--- a/src/ZF/Integ/IntDiv.ML Mon May 21 14:52:04 2001 +0200
+++ b/src/ZF/Integ/IntDiv.ML Mon May 21 14:52:27 2001 +0200
@@ -30,7 +30,7 @@
else negateSnd (posDivAlg (~a,~b));
*)
-Goal "[| #0 $< k; k: int |] ==> 0 < zmagnitude(k)";
+Goal "[| #0 $< k; k \\<in> int |] ==> 0 < zmagnitude(k)";
by (dtac zero_zless_imp_znegative_zminus 1);
by (dtac zneg_int_of 2);
by (auto_tac (claset(), simpset() addsimps [inst "x" "k" zminus_equation]));
@@ -57,7 +57,7 @@
by Auto_tac;
qed "zless_add_succ_iff";
-Goal "z : int ==> (w $+ $# succ(m) $<= z) <-> (w $+ $#m $< z)";
+Goal "z \\<in> int ==> (w $+ $# succ(m) $<= z) <-> (w $+ $#m $< z)";
by (asm_simp_tac (simpset_of Int.thy addsimps
[not_zless_iff_zle RS iff_sym, zless_add_succ_iff]) 1);
by (auto_tac (claset() addIs [zle_anti_sym] addEs [zless_asym],
@@ -137,7 +137,7 @@
(*** Monotonicity of Multiplication ***)
-Goal "k : nat ==> i $<= j ==> i $* $#k $<= j $* $#k";
+Goal "k \\<in> nat ==> i $<= j ==> i $* $#k $<= j $* $#k";
by (induct_tac "k" 1);
by (stac int_succ_int_1 2);
by (ALLGOALS
@@ -182,7 +182,7 @@
(** strict, in 1st argument; proof is by induction on k>0 **)
-Goal "[| i$<j; k : nat |] ==> 0<k --> $#k $* i $< $#k $* j";
+Goal "[| i$<j; k \\<in> nat |] ==> 0<k --> $#k $* i $< $#k $* j";
by (induct_tac "k" 1);
by (stac int_succ_int_1 2);
by (etac natE 2);
@@ -235,7 +235,7 @@
zmult_zless_mono2, zless_zminus]) 1);
qed "zmult_zless_mono2_neg";
-Goal "[| m: int; n: int |] ==> (m$*n = #0) <-> (m = #0 | n = #0)";
+Goal "[| m \\<in> int; n \\<in> int |] ==> (m$*n = #0) <-> (m = #0 | n = #0)";
by (case_tac "m $< #0" 1);
by (auto_tac (claset(),
simpset() addsimps [not_zless_iff_zle, zle_def, neq_iff_zless]));
@@ -252,7 +252,7 @@
(** Cancellation laws for k*m < k*n and m*k < n*k, also for <= and =,
but not (yet?) for k*m < n*k. **)
-Goal "[| k: int; m: int; n: int |] \
+Goal "[| k \\<in> int; m \\<in> int; n \\<in> int |] \
\ ==> (m$*k $< n$*k) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))";
by (case_tac "k = #0" 1);
by (auto_tac (claset(), simpset() addsimps [neq_iff_zless,
@@ -289,11 +289,11 @@
by Auto_tac;
qed "zmult_zle_cancel1";
-Goal "[| m: int; n: int |] ==> m=n <-> (m $<= n & n $<= m)";
+Goal "[| m \\<in> int; n \\<in> int |] ==> m=n <-> (m $<= n & n $<= m)";
by (blast_tac (claset() addIs [zle_refl,zle_anti_sym]) 1);
qed "int_eq_iff_zle";
-Goal "[| k: int; m: int; n: int |] ==> (m$*k = n$*k) <-> (k=#0 | m=n)";
+Goal "[| k \\<in> int; m \\<in> int; n \\<in> int |] ==> (m$*k = n$*k) <-> (k=#0 | m=n)";
by (asm_simp_tac (simpset() addsimps [inst "m" "m$*k" int_eq_iff_zle,
inst "m" "m" int_eq_iff_zle]) 1);
by (auto_tac (claset(),
@@ -348,8 +348,8 @@
qed "unique_quotient_lemma_neg";
-Goal "[| quorem (<a,b>, <q,r>); quorem (<a,b>, <q',r'>); b: int; b ~= #0; \
-\ q: int; q' : int |] ==> q = q'";
+Goal "[| quorem (<a,b>, <q,r>); quorem (<a,b>, <q',r'>); b \\<in> int; b ~= #0; \
+\ q \\<in> int; q' \\<in> int |] ==> q = q'";
by (asm_full_simp_tac
(simpset() addsimps split_ifs@
[quorem_def, neq_iff_zless]) 1);
@@ -362,9 +362,9 @@
sym]) 1));
qed "unique_quotient";
-Goal "[| quorem (<a,b>, <q,r>); quorem (<a,b>, <q',r'>); b: int; b ~= #0; \
-\ q: int; q' : int; \
-\ r: int; r' : int |] ==> r = r'";
+Goal "[| quorem (<a,b>, <q,r>); quorem (<a,b>, <q',r'>); b \\<in> int; b ~= #0; \
+\ q \\<in> int; q' \\<in> int; \
+\ r \\<in> int; r' \\<in> int |] ==> r = r'";
by (subgoal_tac "q = q'" 1);
by (blast_tac (claset() addIs [unique_quotient]) 2);
by (asm_full_simp_tac (simpset() addsimps [quorem_def]) 1);
@@ -374,7 +374,6 @@
(*** Correctness of posDivAlg, the division algorithm for a>=0 and b>0 ***)
-
Goal "adjust(a, b, <q,r>) = (let diff = r$-b in \
\ if #0 $<= diff then <#2$*q $+ #1,diff> \
\ else <#2$*q,r>)";
@@ -390,21 +389,21 @@
zless_add1_iff_zle]@zcompare_rls) 1);
qed "posDivAlg_termination";
-val lemma = wf_measure RS (posDivAlg_def RS def_wfrec RS trans);
+val posDivAlg_unfold = wf_measure RS (posDivAlg_def RS def_wfrec);
-Goal "[| #0 $< b; a: int; b: int |] ==> \
+Goal "[| #0 $< b; a \\<in> int; b \\<in> int |] ==> \
\ posDivAlg(<a,b>) = \
\ (if a$<b then <#0,a> else adjust(a, b, posDivAlg (<a, #2$*b>)))";
-by (rtac lemma 1);
+by (rtac (posDivAlg_unfold RS trans) 1);
by (asm_simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym]) 1);
by (asm_simp_tac (simpset() addsimps [vimage_iff, posDivAlg_termination]) 1);
qed "posDivAlg_eqn";
val [prem] =
-Goal "[| !!a b. [| a: int; b: int; \
+Goal "[| !!a b. [| a \\<in> int; b \\<in> int; \
\ ~ (a $< b | b $<= #0) --> P(<a, #2 $* b>) |] \
\ ==> P(<a,b>) |] \
-\ ==> <u,v>: int*int --> P(<u,v>)";
+\ ==> <u,v> \\<in> int*int --> P(<u,v>)";
by (res_inst_tac [("a","<u,v>")] wf_induct 1);
by (res_inst_tac [("A","int*int"), ("f","%<a,b>.nat_of (a $- b $+ #1)")]
wf_measure 1);
@@ -418,8 +417,8 @@
val prems =
-Goal "[| u: int; v: int; \
-\ !!a b. [| a: int; b: int; ~ (a $< b | b $<= #0) --> P(a, #2 $* b) |] \
+Goal "[| u \\<in> int; v \\<in> int; \
+\ !!a b. [| a \\<in> int; b \\<in> int; ~ (a $< b | b $<= #0) --> P(a, #2 $* b) |] \
\ ==> P(a,b) |] \
\ ==> P(u,v)";
by (subgoal_tac "(%<x,y>. P(x,y))(<u,v>)" 1);
@@ -431,5 +430,1380 @@
by Auto_tac;
qed "posDivAlg_induct";
-(**TO BE COMPLETED**)
+(*FIXME: use intify in integ_of so that we always have integ_of w \\<in> int.
