--- a/src/HOL/Integ/Int.thy Thu Nov 20 10:41:39 2003 +0100
+++ b/src/HOL/Integ/Int.thy Thu Nov 20 10:42:00 2003 +0100
@@ -2,12 +2,11 @@
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
-
-Type "int" is a linear order
*)
-theory Int = IntDef
-files ("Int_lemmas.ML"):
+header {*Type "int" is a Linear Order and Other Lemmas*}
+
+theory Int = IntDef:
instance int :: order
proof qed (assumption | rule zle_refl zle_trans zle_anti_sym int_less_le)+
@@ -19,12 +18,527 @@
proof qed (rule zle_linear)
constdefs
- nat :: "int => nat"
-"nat(Z) == if neg Z then 0 else (THE m. Z = int m)"
+ nat :: "int => nat"
+ "nat(Z) == if neg Z then 0 else (THE m. Z = int m)"
defs (overloaded)
-zabs_def: "abs(i::int) == if i < 0 then -i else i"
+ zabs_def: "abs(i::int) == if i < 0 then -i else i"
+
+
+lemma int_0 [simp]: "int 0 = (0::int)"
+by (simp add: Zero_int_def)
+
+lemma int_1 [simp]: "int 1 = 1"
+by (simp add: One_int_def)
+
+lemma int_Suc0_eq_1: "int (Suc 0) = 1"
+by (simp add: One_int_def One_nat_def)
+
+lemma neg_eq_less_0: "neg x = (x < 0)"
+by (unfold zdiff_def zless_def, auto)
+
+lemma not_neg_eq_ge_0: "(~neg x) = (0 <= x)"
+apply (unfold zle_def)
+apply (simp add: neg_eq_less_0)
+done
+
+subsection{*To simplify inequalities when Numeral1 can get simplified to 1*}
+
+lemma not_neg_0: "~ neg 0"
+by (simp add: One_int_def neg_eq_less_0)
+
+lemma not_neg_1: "~ neg 1"
+by (simp add: One_int_def neg_eq_less_0)
+
+lemma iszero_0: "iszero 0"
+by (simp add: iszero_def)
+
+lemma not_iszero_1: "~ iszero 1"
+by (simp only: Zero_int_def One_int_def One_nat_def iszero_def int_int_eq)
+
+lemma int_0_less_1: "0 < (1::int)"
+by (simp only: Zero_int_def One_int_def One_nat_def zless_int)
+
+lemma int_0_neq_1 [simp]: "0 \<noteq> (1::int)"
+by (simp only: Zero_int_def One_int_def One_nat_def int_int_eq)
+
+
+
+subsection{*@{text Abel_Cancel} simproc on the integers*}
+
+(* Lemmas needed for the simprocs *)
+
+(*Deletion of other terms in the formula, seeking the -x at the front of z*)
+lemma zadd_cancel_21: "((x::int) + (y + z) = y + u) = ((x + z) = u)"
+apply (subst zadd_left_commute)
+apply (rule zadd_left_cancel)
+done
+
+(*A further rule to deal with the case that
+ everything gets cancelled on the right.*)
+lemma zadd_cancel_end: "((x::int) + (y + z) = y) = (x = -z)"
+apply (subst zadd_left_commute)
+apply (rule_tac t = y in zadd_0_right [THEN subst], subst zadd_left_cancel)
+apply (simp add: eq_zdiff_eq [symmetric])
+done
+
+(*Legacy ML bindings, but no longer the structure Int.*)
+ML
+{*
+val Int_thy = the_context ()
+val zabs_def = thm "zabs_def"
+val nat_def = thm "nat_def"
+
+val int_0 = thm "int_0";
+val int_1 = thm "int_1";
+val int_Suc0_eq_1 = thm "int_Suc0_eq_1";
+val neg_eq_less_0 = thm "neg_eq_less_0";
+val not_neg_eq_ge_0 = thm "not_neg_eq_ge_0";
+val not_neg_0 = thm "not_neg_0";
+val not_neg_1 = thm "not_neg_1";
+val iszero_0 = thm "iszero_0";
+val not_iszero_1 = thm "not_iszero_1";
+val int_0_less_1 = thm "int_0_less_1";
