+ add: compare_rls of_nat_add [symmetric] split add: nat_diff_split)
subsection{*The Set of Natural Numbers*}
@@ -725,29 +644,29 @@
syntax (xsymbols) Nats :: "'a set" ("\<nat>")
lemma of_nat_in_Nats [simp]: "of_nat n \<in> Nats"
-by (simp add: Nats_def)
+by (simp add: Nats_def)
lemma Nats_0 [simp]: "0 \<in> Nats"
-apply (simp add: Nats_def)
-apply (rule range_eqI)
+apply (simp add: Nats_def)
+apply (rule range_eqI)
apply (rule of_nat_0 [symmetric])
done
lemma Nats_1 [simp]: "1 \<in> Nats"
-apply (simp add: Nats_def)
-apply (rule range_eqI)
+apply (simp add: Nats_def)
+apply (rule range_eqI)
apply (rule of_nat_1 [symmetric])
done
lemma Nats_add [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a+b \<in> Nats"
-apply (auto simp add: Nats_def)
-apply (rule range_eqI)
+apply (auto simp add: Nats_def)
+apply (rule range_eqI)
apply (rule of_nat_add [symmetric])
done
lemma Nats_mult [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> a*b \<in> Nats"
-apply (auto simp add: Nats_def)
-apply (rule range_eqI)
+apply (auto simp add: Nats_def)
+apply (rule range_eqI)
apply (rule of_nat_mult [symmetric])
done
@@ -755,7 +674,7 @@
lemma int_eq_of_nat: "int = (of_nat :: nat => int)"
proof
fix n
- show "int n = of_nat n" by (induct n, simp_all add: int_Suc add_ac)
+ show "int n = of_nat n" by (induct n, simp_all add: int_Suc add_ac)
qed
@@ -769,9 +688,9 @@
lemma of_int: "of_int (Abs_Integ (intrel `` {(i,j)})) = of_nat i - of_nat j"
apply (simp add: of_int_def)
-apply (rule the_equality, auto)
+apply (rule the_equality, auto)
apply (simp add: compare_rls add_ac of_nat_add [symmetric]
- del: of_nat_add)
+ del: of_nat_add)
done
lemma of_int_0 [simp]: "of_int 0 = 0"
@@ -781,32 +700,26 @@
by (simp add: of_int One_int_def int_def)
lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
-apply (rule eq_Abs_Integ [of w])
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int zadd)
-done
+by (cases w, cases z, simp add: compare_rls of_int add)
lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int zminus)
-done
+by (cases z, simp add: compare_rls of_int minus)
lemma of_int_diff [simp]: "of_int (w-z) = of_int w - of_int z"
by (simp add: diff_minus)
lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
-apply (rule eq_Abs_Integ [of w])
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
- zmult add_ac)
+apply (cases w, cases z)
+apply (simp add: compare_rls of_int left_diff_distrib right_diff_distrib
+ mult add_ac)
done
lemma of_int_le_iff [simp]:
"(of_int w \<le> (of_int z::'a::ordered_ring)) = (w \<le> z)"
-apply (rule eq_Abs_Integ [of w])
-apply (rule eq_Abs_Integ [of z])
-apply (simp add: compare_rls of_int zle zdiff_def zadd zminus
- of_nat_add [symmetric] del: of_nat_add)
+apply (cases w)
+apply (cases z)
+apply (simp add: compare_rls of_int le diff_int_def add minus
+ of_nat_add [symmetric] del: of_nat_add)
done
text{*Special cases where either operand is zero*}
@@ -824,7 +737,7 @@
text{*The ordering on the ring is necessary. See @{text of_nat_eq_iff} above.*}
lemma of_int_eq_iff [simp]:
"(of_int w = (of_int z::'a::ordered_ring)) = (w = z)"
-by (simp add: order_eq_iff)
+by (simp add: order_eq_iff)
text{*Special cases where either operand is zero*}
declare of_int_eq_iff [of 0, simplified, simp]
@@ -842,52 +755,52 @@
Ints :: "'a set" ("\<int>")
lemma Ints_0 [simp]: "0 \<in> Ints"
-apply (simp add: Ints_def)
-apply (rule range_eqI)
+apply (simp add: Ints_def)
+apply (rule range_eqI)
apply (rule of_int_0 [symmetric])
done
lemma Ints_1 [simp]: "1 \<in> Ints"
-apply (simp add: Ints_def)
-apply (rule range_eqI)
+apply (simp add: Ints_def)
+apply (rule range_eqI)
apply (rule of_int_1 [symmetric])
done
lemma Ints_add [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a+b \<in> Ints"
-apply (auto simp add: Ints_def)
-apply (rule range_eqI)
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
apply (rule of_int_add [symmetric])
done
lemma Ints_minus [simp]: "a \<in> Ints ==> -a \<in> Ints"
-apply (auto simp add: Ints_def)
-apply (rule range_eqI)
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
apply (rule of_int_minus [symmetric])
done
lemma Ints_diff [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a-b \<in> Ints"
-apply (auto simp add: Ints_def)
-apply (rule range_eqI)
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
apply (rule of_int_diff [symmetric])
done
lemma Ints_mult [simp]: "[|a \<in> Ints; b \<in> Ints|] ==> a*b \<in> Ints"
-apply (auto simp add: Ints_def)
-apply (rule range_eqI)
+apply (auto simp add: Ints_def)
+apply (rule range_eqI)
apply (rule of_int_mult [symmetric])
done
text{*Collapse nested embeddings*}
lemma of_int_of_nat_eq [simp]: "of_int (of_nat n) = of_nat n"
-by (induct n, auto)
+by (induct n, auto)
lemma of_int_int_eq [simp]: "of_int (int n) = int n"
-by (simp add: int_eq_of_nat)
+by (simp add: int_eq_of_nat)
lemma Ints_cases [case_names of_int, cases set: Ints]:
"q \<in> \<int> ==> (!!z. q = of_int z ==> C) ==> C"
-proof (unfold Ints_def)
+proof (simp add: Ints_def)
assume "!!z. q = of_int z ==> C"
assume "q \<in> range of_int" thus C ..