+ then this rewrite can work for ALL constants!!*)
+Goal "intify(m) = #0 <-> (m $<= #0 & #0 $<= m)";
+by (simp_tac (simpset() addsimps [int_eq_iff_zle]) 1);
+qed "intify_eq_0_iff_zle";
+
+
+
+(*** Products of zeroes ***)
+
+Goal "[| x \\<in> int; y \\<in> int |] \
+\ ==> (x $* y = #0) <-> (intify(x) = #0 | intify(y) = #0)";
+by (case_tac "x $< #0" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [not_zless_iff_zle, zle_def, neq_iff_zless]));
+by (REPEAT
+ (force_tac (claset() addDs [zmult_zless_mono1_neg, zmult_zless_mono1],
+ simpset()) 1));
+qed "zmult_eq_0_iff_lemma";
+
+Goal "(x $* y = #0) <-> (intify(x) = #0 | intify(y) = #0)";
+by (cut_inst_tac [("x","intify(x)"),("y","intify(y)")]
+ zmult_eq_0_iff_lemma 1);
+by Auto_tac;
+qed "zmult_eq_0_iff";
+AddIffs [zmult_eq_0_iff];
+
+
+(*** Some convenient biconditionals for products of signs ***)
+
+Goal "[| #0 $< i; #0 $< j |] ==> #0 $< i $* j";
+by (dtac zmult_zless_mono1 1);
+by Auto_tac;
+qed "zmult_pos";
+
+Goal "[| i $< #0; j $< #0 |] ==> #0 $< i $* j";
+by (dtac zmult_zless_mono1_neg 1);
+by Auto_tac;
+qed "zmult_neg";
+
+Goal "[| #0 $< i; j $< #0 |] ==> i $* j $< #0";
+by (dtac zmult_zless_mono1_neg 1);
+by Auto_tac;
+qed "zmult_pos_neg";
+
+(** Inequality reasoning **)
+
+Goal "[| x \\<in> int; y \\<in> int |] \
+\ ==> (#0 $< x $* y) <-> (#0 $< x & #0 $< y | x $< #0 & y $< #0)";
+by (auto_tac (claset(),
+ simpset() addsimps [zle_def, not_zless_iff_zle,
+ zmult_pos, zmult_neg]));
+by (ALLGOALS (rtac ccontr));
+by (auto_tac (claset(),
+ simpset() addsimps [zle_def, not_zless_iff_zle]));
+by (ALLGOALS (eres_inst_tac [("P","#0$< x$* y")] rev_mp));
+by (ALLGOALS (dtac zmult_pos_neg THEN' assume_tac));
+by (auto_tac (claset() addDs [zless_not_sym],
+ simpset() addsimps [zmult_commute]));
+val lemma = result();
+
+Goal "(#0 $< x $* y) <-> (#0 $< x & #0 $< y | x $< #0 & y $< #0)";
+by (cut_inst_tac [("x","intify(x)"),("y","intify(y)")] lemma 1);
+by Auto_tac;
+qed "int_0_less_mult_iff";
+
+Goal "[| x \\<in> int; y \\<in> int |] \
+\ ==> (#0 $<= x $* y) <-> (#0 $<= x & #0 $<= y | x $<= #0 & y $<= #0)";
+by (auto_tac (claset(),
+ simpset() addsimps [zle_def, not_zless_iff_zle,
+ int_0_less_mult_iff]));
+val lemma = result();
+
+Goal "(#0 $<= x $* y) <-> ((#0 $<= x & #0 $<= y) | (x $<= #0 & y $<= #0))";
+by (cut_inst_tac [("x","intify(x)"),("y","intify(y)")] lemma 1);
+by Auto_tac;
+qed "int_0_le_mult_iff";
+
+Goal "(x $* y $< #0) <-> (#0 $< x & y $< #0 | x $< #0 & #0 $< y)";
+by (auto_tac (claset(),
+ simpset() addsimps [int_0_le_mult_iff,
+ not_zle_iff_zless RS iff_sym]));
+by (auto_tac (claset() addDs [zless_not_sym],
+ simpset() addsimps [not_zle_iff_zless]));
+qed "zmult_less_0_iff";
+
+Goal "(x $* y $<= #0) <-> (#0 $<= x & y $<= #0 | x $<= #0 & #0 $<= y)";
+by (auto_tac (claset() addDs [zless_not_sym],
+ simpset() addsimps [int_0_less_mult_iff,
+ not_zless_iff_zle RS iff_sym]));
+qed "zmult_le_0_iff";
+
+
+(*Typechecking for posDivAlg*)
+Goal "[| a \\<in> int; b \\<in> int |] ==> posDivAlg(<a,b>) \\<in> int * int";
+by (res_inst_tac [("u", "a"), ("v", "b")] posDivAlg_induct 1);
+by (TRYALL assume_tac);
+by (case_tac "#0 $< ba" 1);
+by (asm_simp_tac (simpset() addsimps [posDivAlg_eqn,adjust_def,integ_of_type]
+ addsplits [split_if_asm]) 1);
+by (Clarify_tac 1);
+by (asm_full_simp_tac
+ (simpset() addsimps [int_0_less_mult_iff, not_zle_iff_zless]) 1);
+by (asm_full_simp_tac (simpset() addsimps [not_zless_iff_zle]) 1);
+by (stac posDivAlg_unfold 1);
+by (Asm_full_simp_tac 1);
+qed_spec_mp "posDivAlg_type";
+
+(*Correctness of posDivAlg: it computes quotients correctly*)
+Goal "[| a \\<in> int; b \\<in> int |] \
+\ ==> #0 $<= a --> #0 $< b --> quorem (<a,b>, posDivAlg(<a,b>))";
+by (res_inst_tac [("u", "a"), ("v", "b")] posDivAlg_induct 1);
+by Auto_tac;
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
+(*base case: a<b*)
+by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 3);
+by (asm_full_simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym]) 2);
+by (asm_full_simp_tac (simpset() addsimps [posDivAlg_eqn]) 1);
+(*main argument*)
+by (stac posDivAlg_eqn 1);
+by (ALLGOALS Asm_simp_tac);
+by (etac splitE 1);
+by (rtac posDivAlg_type 1);
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff])));
+by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
+(*now just linear arithmetic*)
+by (asm_full_simp_tac
+ (simpset() addsimps [not_zle_iff_zless, zdiff_zless_iff]) 1);
+qed_spec_mp "posDivAlg_correct";
+
+
+(*** Correctness of negDivAlg, the division algorithm for a<0 and b>0 ***)
+
+Goal "[| #0 $< b; \\<not> #0 $<= a $+ b |] \
+\ ==> nat_of($- a $- #2 $\\<times> b) < nat_of($- a $- b)";
+by (simp_tac (simpset() addsimps [zless_nat_conj]) 1);
+by (asm_full_simp_tac (simpset() addsimps zcompare_rls @
+ [not_zle_iff_zless, zless_zdiff_iff RS iff_sym, zless_zminus]) 1);
+qed "negDivAlg_termination";
+
+val negDivAlg_unfold = wf_measure RS (negDivAlg_def RS def_wfrec);
+
+Goal "[| #0 $< b; a \\<in> int; b \\<in> int |] ==> \
+\ negDivAlg(<a,b>) = \
+\ (if #0 $<= a$+b then <#-1,a$+b> \
+\ else adjust(a, b, negDivAlg (<a, #2$*b>)))";
+by (rtac (negDivAlg_unfold RS trans) 1);
+by (asm_simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym]) 1);
+by (asm_simp_tac (simpset() addsimps [not_zle_iff_zless, vimage_iff,
+ negDivAlg_termination]) 1);
+qed "negDivAlg_eqn";
+
+val [prem] =
+Goal "[| !!a b. [| a \\<in> int; b \\<in> int; \
+\ ~ (#0 $<= a $+ b | b $<= #0) --> P(<a, #2 $* b>) |] \
+\ ==> P(<a,b>) |] \
+\ ==> <u,v> \\<in> int*int --> P(<u,v>)";
+by (res_inst_tac [("a","<u,v>")] wf_induct 1);
+by (res_inst_tac [("A","int*int"), ("f","%<a,b>.nat_of ($- a $- b)")]
+ wf_measure 1);
+by (Clarify_tac 1);
+by (rtac prem 1);
+by (dres_inst_tac [("x","<xa, #2 $\\<times> y>")] spec 3);
+by Auto_tac;
+by (asm_full_simp_tac (simpset() addsimps [not_zle_iff_zless,
+ negDivAlg_termination]) 1);
+val lemma = result() RS mp;
+
+val prems =
+Goal "[| u \\<in> int; v \\<in> int; \
+\ !!a b. [| a \\<in> int; b \\<in> int; \
+\ ~ (#0 $<= a $+ b | b $<= #0) --> P(a, #2 $* b) |] \
+\ ==> P(a,b) |] \
+\ ==> P(u,v)";
+by (subgoal_tac "(%<x,y>. P(x,y))(<u,v>)" 1);
+by (Asm_full_simp_tac 1);
+by (rtac lemma 1);
+by (simp_tac (simpset() addsimps prems) 2);
+by (Full_simp_tac 1);
+by (resolve_tac prems 1);
+by Auto_tac;
+qed "negDivAlg_induct";
+
+
+(*Typechecking for negDivAlg*)
+Goal "[| a \\<in> int; b \\<in> int |] ==> negDivAlg(<a,b>) \\<in> int * int";
+by (res_inst_tac [("u", "a"), ("v", "b")] negDivAlg_induct 1);
+by (TRYALL assume_tac);
+by (case_tac "#0 $< ba" 1);
+by (asm_simp_tac (simpset() addsimps [negDivAlg_eqn,adjust_def,integ_of_type]
+ addsplits [split_if_asm]) 1);
+by (Clarify_tac 1);
+by (asm_full_simp_tac
+ (simpset() addsimps [int_0_less_mult_iff, not_zle_iff_zless]) 1);
+by (asm_full_simp_tac (simpset() addsimps [not_zless_iff_zle]) 1);