+val int_0_neq_1 = thm "int_0_neq_1";
+val zadd_cancel_21 = thm "zadd_cancel_21";
+val zadd_cancel_end = thm "zadd_cancel_end";
+
+structure Int_Cancel_Data =
+struct
+ val ss = HOL_ss
+ val eq_reflection = eq_reflection
+
+ val sg_ref = Sign.self_ref (Theory.sign_of (the_context()))
+ val T = HOLogic.intT
+ val zero = Const ("0", HOLogic.intT)
+ val restrict_to_left = restrict_to_left
+ val add_cancel_21 = zadd_cancel_21
+ val add_cancel_end = zadd_cancel_end
+ val add_left_cancel = zadd_left_cancel
+ val add_assoc = zadd_assoc
+ val add_commute = zadd_commute
+ val add_left_commute = zadd_left_commute
+ val add_0 = zadd_0
+ val add_0_right = zadd_0_right
+
+ val eq_diff_eq = eq_zdiff_eq
+ val eqI_rules = [zless_eqI, zeq_eqI, zle_eqI]
+ fun dest_eqI th =
+ #1 (HOLogic.dest_bin "op =" HOLogic.boolT
+ (HOLogic.dest_Trueprop (concl_of th)))
+
+ val diff_def = zdiff_def
+ val minus_add_distrib = zminus_zadd_distrib
+ val minus_minus = zminus_zminus
+ val minus_0 = zminus_0
+ val add_inverses = [zadd_zminus_inverse, zadd_zminus_inverse2]
+ val cancel_simps = [zadd_zminus_cancel, zminus_zadd_cancel]
+end;
+
+structure Int_Cancel = Abel_Cancel (Int_Cancel_Data);
+
+Addsimprocs [Int_Cancel.sum_conv, Int_Cancel.rel_conv];
+*}
+
+
+subsection{*Misc Results*}
+
+lemma zminus_zdiff_eq [simp]: "- (z - y) = y - (z::int)"
+by simp
+
+lemma zless_eq_neg: "(w<z) = neg(w-z)"
+by (simp add: zless_def)
+
+lemma eq_eq_iszero: "(w=z) = iszero(w-z)"
+by (simp add: iszero_def zdiff_eq_eq)
+
+lemma zle_eq_not_neg: "(w<=z) = (~ neg(z-w))"
+by (simp add: zle_def zless_def)
+
+subsection{*Inequality reasoning*}
+
+lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
+apply (auto simp add: zless_iff_Suc_zadd int_Suc gr0_conv_Suc zero_reorient)
+apply (rule_tac x = "Suc n" in exI)
+apply (simp add: int_Suc)
+done
+
+lemma add1_zle_eq: "(w + (1::int) <= z) = (w<z)"
+apply (simp add: zle_def zless_add1_eq)
+apply (auto intro: zless_asym zle_anti_sym
+ simp add: order_less_imp_le symmetric zle_def)
+done
+
+lemma add1_left_zle_eq: "((1::int) + w <= z) = (w<z)"
+apply (subst zadd_commute)
+apply (rule add1_zle_eq)
+done
+
+
+subsection{*Monotonicity results*}
+
+lemma zadd_right_cancel_zless [simp]: "(v+z < w+z) = (v < (w::int))"
+by simp
+
+lemma zadd_left_cancel_zless [simp]: "(z+v < z+w) = (v < (w::int))"
+by simp
+
+lemma zadd_right_cancel_zle [simp] : "(v+z <= w+z) = (v <= (w::int))"
+by simp
+
+lemma zadd_left_cancel_zle [simp] : "(z+v <= z+w) = (v <= (w::int))"
+by simp
+
+(*"v<=w ==> v+z <= w+z"*)
+lemmas zadd_zless_mono1 = zadd_right_cancel_zless [THEN iffD2, standard]
+
+(*"v<=w ==> z+v <= z+w"*)
+lemmas zadd_zless_mono2 = zadd_left_cancel_zless [THEN iffD2, standard]
+
+(*"v<=w ==> v+z <= w+z"*)
+lemmas zadd_zle_mono1 = zadd_right_cancel_zle [THEN iffD2, standard]
+
+(*"v<=w ==> z+v <= z+w"*)
+lemmas zadd_zle_mono2 = zadd_left_cancel_zle [THEN iffD2, standard]
+