qed
@@ -963,11 +876,41 @@
by (rule setprod_zero_eq, auto)
+text{*Now we replace the case analysis rule by a more conventional one:
+whether an integer is negative or not.*}
+
+lemma zless_iff_Suc_zadd:
+ "(w < z) = (\<exists>n. z = w + int(Suc n))"
+apply (cases z, cases w)
+apply (auto simp add: le add int_def linorder_not_le [symmetric])
+apply (rename_tac a b c d)
+apply (rule_tac x="a+d - Suc(c+b)" in exI)
+apply arith
+done
+
+lemma negD: "x<0 ==> \<exists>n. x = - (int (Suc n))"
+apply (cases x)
+apply (auto simp add: le minus Zero_int_def int_def order_less_le)
+apply (rule_tac x="y - Suc x" in exI)
+apply arith
+done
+
+theorem int_cases [cases type: int, case_names nonneg neg]:
+ "[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P"
+apply (case_tac "z < 0", blast dest!: negD)
+apply (simp add: linorder_not_less)
+apply (blast dest: nat_0_le [THEN sym])
+done
+
+theorem int_induct [induct type: int, case_names nonneg neg]:
+ "[|!! n. P (int n); !!n. P (- (int (Suc n))) |] ==> P z"
+ by (cases z) auto
+
+
(*Legacy ML bindings, but no longer the structure Int.*)
ML
{*
val zabs_def = thm "zabs_def"
-val nat_def = thm "nat_def"
val int_0 = thm "int_0";
val int_1 = thm "int_1";
@@ -1002,11 +945,7 @@
val int_def = thm "int_def";
val Zero_int_def = thm "Zero_int_def";
val One_int_def = thm "One_int_def";
-val zadd_def = thm "zadd_def";
-val zdiff_def = thm "zdiff_def";
-val zless_def = thm "zless_def";
-val zle_def = thm "zle_def";
-val zmult_def = thm "zmult_def";
+val diff_int_def = thm "diff_int_def";
val intrel_iff = thm "intrel_iff";
val equiv_intrel = thm "equiv_intrel";
@@ -1015,13 +954,8 @@
val inj_on_Abs_Integ = thm "inj_on_Abs_Integ";
val inj_Rep_Integ = thm "inj_Rep_Integ";
val inj_int = thm "inj_int";
-val zminus_congruent = thm "zminus_congruent";
-val zminus = thm "zminus";
-val eq_Abs_Integ = thm "eq_Abs_Integ";
val zminus_zminus = thm "zminus_zminus";
-val inj_zminus = thm "inj_zminus";
val zminus_0 = thm "zminus_0";
-val zadd = thm "zadd";
val zminus_zadd_distrib = thm "zminus_zadd_distrib";
val zadd_commute = thm "zadd_commute";
val zadd_assoc = thm "zadd_assoc";
@@ -1033,30 +967,17 @@
val int_Suc = thm "int_Suc";
val zadd_0 = thm "zadd_0";
val zadd_0_right = thm "zadd_0_right";
-val zadd_zminus_inverse = thm "zadd_zminus_inverse";
-val zadd_zminus_inverse2 = thm "zadd_zminus_inverse2";
-val zadd_zminus_cancel = thm "zadd_zminus_cancel";
-val zminus_zadd_cancel = thm "zminus_zadd_cancel";
-val zdiff0 = thm "zdiff0";
-val zdiff0_right = thm "zdiff0_right";
-val zdiff_self = thm "zdiff_self";
-val zmult_congruent2 = thm "zmult_congruent2";
-val zmult = thm "zmult";
val zmult_zminus = thm "zmult_zminus";
val zmult_commute = thm "zmult_commute";
val zmult_assoc = thm "zmult_assoc";
val zadd_zmult_distrib = thm "zadd_zmult_distrib";
-val zmult_zminus_right = thm "zmult_zminus_right";
val zadd_zmult_distrib2 = thm "zadd_zmult_distrib2";
val zdiff_zmult_distrib = thm "zdiff_zmult_distrib";
val zdiff_zmult_distrib2 = thm "zdiff_zmult_distrib2";
val int_distrib = thms "int_distrib";
val zmult_int = thm "zmult_int";
-val zmult_0 = thm "zmult_0";
val zmult_1 = thm "zmult_1";
-val zmult_0_right = thm "zmult_0_right";
val zmult_1_right = thm "zmult_1_right";
-val zless_iff_Suc_zadd = thm "zless_iff_Suc_zadd";
val int_int_eq = thm "int_int_eq";
val int_eq_0_conv = thm "int_eq_0_conv";
val zless_int = thm "zless_int";