+by (stac negDivAlg_unfold 1);
+by (Asm_full_simp_tac 1);
+qed "negDivAlg_type";
+
+
+(*Correctness of negDivAlg: it computes quotients correctly
+ It doesn't work if a=0 because the 0/b=0 rather than -1*)
+Goal "[| a \\<in> int; b \\<in> int |] \
+\ ==> a $< #0 --> #0 $< b --> quorem (<a,b>, negDivAlg(<a,b>))";
+by (res_inst_tac [("u", "a"), ("v", "b")] negDivAlg_induct 1);
+by Auto_tac;
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [quorem_def])));
+(*base case: 0$<=a$+b*)
+by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 3);
+by (asm_full_simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym]) 2);
+by (asm_full_simp_tac (simpset() addsimps [negDivAlg_eqn]) 1);
+(*main argument*)
+by (stac negDivAlg_eqn 1);
+by (ALLGOALS Asm_simp_tac);
+by (etac splitE 1);
+by (rtac negDivAlg_type 1);
+by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff])));
+by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
+(*now just linear arithmetic*)
+by (asm_full_simp_tac
+ (simpset() addsimps [not_zle_iff_zless, zdiff_zless_iff]) 1);
+qed_spec_mp "negDivAlg_correct";
+
+
+(*** Existence shown by proving the division algorithm to be correct ***)
+
+(*the case a=0*)
+Goal "[|b \\<noteq> #0; b \\<in> int|] ==> quorem (<#0,b>, <#0,#0>)";
+by (rotate_tac ~1 1);
+by (auto_tac (claset(),
+ simpset() addsimps [quorem_def, neq_iff_zless]));
+qed "quorem_0";
+
+Goal "posDivAlg(<a,#0>) = <#0,a>";
+by (stac posDivAlg_unfold 1);
+by (Simp_tac 1);
+qed "posDivAlg_zero_divisor";
+
+Goal "posDivAlg (<#0,b>) = <#0,#0>";
+by (stac posDivAlg_unfold 1);
+by (simp_tac (simpset() addsimps [not_zle_iff_zless]) 1);
+qed "posDivAlg_0";
+Addsimps [posDivAlg_0];
+
+Goal "negDivAlg (<#-1,b>) = <#-1, b$-#1>";
+by (stac negDivAlg_unfold 1);
+by Auto_tac;
+(*ALL the rest is linear arithmetic: to notice the contradiction*)
+by (asm_full_simp_tac (simpset() addsimps [not_zle_iff_zless]) 1);
+by (dtac (zminus_zless_zminus RS iffD2) 1);
+by (asm_full_simp_tac (simpset() addsimps [zadd_commute, zless_add1_iff_zle,
+ zle_zminus]) 1);
+by (asm_full_simp_tac (simpset() addsimps [not_zle_iff_zless RS iff_sym]) 1);
+qed "negDivAlg_minus1";
+Addsimps [negDivAlg_minus1];
+
+Goalw [negateSnd_def] "negateSnd (<q,r>) = <q, $-r>";
+by Auto_tac;
+qed "negateSnd_eq";
+Addsimps [negateSnd_eq];
+
+Goalw [negateSnd_def] "qr \\<in> int * int ==> negateSnd (qr) \\<in> int * int";
+by Auto_tac;
+qed "negateSnd_type";
+
+Goal "[|quorem (<$-a,$-b>, qr); a \\<in> int; b \\<in> int; qr \\<in> int * int|] \
+\ ==> quorem (<a,b>, negateSnd(qr))";
+by (Clarify_tac 1);
+by (auto_tac (claset() addEs [zless_asym],
+ simpset() addsimps [quorem_def, zless_zminus]));
+(*linear arithmetic from here on*)
+by (ALLGOALS
+ (asm_full_simp_tac
+ (simpset() addsimps [inst "x" "a" zminus_equation, zminus_zless])));
+by (ALLGOALS (cut_inst_tac [("z","b"),("w","#0")] zless_linear));
+by Auto_tac;
+by (ALLGOALS (blast_tac (claset() addDs [zle_zless_trans])));
+qed "quorem_neg";
+
+Goal "[|b \\<noteq> #0; a \\<in> int; b \\<in> int|] ==> quorem (<a,b>, divAlg(<a,b>))";
+by (rotate_tac 1 1);
+by (auto_tac (claset(),
+ simpset() addsimps [quorem_0, divAlg_def]));
+by (REPEAT_FIRST (ares_tac [quorem_neg, posDivAlg_correct, negDivAlg_correct,
+ posDivAlg_type, negDivAlg_type]));
+by (auto_tac (claset(),
+ simpset() addsimps [quorem_def, neq_iff_zless]));
+(*linear arithmetic from here on*)
+by (auto_tac (claset(), simpset() addsimps [zle_def]));
+qed "divAlg_correct";
+
+Goal "[|a \\<in> int; b \\<in> int|] ==> divAlg(<a,b>) \\<in> int * int";
+by (auto_tac (claset(), simpset() addsimps [divAlg_def]));
+by (auto_tac (claset(),
+ simpset() addsimps [posDivAlg_type, negDivAlg_type, negateSnd_type]));
+qed "divAlg_type";
+
+
+(** intify cancellation **)
+
+Goal "intify(x) zdiv y = x zdiv y";
+by (simp_tac (simpset() addsimps [zdiv_def]) 1);
+qed "zdiv_intify1";
+
+Goal "x zdiv intify(y) = x zdiv y";
+by (simp_tac (simpset() addsimps [zdiv_def]) 1);
+qed "zdiv_intify2";
+Addsimps [zdiv_intify1, zdiv_intify2];
+
+Goalw [zdiv_def] "z zdiv w \\<in> int";
+by (blast_tac (claset() addIs [fst_type, divAlg_type]) 1);
+qed "zdiv_type";
+AddIffs [zdiv_type]; AddTCs [zdiv_type];
+
+Goal "intify(x) zmod y = x zmod y";
+by (simp_tac (simpset() addsimps [zmod_def]) 1);
+qed "zmod_intify1";
+
+Goal "x zmod intify(y) = x zmod y";
+by (simp_tac (simpset() addsimps [zmod_def]) 1);
+qed "zmod_intify2";
+Addsimps [zmod_intify1, zmod_intify2];
+
+Goalw [zmod_def] "z zmod w \\<in> int";
+by (rtac snd_type 1);
+by (blast_tac (claset() addIs [divAlg_type]) 1);
+qed "zmod_type";
+AddIffs [zmod_type]; AddTCs [zmod_type];
+
+
+(** Arbitrary definitions for division by zero. Useful to simplify
+ certain equations **)
+
+Goal "a zdiv #0 = #0";
+by (simp_tac
+ (simpset() addsimps [zdiv_def, divAlg_def, posDivAlg_zero_divisor]) 1);
+qed "DIVISION_BY_ZERO_ZDIV"; (*NOT for adding to default simpset*)
+
+Goal "a zmod #0 = intify(a)";
+by (simp_tac
+ (simpset() addsimps [zmod_def, divAlg_def, posDivAlg_zero_divisor]) 1);
+qed "DIVISION_BY_ZERO_ZMOD"; (*NOT for adding to default simpset*)
+
+fun zdiv_undefined_case_tac s i =
+ case_tac s i THEN
+ asm_simp_tac (simpset() addsimps [DIVISION_BY_ZERO_ZDIV,
+ DIVISION_BY_ZERO_ZMOD]) i;
+
+
+(** Basic laws about division and remainder **)
+
+Goal "[| a \\<in> int; b \\<in> int |] ==> a = b $* (a zdiv b) $+ (a zmod b)";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
+by (auto_tac (claset(),
+ simpset() addsimps [quorem_def, zdiv_def, zmod_def, split_def]));
+qed "raw_zmod_zdiv_equality";
+
+Goal "intify(a) = b $* (a zdiv b) $+ (a zmod b)";
+by (rtac trans 1);
+by (res_inst_tac [("b","intify(b)")] raw_zmod_zdiv_equality 1);
+by Auto_tac;
+qed "zmod_zdiv_equality";
+
+Goal "#0 $< b ==> #0 $<= a zmod b & a zmod b $< b";
+by (cut_inst_tac [("a","intify(a)"),("b","intify(b)")] divAlg_correct 1);
+by (auto_tac (claset(),
+ simpset() addsimps [intify_eq_0_iff_zle, quorem_def, zmod_def,
+ split_def]));
+by (ALLGOALS (blast_tac (claset() addDs [zle_zless_trans])));
+bind_thm ("pos_mod_sign", result() RS conjunct1);
+bind_thm ("pos_mod_bound", result() RS conjunct2);
+
+Goal "b $< #0 ==> a zmod b $<= #0 & b $< a zmod b";
+by (cut_inst_tac [("a","intify(a)"),("b","intify(b)")] divAlg_correct 1);
+by (auto_tac (claset(),
+ simpset() addsimps [intify_eq_0_iff_zle, quorem_def, zmod_def,
+ split_def]));
+by (blast_tac (claset() addDs [zle_zless_trans]) 1);
+by (ALLGOALS (blast_tac (claset() addDs [zless_trans])));
+bind_thm ("neg_mod_sign", result() RS conjunct1);
+bind_thm ("neg_mod_bound", result() RS conjunct2);
+
+
+(** proving general properties of zdiv and zmod **)
+
+Goal "[|b \\<noteq> #0; a \\<in> int; b \\<in> int |] \
+\ ==> quorem (<a,b>, <a zdiv b, a zmod b>)";
+by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
+by (rotate_tac 1 1);
+by (auto_tac
+ (claset(),