+lemma zadd_zle_mono: "[| w'<=w; z'<=z |] ==> w' + z' <= w + (z::int)"
+by (erule zadd_zle_mono1 [THEN zle_trans], simp)
+
+lemma zadd_zless_mono: "[| w'<w; z'<=z |] ==> w' + z' < w + (z::int)"
+by (erule zadd_zless_mono1 [THEN order_less_le_trans], simp)
+
+
+subsection{*Comparison laws*}
+
+lemma zminus_zless_zminus [simp]: "(- x < - y) = (y < (x::int))"
+by (simp add: zless_def zdiff_def zadd_ac)
+
+lemma zminus_zle_zminus [simp]: "(- x <= - y) = (y <= (x::int))"
+by (simp add: zle_def)
+
+text{*The next several equations can make the simplifier loop!*}
+
+lemma zless_zminus: "(x < - y) = (y < - (x::int))"
+by (simp add: zless_def zdiff_def zadd_ac)
+
+lemma zminus_zless: "(- x < y) = (- y < (x::int))"
+by (simp add: zless_def zdiff_def zadd_ac)
+
+lemma zle_zminus: "(x <= - y) = (y <= - (x::int))"
+by (simp add: zle_def zminus_zless)
+
+lemma zminus_zle: "(- x <= y) = (- y <= (x::int))"
+by (simp add: zle_def zless_zminus)
+
+lemma equation_zminus: "(x = - y) = (y = - (x::int))"
+by auto
+
+lemma zminus_equation: "(- x = y) = (- (y::int) = x)"
+by auto
+
+
+subsection{*Instances of the equations above, for zero*}
+
+(*instantiate a variable to zero and simplify*)
+
+declare zless_zminus [of 0, simplified, simp]
+declare zminus_zless [of _ 0, simplified, simp]
+declare zle_zminus [of 0, simplified, simp]
+declare zminus_zle [of _ 0, simplified, simp]
+declare equation_zminus [of 0, simplified, simp]
+declare zminus_equation [of _ 0, simplified, simp]
+
+lemma negative_zless_0: "- (int (Suc n)) < 0"
+by (simp add: zless_def)
+
+lemma negative_zless [iff]: "- (int (Suc n)) < int m"
+by (rule negative_zless_0 [THEN order_less_le_trans], simp)
+
+lemma negative_zle_0: "- int n <= 0"
+by (simp add: zminus_zle)
+
+lemma negative_zle [iff]: "- int n <= int m"
+by (simp add: zless_def zle_def zdiff_def zadd_int)
+
+lemma not_zle_0_negative [simp]: "~(0 <= - (int (Suc n)))"
+by (subst zle_zminus, simp)
+
+lemma int_zle_neg: "(int n <= - int m) = (n = 0 & m = 0)"
+apply safe
+apply (drule_tac [2] zle_zminus [THEN iffD1])
+apply (auto dest: zle_trans [OF _ negative_zle_0])
+done
+
+lemma not_int_zless_negative [simp]: "~(int n < - int m)"
+by (simp add: zle_def [symmetric])
+
+lemma negative_eq_positive [simp]: "(- int n = int m) = (n = 0 & m = 0)"
+apply (rule iffI)
+apply (rule int_zle_neg [THEN iffD1])
+apply (drule sym)
+apply (simp_all (no_asm_simp))
+done
-use "Int_lemmas.ML"
+lemma zle_iff_zadd: "(w <= z) = (EX 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 int_le_less)
+
+lemma abs_int_eq [simp]: "abs (int m) = int m"
+by (simp add: zabs_def)
+
+
+subsection{*nat: magnitide of an integer, as a natural number*}
+
+lemma nat_int [simp]: "nat(int n) = n"
+by (unfold nat_def, auto)
+
+lemma nat_zminus_int [simp]: "nat(- (int n)) = 0"
+apply (unfold nat_def)
+apply (auto simp add: neg_eq_less_0 zero_reorient zminus_zless)
+done
+
+lemma nat_zero [simp]: "nat 0 = 0"
+apply (unfold Zero_int_def)
+apply (rule nat_int)
+done
+
+lemma not_neg_nat: "~ neg z ==> int (nat z) = z"
+apply (drule not_neg_eq_ge_0 [THEN iffD1])
+apply (drule zle_imp_zless_or_eq)
+apply (auto simp add: zless_iff_Suc_zadd)
+done
+
+lemma negD: "neg x ==> EX n. x = - (int (Suc n))"
+by (auto simp add: neg_eq_less_0 zless_iff_Suc_zadd zdiff_eq_eq [symmetric] zdiff_def)
+
+lemma neg_nat: "neg z ==> nat z = 0"
+by (unfold nat_def, auto)
+
+lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
+apply (case_tac "neg z")
+apply (erule_tac [2] not_neg_nat [THEN subst])
+apply (auto simp add: neg_nat)
+apply (auto dest: order_less_trans simp add: neg_eq_less_0)
+done
+
+lemma nat_0_le [simp]: "0 <= z ==> int (nat z) = z"
+by (simp add: neg_eq_less_0 zle_def not_neg_nat)
+
+lemma nat_le_0 [simp]: "z <= 0 ==> nat z = 0"
+by (auto simp add: order_le_less neg_eq_less_0 zle_def neg_nat)
+
+(*An alternative condition is 0 <= w *)
+lemma nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
+apply (subst zless_int [symmetric])
+apply (simp (no_asm_simp) add: not_neg_nat not_neg_eq_ge_0 order_le_less)
+apply (case_tac "neg w")
+ apply (simp add: neg_eq_less_0 neg_nat)
+ apply (blast intro: order_less_trans)
+apply (simp add: not_neg_nat)
+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)
+done
+
+(* a case theorem distinguishing non-negative and negative int *)
+
+lemma int_cases:
+ "[|!! n. z = int n ==> P; !! n. z = - (int (Suc n)) ==> P |] ==> P"
+apply (case_tac "neg z")
+apply (fast dest!: negD)
+apply (drule not_neg_nat [symmetric], auto)
+done
+
+
+subsection{*Monotonicity of Multiplication*}
+
+lemma zmult_zle_mono1_lemma: "i <= (j::int) ==> i * int k <= j * int k"
+apply (induct_tac "k")
+apply (simp_all (no_asm_simp) add: int_Suc zadd_zmult_distrib2 zadd_zle_mono int_Suc0_eq_1)
+done
+
+lemma zmult_zle_mono1: "[| i <= j; (0::int) <= k |] ==> i*k <= j*k"
+apply (rule_tac t = k in not_neg_nat [THEN subst])
+apply (erule_tac [2] zmult_zle_mono1_lemma)
+apply (simp (no_asm_use) add: not_neg_eq_ge_0)
+done
+
+lemma zmult_zle_mono1_neg: "[| i <= j; k <= (0::int) |] ==> j*k <= i*k"
+apply (rule zminus_zle_zminus [THEN iffD1])
+apply (simp add: zmult_zminus_right [symmetric] zmult_zle_mono1 zle_zminus)
+done
+
+lemma zmult_zle_mono2: "[| i <= j; (0::int) <= k |] ==> k*i <= k*j"
+apply (drule zmult_zle_mono1)
+apply (simp_all add: zmult_commute)
+done
+
+lemma zmult_zle_mono2_neg: "[| i <= j; k <= (0::int) |] ==> k*j <= k*i"
+apply (drule zmult_zle_mono1_neg)
+apply (simp_all add: zmult_commute)
+done
+
+(* <= monotonicity, BOTH arguments*)
+lemma zmult_zle_mono: "[| i <= j; k <= l; (0::int) <= j; (0::int) <= k |] ==> i*k <= j*l"
+apply (erule zmult_zle_mono1 [THEN order_trans], assumption)
+apply (erule zmult_zle_mono2, assumption)
+done
+
+
+subsection{*strict, in 1st argument; proof is by induction on k>0*}
+
+lemma zmult_zless_mono2_lemma: "i<j ==> 0<k --> int k * i < int k * j"
+apply (induct_tac "k", simp)
+apply (simp add: int_Suc)
+apply (case_tac "n=0")