+ simpset() addsimps [quorem_def, neq_iff_zless,
+ pos_mod_sign,pos_mod_bound,
+ neg_mod_sign, neg_mod_bound]));
+qed "quorem_div_mod";
+
+(*Surely quorem(<a,b>,<q,r>) implies a \\<in> int, but it doesn't matter*)
+Goal "[| quorem(<a,b>,<q,r>); b \\<noteq> #0; a \\<in> int; b \\<in> int; q \\<in> int |] \
+\ ==> a zdiv b = q";
+by (blast_tac (claset() addIs [quorem_div_mod RS unique_quotient]) 1);
+qed "quorem_div";
+
+Goal "[| quorem(<a,b>,<q,r>); b \\<noteq> #0; a \\<in> int; b \\<in> int; q \\<in> int; r \\<in> int |] ==> a zmod b = r";
+by (blast_tac (claset() addIs [quorem_div_mod RS unique_remainder]) 1);
+qed "quorem_mod";
+
+Goal "[| a \\<in> int; b \\<in> int; #0 $<= a; a $< b |] ==> a zdiv b = #0";
+by (rtac quorem_div 1);
+by (auto_tac (claset(), simpset() addsimps [quorem_def]));
+(*linear arithmetic*)
+by (ALLGOALS (blast_tac (claset() addDs [zle_zless_trans])));
+qed "zdiv_pos_pos_trivial_raw";
+
+Goal "[| #0 $<= a; a $< b |] ==> a zdiv b = #0";
+by (cut_inst_tac [("a", "intify(a)"), ("b", "intify(b)")]
+ zdiv_pos_pos_trivial_raw 1);
+by Auto_tac;
+qed "zdiv_pos_pos_trivial";
+
+Goal "[| a \\<in> int; b \\<in> int; a $<= #0; b $< a |] ==> a zdiv b = #0";
+by (res_inst_tac [("r","a")] quorem_div 1);
+by (auto_tac (claset(), simpset() addsimps [quorem_def]));
+(*linear arithmetic*)
+by (ALLGOALS (blast_tac (claset() addDs [zle_zless_trans, zless_trans])));
+qed "zdiv_neg_neg_trivial_raw";
+
+Goal "[| a $<= #0; b $< a |] ==> a zdiv b = #0";
+by (cut_inst_tac [("a", "intify(a)"), ("b", "intify(b)")]
+ zdiv_neg_neg_trivial_raw 1);
+by Auto_tac;
+qed "zdiv_neg_neg_trivial";
+
+Goal "[| a$+b $<= #0; #0 $< a; #0 $< b |] ==> False";
+by (dres_inst_tac [("z'","#0"), ("z","b")] zadd_zless_mono 1);
+by (auto_tac (claset(), simpset() addsimps [zle_def]));
+by (blast_tac (claset() addDs [zless_trans]) 1);
+qed "zadd_le_0_lemma";
+
+Goal "[| a \\<in> int; b \\<in> int; #0 $< a; a$+b $<= #0 |] ==> a zdiv b = #-1";
+by (res_inst_tac [("r","a $+ b ")] quorem_div 1);
+by (auto_tac (claset(), simpset() addsimps [quorem_def]));
+(*linear arithmetic*)
+by (ALLGOALS (blast_tac (claset() addDs [zadd_le_0_lemma, zle_zless_trans])));
+qed "zdiv_pos_neg_trivial_raw";
+
+Goal "[| #0 $< a; a$+b $<= #0 |] ==> a zdiv b = #-1";
+by (cut_inst_tac [("a", "intify(a)"), ("b", "intify(b)")]
+ zdiv_pos_neg_trivial_raw 1);
+by Auto_tac;
+qed "zdiv_pos_neg_trivial";
+
+(*There is no zdiv_neg_pos_trivial because #0 zdiv b = #0 would supersede it*)
+
+
+Goal "[| a \\<in> int; b \\<in> int; #0 $<= a; a $< b |] ==> a zmod b = a";
+by (res_inst_tac [("q","#0")] quorem_mod 1);
+by (auto_tac (claset(), simpset() addsimps [quorem_def]));
+(*linear arithmetic*)
+by (ALLGOALS (blast_tac (claset() addDs [zle_zless_trans])));
+qed "zmod_pos_pos_trivial_raw";
+
+Goal "[| #0 $<= a; a $< b |] ==> a zmod b = intify(a)";
+by (cut_inst_tac [("a", "intify(a)"), ("b", "intify(b)")]
+ zmod_pos_pos_trivial_raw 1);
+by Auto_tac;
+qed "zmod_pos_pos_trivial";
+
+Goal "[| a \\<in> int; b \\<in> int; a $<= #0; b $< a |] ==> a zmod b = a";
+by (res_inst_tac [("q","#0")] quorem_mod 1);
+by (auto_tac (claset(), simpset() addsimps [quorem_def]));
+(*linear arithmetic*)
+by (ALLGOALS (blast_tac (claset() addDs [zle_zless_trans, zless_trans])));
+qed "zmod_neg_neg_trivial_raw";
+
+Goal "[| a $<= #0; b $< a |] ==> a zmod b = intify(a)";
+by (cut_inst_tac [("a", "intify(a)"), ("b", "intify(b)")]
+ zmod_neg_neg_trivial_raw 1);
+by Auto_tac;
+qed "zmod_neg_neg_trivial";
+
+Goal "[| a \\<in> int; b \\<in> int; #0 $< a; a$+b $<= #0 |] ==> a zmod b = a$+b";
+by (res_inst_tac [("q","#-1")] quorem_mod 1);
+by (auto_tac (claset(), simpset() addsimps [quorem_def]));
+(*linear arithmetic*)
+by (ALLGOALS (blast_tac (claset() addDs [zadd_le_0_lemma, zle_zless_trans])));
+qed "zmod_pos_neg_trivial_raw";
+
+Goal "[| #0 $< a; a$+b $<= #0 |] ==> a zmod b = a$+b";
+by (cut_inst_tac [("a", "intify(a)"), ("b", "intify(b)")]
+ zmod_pos_neg_trivial_raw 1);
+by Auto_tac;
+qed "zmod_pos_neg_trivial";
+
+(*There is no zmod_neg_pos_trivial...*)
+
+
+(*Simpler laws such as -a zdiv b = -(a zdiv b) FAIL*)
+
+Goal "[|a \\<in> int; b \\<in> int|] ==> ($-a) zdiv ($-b) = a zdiv b";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg))
+ RS quorem_div) 1);
+by Auto_tac;
+qed "zdiv_zminus_zminus_raw";
+
+Goal "($-a) zdiv ($-b) = a zdiv b";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)")]
+ zdiv_zminus_zminus_raw 1);
+by Auto_tac;
+qed "zdiv_zminus_zminus";
+Addsimps [zdiv_zminus_zminus];
+
+(*Simpler laws such as -a zmod b = -(a zmod b) FAIL*)
+Goal "[|a \\<in> int; b \\<in> int|] ==> ($-a) zmod ($-b) = $- (a zmod b)";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (stac ((simplify(simpset()) (quorem_div_mod RS quorem_neg))
+ RS quorem_mod) 1);
+by Auto_tac;
+qed "zmod_zminus_zminus_raw";
+
+Goal "($-a) zmod ($-b) = $- (a zmod b)";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)")]
+ zmod_zminus_zminus_raw 1);
+by Auto_tac;
+qed "zmod_zminus_zminus";
+Addsimps [zmod_zminus_zminus];
+
+
+(*** division of a number by itself ***)
+
+Goal "[| #0 $< a; a = r $+ a$*q; r $< a |] ==> #1 $<= q";
+by (subgoal_tac "#0 $< a$*q" 1);
+by (cut_inst_tac [("w","#0"),("z","q")] add1_zle_iff 1);
+by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 1);
+by (blast_tac (claset() addDs [zless_trans]) 1);
+(*linear arithmetic...*)
+by (dres_inst_tac [("t","%x. x $- r")] subst_context 1);
+by (dtac sym 1);
+by (asm_full_simp_tac (simpset() addsimps zcompare_rls) 1);
+val lemma1 = result();
+
+Goal "[| #0 $< a; a = r $+ a$*q; #0 $<= r |] ==> q $<= #1";
+by (subgoal_tac "#0 $<= a$*(#1$-q)" 1);
+by (asm_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 2);
+by (dres_inst_tac [("t","%x. x $- a $* q")] subst_context 2);
+by (asm_full_simp_tac (simpset() addsimps zcompare_rls) 2);
+by (asm_full_simp_tac (simpset() addsimps int_0_le_mult_iff::zcompare_rls) 1);
+by (blast_tac (claset() addDs [zle_zless_trans]) 1);
+val lemma2 = result();
+
+Goal "[| quorem(<a,a>,<q,r>); a \\<in> int; q \\<in> int; a \\<noteq> #0|] ==> q = #1";
+by (asm_full_simp_tac
+ (simpset() addsimps split_ifs@[quorem_def, neq_iff_zless]) 1);
+by (rtac zle_anti_sym 1);
+by Safe_tac;
+by Auto_tac;
+by (blast_tac (claset() addDs [zless_trans]) 4);
+by (blast_tac (claset() addDs [zless_trans]) 1);
+by (res_inst_tac [("a", "$-a"),("r", "$-r")] lemma1 3);
+by (res_inst_tac [("a", "$-a"),("r", "$-r")] lemma2 1);
+by (rtac (zminus_equation RS iffD1) 6);
+by (rtac (zminus_equation RS iffD1) 2);
+by (REPEAT (force_tac (claset() addIs [lemma1,lemma2],
+ simpset() addsimps [zadd_commute, zmult_zminus]) 1));
+qed "self_quotient";
+
+Goal "[|quorem(<a,a>,<q,r>); a \\<in> int; q \\<in> int; r \\<in> int; a \\<noteq> #0|] ==> r = #0";
+by (ftac self_quotient 1);
+by (auto_tac (claset(), simpset() addsimps [quorem_def]));