+apply (simp_all add: zadd_zmult_distrib zadd_zless_mono int_Suc0_eq_1 order_le_less)
+done
+
+lemma zmult_zless_mono2: "[| i<j; (0::int) < k |] ==> k*i < k*j"
+apply (rule_tac t = k in not_neg_nat [THEN subst])
+apply (erule_tac [2] zmult_zless_mono2_lemma [THEN mp])
+apply (simp add: not_neg_eq_ge_0 order_le_less)
+apply (frule conjI [THEN zless_nat_conj [THEN iffD2]], auto)
+done
+
+lemma zmult_zless_mono1: "[| i<j; (0::int) < k |] ==> i*k < j*k"
+apply (drule zmult_zless_mono2)
+apply (simp_all add: zmult_commute)
+done
+
+(* < monotonicity, BOTH arguments*)
+lemma zmult_zless_mono: "[| i < j; k < l; (0::int) < j; (0::int) < k |] ==> i*k < j*l"
+apply (erule zmult_zless_mono1 [THEN order_less_trans], assumption)
+apply (erule zmult_zless_mono2, assumption)
+done
+
+lemma zmult_zless_mono1_neg: "[| i<j; k < (0::int) |] ==> j*k < i*k"
+apply (rule zminus_zless_zminus [THEN iffD1])
+apply (simp add: zmult_zminus_right [symmetric] zmult_zless_mono1 zless_zminus)
+done
+
+lemma zmult_zless_mono2_neg: "[| i<j; k < (0::int) |] ==> k*j < k*i"
+apply (rule zminus_zless_zminus [THEN iffD1])
+apply (simp add: zmult_zminus [symmetric] zmult_zless_mono2 zless_zminus)
+done
+
+lemma zmult_eq_0_iff [iff]: "(m*n = (0::int)) = (m = 0 | n = 0)"
+apply (case_tac "m < (0::int) ")
+apply (auto simp add: linorder_not_less order_le_less linorder_neq_iff)
+apply (force dest: zmult_zless_mono1_neg zmult_zless_mono1)+
+done
+
+
+text{*Cancellation laws for k*m < k*n and m*k < n*k, also for <= and =,
+ but not (yet?) for k*m < n*k.*}
+
+lemma zmult_zless_cancel2: "(m*k < n*k) = (((0::int) < k & m<n) | (k < 0 & n<m))"
+apply (case_tac "k = (0::int) ")
+apply (auto simp add: linorder_neq_iff zmult_zless_mono1 zmult_zless_mono1_neg)
+apply (auto simp add: linorder_not_less
+ linorder_not_le [symmetric, of "m*k"]
+ linorder_not_le [symmetric, of m])
+apply (erule_tac [!] notE)
+apply (auto simp add: order_less_imp_le zmult_zle_mono1 zmult_zle_mono1_neg)
+done
+
+
+lemma zmult_zless_cancel1:
+ "(k*m < k*n) = (((0::int) < k & m<n) | (k < 0 & n<m))"
+by (simp add: zmult_commute [of k] zmult_zless_cancel2)
+
+lemma zmult_zle_cancel2:
+ "(m*k <= n*k) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))"
+by (simp add: linorder_not_less [symmetric] zmult_zless_cancel2)
+
+lemma zmult_zle_cancel1:
+ "(k*m <= k*n) = (((0::int) < k --> m<=n) & (k < 0 --> n<=m))"
+by (simp add: linorder_not_less [symmetric] zmult_zless_cancel1)
+
+lemma zmult_cancel2 [simp]: "(m*k = n*k) = (k = (0::int) | m=n)"
+apply (cut_tac linorder_less_linear [of 0 k])
+apply (force dest: zmult_zless_mono1_neg zmult_zless_mono1
+ simp add: linorder_neq_iff)
+done
+
+lemma zmult_cancel1 [simp]: "(k*m = k*n) = (k = (0::int) | m=n)"
+by (simp add: zmult_commute [of k] zmult_cancel2)
+
+(*Analogous to zadd_int*)
+lemma zdiff_int [rule_format (no_asm)]: "n<=m --> int m - int n = int (m-n)"
+apply (induct_tac m n rule: diff_induct)
+apply (auto simp add: int_Suc symmetric zdiff_def)
+done
+
+ML
+{*
+val zminus_zdiff_eq = thm "zminus_zdiff_eq";
+val zless_eq_neg = thm "zless_eq_neg";
+val eq_eq_iszero = thm "eq_eq_iszero";
+val zle_eq_not_neg = thm "zle_eq_not_neg";
+val zless_add1_eq = thm "zless_add1_eq";
+val add1_zle_eq = thm "add1_zle_eq";
+val add1_left_zle_eq = thm "add1_left_zle_eq";
+val zadd_right_cancel_zless = thm "zadd_right_cancel_zless";
+val zadd_left_cancel_zless = thm "zadd_left_cancel_zless";
+val zadd_right_cancel_zle = thm "zadd_right_cancel_zle";
+val zadd_left_cancel_zle = thm "zadd_left_cancel_zle";
+val zadd_zless_mono1 = thm "zadd_zless_mono1";
+val zadd_zless_mono2 = thm "zadd_zless_mono2";
+val zadd_zle_mono1 = thm "zadd_zle_mono1";
+val zadd_zle_mono2 = thm "zadd_zle_mono2";
+val zadd_zle_mono = thm "zadd_zle_mono";
+val zadd_zless_mono = thm "zadd_zless_mono";
+val zminus_zless_zminus = thm "zminus_zless_zminus";
+val zminus_zle_zminus = thm "zminus_zle_zminus";
+val zless_zminus = thm "zless_zminus";
+val zminus_zless = thm "zminus_zless";
+val zle_zminus = thm "zle_zminus";
+val zminus_zle = thm "zminus_zle";
+val equation_zminus = thm "equation_zminus";
+val zminus_equation = thm "zminus_equation";
+val negative_zless_0 = thm "negative_zless_0";
+val negative_zless = thm "negative_zless";
+val negative_zle_0 = thm "negative_zle_0";
+val negative_zle = thm "negative_zle";
+val not_zle_0_negative = thm "not_zle_0_negative";
+val int_zle_neg = thm "int_zle_neg";
+val not_int_zless_negative = thm "not_int_zless_negative";
+val negative_eq_positive = thm "negative_eq_positive";
+val zle_iff_zadd = thm "zle_iff_zadd";
+val abs_int_eq = thm "abs_int_eq";
+val nat_int = thm "nat_int";
+val nat_zminus_int = thm "nat_zminus_int";
+val nat_zero = thm "nat_zero";
+val not_neg_nat = thm "not_neg_nat";
+val negD = thm "negD";
+val neg_nat = thm "neg_nat";
+val zless_nat_eq_int_zless = thm "zless_nat_eq_int_zless";
+val nat_0_le = thm "nat_0_le";
+val nat_le_0 = thm "nat_le_0";
+val zless_nat_conj = thm "zless_nat_conj";
+val int_cases = thm "int_cases";
+val zmult_zle_mono1 = thm "zmult_zle_mono1";
+val zmult_zle_mono1_neg = thm "zmult_zle_mono1_neg";
+val zmult_zle_mono2 = thm "zmult_zle_mono2";
+val zmult_zle_mono2_neg = thm "zmult_zle_mono2_neg";
+val zmult_zle_mono = thm "zmult_zle_mono";
+val zmult_zless_mono2 = thm "zmult_zless_mono2";
+val zmult_zless_mono1 = thm "zmult_zless_mono1";
+val zmult_zless_mono = thm "zmult_zless_mono";
+val zmult_zless_mono1_neg = thm "zmult_zless_mono1_neg";
+val zmult_zless_mono2_neg = thm "zmult_zless_mono2_neg";
+val zmult_eq_0_iff = thm "zmult_eq_0_iff";
+val zmult_zless_cancel2 = thm "zmult_zless_cancel2";
+val zmult_zless_cancel1 = thm "zmult_zless_cancel1";
+val zmult_zle_cancel2 = thm "zmult_zle_cancel2";
+val zmult_zle_cancel1 = thm "zmult_zle_cancel1";
+val zmult_cancel2 = thm "zmult_cancel2";
+val zmult_cancel1 = thm "zmult_cancel1";
+val zdiff_int = thm "zdiff_int";
+*}
end