+qed "self_remainder";
+
+Goal "[|a \\<noteq> #0; a \\<in> int|] ==> a zdiv a = #1";
+by (blast_tac (claset() addIs [quorem_div_mod RS self_quotient]) 1);
+qed "zdiv_self_raw";
+
+Goal "intify(a) \\<noteq> #0 ==> a zdiv a = #1";
+by (dtac zdiv_self_raw 1);
+by Auto_tac;
+qed "zdiv_self";
+Addsimps [zdiv_self];
+
+(*Here we have 0 zmod 0 = 0, also assumed by Knuth (who puts m zmod 0 = 0) *)
+Goal "a \\<in> int ==> a zmod a = #0";
+by (zdiv_undefined_case_tac "a = #0" 1);
+by (blast_tac (claset() addIs [quorem_div_mod RS self_remainder]) 1);
+qed "zmod_self_raw";
+
+Goal "a zmod a = #0";
+by (cut_inst_tac [("a","intify(a)")] zmod_self_raw 1);
+by Auto_tac;
+qed "zmod_self";
+Addsimps [zmod_self];
+
+
+(*** Computation of division and remainder ***)
+
+Goal "#0 zdiv b = #0";
+by (simp_tac (simpset() addsimps [zdiv_def, divAlg_def]) 1);
+qed "zdiv_zero";
+
+Goal "#0 $< b ==> #-1 zdiv b = #-1";
+by (asm_simp_tac (simpset() addsimps [zdiv_def, divAlg_def]) 1);
+qed "zdiv_eq_minus1";
+
+Goal "#0 zmod b = #0";
+by (simp_tac (simpset() addsimps [zmod_def, divAlg_def]) 1);
+qed "zmod_zero";
+
+Addsimps [zdiv_zero, zmod_zero];
+
+Goal "#0 $< b ==> #-1 zdiv b = #-1";
+by (asm_simp_tac (simpset() addsimps [zdiv_def, divAlg_def]) 1);
+qed "zdiv_minus1";
+
+Goal "#0 $< b ==> #-1 zmod b = b $- #1";
+by (asm_simp_tac (simpset() addsimps [zmod_def, divAlg_def]) 1);
+qed "zmod_minus1";
+
+(** a positive, b positive **)
+
+Goal "[| #0 $< a; #0 $<= b |] \
+\ ==> a zdiv b = fst (posDivAlg(<intify(a), intify(b)>))";
+by (asm_simp_tac (simpset() addsimps [zdiv_def, divAlg_def]) 1);
+by (auto_tac (claset(), simpset() addsimps [zle_def]));
+qed "zdiv_pos_pos";
+
+Goal "[| #0 $< a; #0 $<= b |] \
+\ ==> a zmod b = snd (posDivAlg(<intify(a), intify(b)>))";
+by (asm_simp_tac (simpset() addsimps [zmod_def, divAlg_def]) 1);
+by (auto_tac (claset(), simpset() addsimps [zle_def]));
+qed "zmod_pos_pos";
+
+(** a negative, b positive **)
+
+Goal "[| a $< #0; #0 $< b |] \
+\ ==> a zdiv b = fst (negDivAlg(<intify(a), intify(b)>))";
+by (asm_simp_tac (simpset() addsimps [zdiv_def, divAlg_def]) 1);
+by (blast_tac (claset() addDs [zle_zless_trans]) 1);
+qed "zdiv_neg_pos";
+
+Goal "[| a $< #0; #0 $< b |] \
+\ ==> a zmod b = snd (negDivAlg(<intify(a), intify(b)>))";
+by (asm_simp_tac (simpset() addsimps [zmod_def, divAlg_def]) 1);
+by (blast_tac (claset() addDs [zle_zless_trans]) 1);
+qed "zmod_neg_pos";
+
+(** a positive, b negative **)
+
+Goal "[| #0 $< a; b $< #0 |] \
+\ ==> a zdiv b = fst (negateSnd(negDivAlg (<$-a, $-b>)))";
+by (asm_simp_tac
+ (simpset() addsimps [zdiv_def, divAlg_def, intify_eq_0_iff_zle]) 1);
+by Auto_tac;
+by (REPEAT (blast_tac (claset() addDs [zle_zless_trans]) 1));
+by (blast_tac (claset() addDs [zless_trans]) 1);
+by (blast_tac (claset() addIs [zless_imp_zle]) 1);
+qed "zdiv_pos_neg";
+
+Goal "[| #0 $< a; b $< #0 |] \
+\ ==> a zmod b = snd (negateSnd(negDivAlg (<$-a, $-b>)))";
+by (asm_simp_tac
+ (simpset() addsimps [zmod_def, divAlg_def, intify_eq_0_iff_zle]) 1);
+by Auto_tac;
+by (REPEAT (blast_tac (claset() addDs [zle_zless_trans]) 1));
+by (blast_tac (claset() addDs [zless_trans]) 1);
+by (blast_tac (claset() addIs [zless_imp_zle]) 1);
+qed "zmod_pos_neg";
+
+(** a negative, b negative **)
+
+Goal "[| a $< #0; b $<= #0 |] \
+\ ==> a zdiv b = fst (negateSnd(posDivAlg(<$-a, $-b>)))";
+by (asm_simp_tac (simpset() addsimps [zdiv_def, divAlg_def]) 1);
+by Auto_tac;
+by (REPEAT (blast_tac (claset() addSDs [zle_zless_trans]) 1));
+qed "zdiv_neg_neg";
+
+Goal "[| a $< #0; b $<= #0 |] \
+\ ==> a zmod b = snd (negateSnd(posDivAlg(<$-a, $-b>)))";
+by (asm_simp_tac (simpset() addsimps [zmod_def, divAlg_def]) 1);
+by Auto_tac;
+by (REPEAT (blast_tac (claset() addSDs [zle_zless_trans]) 1));
+qed "zmod_neg_neg";
+Addsimps (map (read_instantiate_sg (sign_of (the_context ()))
+ [("a", "integ_of (?v)"), ("b", "integ_of (?w)")])
+ [zdiv_pos_pos, zdiv_neg_pos, zdiv_pos_neg, zdiv_neg_neg,
+ zmod_pos_pos, zmod_neg_pos, zmod_pos_neg, zmod_neg_neg,
+ posDivAlg_eqn, negDivAlg_eqn]);
+
+
+(** Special-case simplification **)
+
+Goal "a zmod #1 = #0";
+by (cut_inst_tac [("a","a"),("b","#1")] pos_mod_sign 1);
+by (cut_inst_tac [("a","a"),("b","#1")] pos_mod_bound 2);
+by Auto_tac;
+(*arithmetic*)
+by (dtac (add1_zle_iff RS iffD2) 1);
+by (rtac zle_anti_sym 1);
+by Auto_tac;
+qed "zmod_1";
+Addsimps [zmod_1];
+
+Goal "a zdiv #1 = intify(a)";
+by (cut_inst_tac [("a","a"),("b","#1")] zmod_zdiv_equality 1);
+by Auto_tac;
+qed "zdiv_1";
+Addsimps [zdiv_1];
+
+Goal "a zmod #-1 = #0";
+by (cut_inst_tac [("a","a"),("b","#-1")] neg_mod_sign 1);
+by (cut_inst_tac [("a","a"),("b","#-1")] neg_mod_bound 2);
+by Auto_tac;
+(*arithmetic*)
+by (dtac (add1_zle_iff RS iffD2) 1);
+by (rtac zle_anti_sym 1);
+by Auto_tac;
+qed "zmod_minus1_right";
+Addsimps [zmod_minus1_right];
+
+Goal "a \\<in> int ==> a zdiv #-1 = $-a";
+by (cut_inst_tac [("a","a"),("b","#-1")] zmod_zdiv_equality 1);
+by Auto_tac;
+by (rtac (equation_zminus RS iffD2) 1);
+by Auto_tac;
+qed "zdiv_minus1_right_raw";
+
+Goal "a zdiv #-1 = $-a";
+by (cut_inst_tac [("a","intify(a)")] zdiv_minus1_right_raw 1);
+by Auto_tac;
+qed "zdiv_minus1_right";
+Addsimps [zdiv_minus1_right];
+
+
+(*** Monotonicity in the first argument (divisor) ***)
+
+Goal "[| a $<= a'; #0 $< b |] ==> a zdiv b $<= a' zdiv b";
+by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
+by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
+by (rtac unique_quotient_lemma 1);
+by (etac subst 1);
+by (etac subst 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
+qed "zdiv_mono1";
+
+Goal "[| a $<= a'; b $< #0 |] ==> a' zdiv b $<= a zdiv b";
+by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
+by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
+by (rtac unique_quotient_lemma_neg 1);
+by (etac subst 1);
+by (etac subst 1);
+by (ALLGOALS (asm_simp_tac (simpset() addsimps [neg_mod_sign,neg_mod_bound])));
+qed "zdiv_mono1_neg";
+
+
+(*** Monotonicity in the second argument (dividend) ***)
+
+Goal "[| b$*q $+ r = b'$*q' $+ r'; #0 $<= b'$*q' $+ r'; \
+\ r' $< b'; #0 $<= r; #0 $< b'; b' $<= b |] \
+\ ==> q $<= q'";
+by (subgoal_tac "#0 $<= q'" 1);
+ by (subgoal_tac "#0 $< b'$*(q' $+ #1)" 2);
+ by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 3);
+ by (etac zle_zless_trans 3);
+ by (etac zadd_zless_mono2 3);
+ by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 2);
+ by (blast_tac (claset() addDs [zless_trans]
+ addIs [zless_add1_iff_zle RS iffD1]) 2);
+by (subgoal_tac "b$*q $< b$*(q' $+ #1)" 1);
+ by (asm_full_simp_tac (simpset() addsimps [zmult_zless_cancel1]) 1);
+ by (force_tac (claset() addDs [zless_add1_iff_zle RS iffD1,
+ zless_trans, zless_zle_trans],
+ simpset()) 1);
+by (subgoal_tac "b$*q = r' $- r $+ b'$*q'" 1);
+ by (asm_simp_tac (simpset() addsimps zcompare_rls) 2);
+by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
+by (stac zadd_commute 1 THEN rtac zadd_zless_mono 1);
+ by (blast_tac (claset() addIs [zmult_zle_mono1]) 2);
+by (subgoal_tac "r' $+ #0 $< b $+ r" 1);
+ by (asm_full_simp_tac (simpset() addsimps zcompare_rls) 1);
+by (rtac zadd_zless_mono 1);
+ by Auto_tac;
+by (blast_tac (claset() addDs [zless_zle_trans]) 1);
+qed "zdiv_mono2_lemma";
+
+Goal "[| #0 $<= a; #0 $< b'; b' $<= b; a \\<in> int |] \
+\ ==> a zdiv b $<= a zdiv b'";
+by (subgoal_tac "#0 $< b" 1);
+ by (blast_tac (claset() addDs [zless_zle_trans]) 2);
+by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
+by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
+by (rtac zdiv_mono2_lemma 1);
+by (etac subst 1);
+by (etac subst 1);
+by (ALLGOALS
+ (asm_full_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
+qed "zdiv_mono2_raw";
+
+Goal "[| #0 $<= a; #0 $< b'; b' $<= b |] \
+\ ==> a zdiv b $<= a zdiv b'";
+by (cut_inst_tac [("a","intify(a)")] zdiv_mono2_raw 1);
+by Auto_tac;
+qed "zdiv_mono2";
+
+Goal "[| b$*q $+ r = b'$*q' $+ r'; b'$*q' $+ r' $< #0; \
+\ r $< b; #0 $<= r'; #0 $< b'; b' $<= b |] \
+\ ==> q' $<= q";
+by (subgoal_tac "#0 $< b" 1);
+ by (blast_tac (claset() addDs [zless_zle_trans]) 2);
+by (subgoal_tac "q' $< #0" 1);
+ by (subgoal_tac "b'$*q' $< #0" 2);
+ by (force_tac (claset() addIs [zle_zless_trans], simpset()) 3);
+ by (asm_full_simp_tac (simpset() addsimps [zmult_less_0_iff]) 2);
+ by (blast_tac (claset() addDs [zless_trans]) 2);
+by (subgoal_tac "b$*q' $< b$*(q $+ #1)" 1);
+ by (asm_full_simp_tac (simpset() addsimps [zmult_zless_cancel1]) 1);
+ by (blast_tac (claset() addDs [zless_trans, zless_add1_iff_zle RS iffD1]) 1);
+by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
+by (subgoal_tac "b$*q' $<= b'$*q'" 1);
+ by (asm_simp_tac (simpset() addsimps [zmult_zle_cancel2]) 2);
+ by (blast_tac (claset() addDs [zless_trans]) 2);
+by (subgoal_tac "b'$*q' $+ r $< b $+ (b$*q $+ r)" 1);
+ by (etac ssubst 2);
+ by (Asm_simp_tac 2);
+ by (dres_inst_tac [("w'","r"),("z'","#0")] zadd_zless_mono 2);
+ by (assume_tac 2);
+ by (Asm_full_simp_tac 2);
+by (full_simp_tac (simpset() addsimps [zadd_commute]) 1);
+by (rtac zle_zless_trans 1);
+by (assume_tac 2);
+ by (asm_simp_tac (simpset() addsimps [zmult_zle_cancel2]) 1);
+by (blast_tac (claset() addDs [zless_trans]) 1);
+qed "zdiv_mono2_neg_lemma";
+
+Goal "[| a $< #0; #0 $< b'; b' $<= b; a \\<in> int |] \
+\ ==> a zdiv b' $<= a zdiv b";
+by (subgoal_tac "#0 $< b" 1);
+ by (blast_tac (claset() addDs [zless_zle_trans]) 2);
+by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
+by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
+by (rtac zdiv_mono2_neg_lemma 1);
+by (etac subst 1);
+by (etac subst 1);
+by (ALLGOALS
+ (asm_full_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
+qed "zdiv_mono2_neg_raw";
+
+Goal "[| a $< #0; #0 $< b'; b' $<= b |] \
+\ ==> a zdiv b' $<= a zdiv b";
+by (cut_inst_tac [("a","intify(a)")] zdiv_mono2_neg_raw 1);
+by Auto_tac;
+qed "zdiv_mono2_neg";
+
+
+
+(*** More algebraic laws for zdiv and zmod ***)
+
+(** proving (a*b) zdiv c = a $* (b zdiv c) $+ a * (b zmod c) **)
+
+Goal "[| quorem(<b,c>, <q,r>); c \\<in> int; c \\<noteq> #0 |] \
+\ ==> quorem (<a$*b, c>, <a$*q $+ (a$*r) zdiv c, (a$*r) zmod c>)";
+by (auto_tac
+ (claset(),
+ simpset() addsimps split_ifs@
+ [quorem_def, neq_iff_zless,
+ zadd_zmult_distrib2,
+ pos_mod_sign,pos_mod_bound,
+ neg_mod_sign, neg_mod_bound]));
+by (ALLGOALS (rtac raw_zmod_zdiv_equality));
+by Auto_tac;
+qed "zmult1_lemma";
+
+Goal "[|b \\<in> int; c \\<in> int|] \
+\ ==> (a$*b) zdiv c = a$*(b zdiv c) $+ a$*(b zmod c) zdiv c";
+by (zdiv_undefined_case_tac "c = #0" 1);
+by (rtac (quorem_div_mod RS zmult1_lemma RS quorem_div) 1);
+by Auto_tac;
+qed "zdiv_zmult1_eq_raw";
+
+Goal "(a$*b) zdiv c = a$*(b zdiv c) $+ a$*(b zmod c) zdiv c";
+by (cut_inst_tac [("b","intify(b)"), ("c","intify(c)")] zdiv_zmult1_eq_raw 1);
+by Auto_tac;
+qed "zdiv_zmult1_eq";
+
+Goal "[|b \\<in> int; c \\<in> int|] ==> (a$*b) zmod c = a$*(b zmod c) zmod c";
+by (zdiv_undefined_case_tac "c = #0" 1);
+by (rtac (quorem_div_mod RS zmult1_lemma RS quorem_mod) 1);
+by Auto_tac;
+qed "zmod_zmult1_eq_raw";
+
+Goal "(a$*b) zmod c = a$*(b zmod c) zmod c";
+by (cut_inst_tac [("b","intify(b)"), ("c","intify(c)")] zmod_zmult1_eq_raw 1);
+by Auto_tac;
+qed "zmod_zmult1_eq";
+
+Goal "(a$*b) zmod c = ((a zmod c) $* b) zmod c";
+by (rtac trans 1);
+by (res_inst_tac [("b", "(b $* a) zmod c")] trans 1);
+by (rtac zmod_zmult1_eq 2);
+by (ALLGOALS (simp_tac (simpset() addsimps [zmult_commute])));
+qed "zmod_zmult1_eq'";
+
+Goal "(a$*b) zmod c = ((a zmod c) $* (b zmod c)) zmod c";
+by (rtac (zmod_zmult1_eq' RS trans) 1);
+by (rtac zmod_zmult1_eq 1);
+qed "zmod_zmult_distrib";
+
+Goal "intify(b) \\<noteq> #0 ==> (a$*b) zdiv b = intify(a)";
+by (asm_simp_tac (simpset() addsimps [zdiv_zmult1_eq]) 1);
+qed "zdiv_zmult_self1";
+Addsimps [zdiv_zmult_self1];
+
+Goal "intify(b) \\<noteq> #0 ==> (b$*a) zdiv b = intify(a)";
+by (stac zmult_commute 1 THEN etac zdiv_zmult_self1 1);
+qed "zdiv_zmult_self2";
+Addsimps [zdiv_zmult_self2];
+
+Goal "(a$*b) zmod b = #0";
+by (simp_tac (simpset() addsimps [zmod_zmult1_eq]) 1);
+qed "zmod_zmult_self1";
+Addsimps [zmod_zmult_self1];
+
+Goal "(b$*a) zmod b = #0";
+by (simp_tac (simpset() addsimps [zmult_commute, zmod_zmult1_eq]) 1);
+qed "zmod_zmult_self2";
+Addsimps [zmod_zmult_self2];
+
+
+(** proving (a$+b) zdiv c =
+ a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c) **)
+
+Goal "[| quorem(<a,c>, <aq,ar>); quorem(<b,c>, <bq,br>); \
+\ c \\<in> int; c \\<noteq> #0 |] \
+\ ==> quorem (<a$+b, c>, <aq $+ bq $+ (ar$+br) zdiv c, (ar$+br) zmod c>)";
+by (auto_tac
+ (claset(),
+ simpset() addsimps split_ifs@
+ [quorem_def, neq_iff_zless,
+ zadd_zmult_distrib2,
+ pos_mod_sign,pos_mod_bound,
+ neg_mod_sign, neg_mod_bound]));
+by (ALLGOALS (rtac raw_zmod_zdiv_equality));
+by Auto_tac;
+val zadd1_lemma = result();
+
+(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
+Goal "[|a \\<in> int; b \\<in> int; c \\<in> int|] ==> \
+\ (a$+b) zdiv c = a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c)";
+by (zdiv_undefined_case_tac "c = #0" 1);
+by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
+ MRS zadd1_lemma RS quorem_div]) 1);
+qed "zdiv_zadd1_eq_raw";
+
+Goal "(a$+b) zdiv c = a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c)";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)"), ("c","intify(c)")]
+ zdiv_zadd1_eq_raw 1);
+by Auto_tac;
+qed "zdiv_zadd1_eq";
+
+Goal "[|a \\<in> int; b \\<in> int; c \\<in> int|] \
+\ ==> (a$+b) zmod c = (a zmod c $+ b zmod c) zmod c";
+by (zdiv_undefined_case_tac "c = #0" 1);
+by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
+ MRS zadd1_lemma RS quorem_mod]) 1);
+qed "zmod_zadd1_eq_raw";
+
+Goal "(a$+b) zmod c = (a zmod c $+ b zmod c) zmod c";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)"), ("c","intify(c)")]
+ zmod_zadd1_eq_raw 1);
+by Auto_tac;
+qed "zmod_zadd1_eq";
+
+Goal "[|a \\<in> int; b \\<in> int|] ==> (a zmod b) zdiv b = #0";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [neq_iff_zless,
+ pos_mod_sign, pos_mod_bound, zdiv_pos_pos_trivial,
+ neg_mod_sign, neg_mod_bound, zdiv_neg_neg_trivial]));
+qed "zmod_div_trivial_raw";
+
+Goal "(a zmod b) zdiv b = #0";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)")]
+ zmod_div_trivial_raw 1);
+by Auto_tac;
+qed "zmod_div_trivial";
+Addsimps [zmod_div_trivial];
+
+Goal "[|a \\<in> int; b \\<in> int|] ==> (a zmod b) zmod b = a zmod b";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (auto_tac (claset(),
+ simpset() addsimps [neq_iff_zless,
+ pos_mod_sign, pos_mod_bound, zmod_pos_pos_trivial,
+ neg_mod_sign, neg_mod_bound, zmod_neg_neg_trivial]));
+qed "zmod_mod_trivial_raw";
+
+Goal "(a zmod b) zmod b = a zmod b";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)")]
+ zmod_mod_trivial_raw 1);
+by Auto_tac;
+qed "zmod_mod_trivial";
+Addsimps [zmod_mod_trivial];
+
+Goal "(a$+b) zmod c = ((a zmod c) $+ b) zmod c";
+by (rtac (trans RS sym) 1);
+by (rtac zmod_zadd1_eq 1);
+by (Simp_tac 1);
+by (rtac (zmod_zadd1_eq RS sym) 1);
+qed "zmod_zadd_left_eq";
+
+Goal "(a$+b) zmod c = (a $+ (b zmod c)) zmod c";
+by (rtac (trans RS sym) 1);
+by (rtac zmod_zadd1_eq 1);
+by (Simp_tac 1);
+by (rtac (zmod_zadd1_eq RS sym) 1);
+qed "zmod_zadd_right_eq";
+
+
+Goal "intify(a) \\<noteq> #0 ==> (a$+b) zdiv a = b zdiv a $+ #1";
+by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
+qed "zdiv_zadd_self1";
+Addsimps [zdiv_zadd_self1];
+
+Goal "intify(a) \\<noteq> #0 ==> (b$+a) zdiv a = b zdiv a $+ #1";
+by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
+qed "zdiv_zadd_self2";
+Addsimps [zdiv_zadd_self2];
+
+Goal "(a$+b) zmod a = b zmod a";
+by (zdiv_undefined_case_tac "a = #0" 1);
+by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
+qed "zmod_zadd_self1";
+
+Goal "(b$+a) zmod a = b zmod a";
+by (zdiv_undefined_case_tac "a = #0" 1);
+by (asm_simp_tac (simpset() addsimps [zmod_zadd1_eq]) 1);
+qed "zmod_zadd_self2";
+Addsimps [zmod_zadd_self1, zmod_zadd_self2];
+
+
+(*** proving a zdiv (b*c) = (a zdiv b) zdiv c ***)
+
+(*The condition c>0 seems necessary. Consider that 7 zdiv ~6 = ~2 but
+ 7 zdiv 2 zdiv ~3 = 3 zdiv ~3 = ~1. The subcase (a zdiv b) zmod c = 0 seems
+ to cause particular problems.*)
+
+(** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)
+
+Goal "[| #0 $< c; b $< r; r $<= #0 |] ==> b$*c $< b$*(q zmod c) $+ r";
+by (subgoal_tac "b $* (c $- q zmod c) $< r $* #1" 1);
+by (asm_full_simp_tac
+ (simpset() addsimps [zdiff_zmult_distrib2, zadd_commute]@zcompare_rls) 1);
+by (rtac zle_zless_trans 1);
+by (etac zmult_zless_mono1 2);
+by (rtac zmult_zle_mono2_neg 1);
+by (auto_tac
+ (claset(),
+ simpset() addsimps zcompare_rls@
+ [zadd_commute, add1_zle_iff, pos_mod_bound]));
+by (blast_tac (claset() addIs [zless_imp_zle]
+ addDs [zless_zle_trans]) 1);
+val lemma1 = result();
+
+Goal "[| #0 $< c; b $< r; r $<= #0 |] ==> b $* (q zmod c) $+ r $<= #0";
+by (subgoal_tac "b $* (q zmod c) $<= #0" 1);
+by (asm_simp_tac (simpset() addsimps [zmult_le_0_iff, pos_mod_sign]) 2);
+by (blast_tac (claset() addIs [zless_imp_zle]
+ addDs [zless_zle_trans]) 2);
+(*arithmetic*)
+by (dtac zadd_zle_mono 1);
+by (assume_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [zadd_commute]) 1);
+val lemma2 = result();
+
+Goal "[| #0 $< c; #0 $<= r; r $< b |] ==> #0 $<= b $* (q zmod c) $+ r";
+by (subgoal_tac "#0 $<= b $* (q zmod c)" 1);
+by (asm_simp_tac (simpset() addsimps [int_0_le_mult_iff, pos_mod_sign]) 2);
+by (blast_tac (claset() addIs [zless_imp_zle]
+ addDs [zle_zless_trans]) 2);
+(*arithmetic*)
+by (dtac zadd_zle_mono 1);
+by (assume_tac 1);
+by (asm_full_simp_tac (simpset() addsimps [zadd_commute]) 1);
+val lemma3 = result();
+
+Goal "[| #0 $< c; #0 $<= r; r $< b |] ==> b $* (q zmod c) $+ r $< b $* c";
+by (subgoal_tac "r $* #1 $< b $* (c $- q zmod c)" 1);
+by (asm_full_simp_tac
+ (simpset() addsimps [zdiff_zmult_distrib2, zadd_commute]@zcompare_rls) 1);
+by (rtac zless_zle_trans 1);
+by (etac zmult_zless_mono1 1);
+by (rtac zmult_zle_mono2 2);
+by (auto_tac
+ (claset(),
+ simpset() addsimps zcompare_rls@
+ [zadd_commute, add1_zle_iff, pos_mod_bound]));
+by (blast_tac (claset() addIs [zless_imp_zle]
+ addDs [zle_zless_trans]) 1);
+val lemma4 = result();
+
+Goal "[| quorem (<a,b>, <q,r>); a \\<in> int; b \\<in> int; b \\<noteq> #0; #0 $< c |] \
+\ ==> quorem (<a,b$*c>, <q zdiv c, b$*(q zmod c) $+ r>)";
+by (auto_tac
+ (claset(),
+ simpset() addsimps zmult_ac@
+ [zmod_zdiv_equality RS sym, quorem_def, neq_iff_zless,
+ int_0_less_mult_iff,
+ zadd_zmult_distrib2 RS sym,
+ lemma1, lemma2, lemma3, lemma4]));
+by (ALLGOALS (blast_tac (claset() addDs [zless_trans])));
+val lemma = result();
+
+Goal "[|#0 $< c; a \\<in> int; b \\<in> int|] ==> a zdiv (b$*c) = (a zdiv b) zdiv c";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (rtac (quorem_div_mod RS lemma RS quorem_div) 1);
+by (auto_tac (claset(), simpset() addsimps [intify_eq_0_iff_zle]));
+by (blast_tac (claset() addDs [zle_zless_trans]) 1);
+qed "zdiv_zmult2_eq_raw";
+
+Goal "#0 $< c ==> a zdiv (b$*c) = (a zdiv b) zdiv c";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)")]
+ zdiv_zmult2_eq_raw 1);
+by Auto_tac;
+qed "zdiv_zmult2_eq";
+
+Goal "[|#0 $< c; a \\<in> int; b \\<in> int|] \
+\ ==> a zmod (b$*c) = b$*(a zdiv b zmod c) $+ a zmod b";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (rtac (quorem_div_mod RS lemma RS quorem_mod) 1);
+by (auto_tac (claset(), simpset() addsimps [intify_eq_0_iff_zle]));
+by (blast_tac (claset() addDs [zle_zless_trans]) 1);
+qed "zmod_zmult2_eq_raw";
+
+Goal "#0 $< c ==> a zmod (b$*c) = b$*(a zdiv b zmod c) $+ a zmod b";
+by (cut_inst_tac [("a","intify(a)"), ("b","intify(b)")]
+ zmod_zmult2_eq_raw 1);
+by Auto_tac;
+qed "zmod_zmult2_eq";
+
+(*** Cancellation of common factors in "zdiv" ***)
+
+Goal "[| #0 $< b; intify(c) \\<noteq> #0 |] ==> (c$*a) zdiv (c$*b) = a zdiv b";
+by (stac zdiv_zmult2_eq 1);
+by Auto_tac;
+val lemma1 = result();
+
+Goal "[| b $< #0; intify(c) \\<noteq> #0 |] ==> (c$*a) zdiv (c$*b) = a zdiv b";
+by (subgoal_tac "(c $* ($-a)) zdiv (c $* ($-b)) = ($-a) zdiv ($-b)" 1);
+by (rtac lemma1 2);
+by Auto_tac;
+val lemma2 = result();
+
+Goal "[|intify(c) \\<noteq> #0; b \\<in> int|] ==> (c$*a) zdiv (c$*b) = a zdiv b";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (auto_tac
+ (claset(),
+ simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless,
+ lemma1, lemma2]));
+qed "zdiv_zmult_zmult1_raw";
+
+Goal "intify(c) \\<noteq> #0 ==> (c$*a) zdiv (c$*b) = a zdiv b";
+by (cut_inst_tac [("b","intify(b)")] zdiv_zmult_zmult1_raw 1);
+by Auto_tac;
+qed "zdiv_zmult_zmult1";
+
+Goal "intify(c) \\<noteq> #0 ==> (a$*c) zdiv (b$*c) = a zdiv b";
+by (dtac zdiv_zmult_zmult1 1);
+by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
+qed "zdiv_zmult_zmult2";
+
+
+(*** Distribution of factors over "zmod" ***)
+
+Goal "[| #0 $< b; intify(c) \\<noteq> #0 |] \
+\ ==> (c$*a) zmod (c$*b) = c $* (a zmod b)";
+by (stac zmod_zmult2_eq 1);
+by Auto_tac;
+val lemma1 = result();
+
+Goal "[| b $< #0; intify(c) \\<noteq> #0 |] \
+\ ==> (c$*a) zmod (c$*b) = c $* (a zmod b)";
+by (subgoal_tac "(c $* ($-a)) zmod (c $* ($-b)) = c $* (($-a) zmod ($-b))" 1);
+by (rtac lemma1 2);
+by Auto_tac;
+val lemma2 = result();
+
+Goal "[|b \\<in> int; c \\<in> int|] ==> (c$*a) zmod (c$*b) = c $* (a zmod b)";
+by (zdiv_undefined_case_tac "b = #0" 1);
+by (zdiv_undefined_case_tac "c = #0" 1);
+by (auto_tac
+ (claset(),
+ simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless,
+ lemma1, lemma2]));
+qed "zmod_zmult_zmult1_raw";
+
+Goal "(c$*a) zmod (c$*b) = c $* (a zmod b)";
+by (cut_inst_tac [("b","intify(b)"),("c","intify(c)")]
+ zmod_zmult_zmult1_raw 1);
+by Auto_tac;
+qed "zmod_zmult_zmult1";
+
+Goal "(a$*c) zmod (b$*c) = (a zmod b) $* c";
+by (cut_inst_tac [("c","c")] zmod_zmult_zmult1 1);
+by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
+qed "zmod_zmult_zmult2";
+
+
+(** Quotients of signs **)
+
+Goal "[| a $< #0; #0 $< b |] ==> a zdiv b $< #0";
+by (subgoal_tac "a zdiv b $<= #-1" 1);
+by (etac zle_zless_trans 1);
+by (Simp_tac 1);
+by (rtac zle_trans 1);
+by (res_inst_tac [("a'","#-1")] zdiv_mono1 1);
+by (rtac (zless_add1_iff_zle RS iffD1) 1);
+by (Simp_tac 1);
+by (auto_tac (claset(), simpset() addsimps [zdiv_minus1]));
+qed "zdiv_neg_pos_less0";
+
+Goal "[| #0 $<= a; b $< #0 |] ==> a zdiv b $<= #0";
+by (dtac zdiv_mono1_neg 1);
+by Auto_tac;
+qed "zdiv_nonneg_neg_le0";
+
+Goal "#0 $< b ==> (#0 $<= a zdiv b) <-> (#0 $<= a)";
+by Auto_tac;
+by (dtac zdiv_mono1 2);
+by (auto_tac (claset(), simpset() addsimps [neq_iff_zless]));
+by (full_simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym]) 1);
+by (blast_tac (claset() addIs [zdiv_neg_pos_less0]) 1);
+qed "pos_imp_zdiv_nonneg_iff";
+
+Goal "b $< #0 ==> (#0 $<= a zdiv b) <-> (a $<= #0)";
+by (stac (zdiv_zminus_zminus RS sym) 1);
+by (rtac iff_trans 1);
+by (rtac pos_imp_zdiv_nonneg_iff 1);
+by Auto_tac;
+qed "neg_imp_zdiv_nonneg_iff";
+
+(*But not (a zdiv b $<= 0 iff a$<=0); consider a=1, b=2 when a zdiv b = 0.*)
+Goal "#0 $< b ==> (a zdiv b $< #0) <-> (a $< #0)";
+by (asm_simp_tac (simpset() addsimps [not_zle_iff_zless RS iff_sym]) 1);
+by (etac pos_imp_zdiv_nonneg_iff 1);
+qed "pos_imp_zdiv_neg_iff";
+
+(*Again the law fails for $<=: consider a = -1, b = -2 when a zdiv b = 0*)
+Goal "b $< #0 ==> (a zdiv b $< #0) <-> (#0 $< a)";
+by (asm_simp_tac (simpset() addsimps [not_zle_iff_zless RS iff_sym]) 1);
+by (etac neg_imp_zdiv_nonneg_iff 1);
+qed "neg_imp_zdiv_neg_iff";
+
+(*
+ THESE REMAIN TO BE CONVERTED -- but aren't that useful!
+
+ (*** Speeding up the division algorithm with shifting ***)
+
+ (** computing "zdiv" by shifting **)
+
+ Goal "#0 $<= a ==> (#1 $+ #2$*b) zdiv (#2$*a) = b zdiv a";
+ by (zdiv_undefined_case_tac "a = #0" 1);
+ by (subgoal_tac "#1 $<= a" 1);
+ by (arith_tac 2);
+ by (subgoal_tac "#1 $< a $* #2" 1);
+ by (arith_tac 2);
+ by (subgoal_tac "#2$*(#1 $+ b zmod a) $<= #2$*a" 1);
+ by (rtac zmult_zle_mono2 2);
+ by (auto_tac (claset(),
+ simpset() addsimps [zadd_commute, zmult_commute,
+ add1_zle_iff, pos_mod_bound]));
+ by (stac zdiv_zadd1_eq 1);
+ by (asm_simp_tac (simpset() addsimps [zdiv_zmult_zmult2, zmod_zmult_zmult2,
+ zdiv_pos_pos_trivial]) 1);
+ by (stac zdiv_pos_pos_trivial 1);
+ by (asm_simp_tac (simpset()
+ addsimps [zmod_pos_pos_trivial,
+ pos_mod_sign RS zadd_zle_mono1 RSN (2,zle_trans)]) 1);
+ by (auto_tac (claset(),
+ simpset() addsimps [zmod_pos_pos_trivial]));
+ by (subgoal_tac "#0 $<= b zmod a" 1);
+ by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
+ by (arith_tac 1);
+ qed "pos_zdiv_mult_2";
+
+
+ Goal "a $<= #0 ==> (#1 $+ #2$*b) zdiv (#2$*a) <-> (b$+#1) zdiv a";
+ by (subgoal_tac "(#1 $+ #2$*($-b-#1)) zdiv (#2 $* ($-a)) <-> ($-b-#1) zdiv ($-a)" 1);
+ by (rtac pos_zdiv_mult_2 2);
+ by (auto_tac (claset(),
+ simpset() addsimps [zmult_zminus_right]));
+ by (subgoal_tac "(#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))" 1);
+ by (Simp_tac 2);
+ by (asm_full_simp_tac (HOL_ss
+ addsimps [zdiv_zminus_zminus, zdiff_def,
+ zminus_zadd_distrib RS sym]) 1);
+ qed "neg_zdiv_mult_2";
+
+
+ (*Not clear why this must be proved separately; probably integ_of causes
+ simplification problems*)
+ Goal "~ #0 $<= x ==> x $<= #0";
+ by Auto_tac;
+ val lemma = result();
+
+ Goal "integ_of (v BIT b) zdiv integ_of (w BIT False) = \
+ \ (if ~b | #0 $<= integ_of w \
+ \ then integ_of v zdiv (integ_of w) \
+ \ else (integ_of v $+ #1) zdiv (integ_of w))";
+ by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, integ_of_BIT]) 1);
+ by (asm_simp_tac (simpset()
+ delsimps bin_arith_extra_simps@bin_rel_simps
+ addsimps [zdiv_zmult_zmult1,
+ pos_zdiv_mult_2, lemma, neg_zdiv_mult_2]) 1);
+ qed "zdiv_integ_of_BIT";
+
+ Addsimps [zdiv_integ_of_BIT];
+
+
+ (** computing "zmod" by shifting (proofs resemble those for "zdiv") **)
+
+ Goal "#0 $<= a ==> (#1 $+ #2$*b) zmod (#2$*a) = #1 $+ #2 $* (b zmod a)";
+ by (zdiv_undefined_case_tac "a = #0" 1);
+ by (subgoal_tac "#1 $<= a" 1);
+ by (arith_tac 2);
+ by (subgoal_tac "#1 $< a $* #2" 1);
+ by (arith_tac 2);
+ by (subgoal_tac "#2$*(#1 $+ b zmod a) $<= #2$*a" 1);
+ by (rtac zmult_zle_mono2 2);
+ by (auto_tac (claset(),
+ simpset() addsimps [zadd_commute, zmult_commute,
+ add1_zle_iff, pos_mod_bound]));
+ by (stac zmod_zadd1_eq 1);
+ by (asm_simp_tac (simpset() addsimps [zmod_zmult_zmult2,
+ zmod_pos_pos_trivial]) 1);
+ by (rtac zmod_pos_pos_trivial 1);
+ by (asm_simp_tac (simpset()
+ # addsimps [zmod_pos_pos_trivial,
+ pos_mod_sign RS zadd_zle_mono1 RSN (2,zle_trans)]) 1);
+ by (auto_tac (claset(),
+ simpset() addsimps [zmod_pos_pos_trivial]));
+ by (subgoal_tac "#0 $<= b zmod a" 1);
+ by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
+ by (arith_tac 1);
+ qed "pos_zmod_mult_2";
+
+
+ Goal "a $<= #0 ==> (#1 $+ #2$*b) zmod (#2$*a) = #2 $* ((b$+#1) zmod a) - #1";
+ by (subgoal_tac
+ "(#1 $+ #2$*($-b-#1)) zmod (#2$*($-a)) = #1 $+ #2$*(($-b-#1) zmod ($-a))" 1);
+ by (rtac pos_zmod_mult_2 2);
+ by (auto_tac (claset(),
+ simpset() addsimps [zmult_zminus_right]));
+ by (subgoal_tac "(#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))" 1);
+ by (Simp_tac 2);
+ by (asm_full_simp_tac (HOL_ss
+ addsimps [zmod_zminus_zminus, zdiff_def,
+ zminus_zadd_distrib RS sym]) 1);
+ by (dtac (zminus_equation RS iffD1 RS sym) 1);
+ by Auto_tac;
+ qed "neg_zmod_mult_2";
+
+ Goal "integ_of (v BIT b) zmod integ_of (w BIT False) = \
+ \ (if b then \
+ \ if #0 $<= integ_of w \
+ \ then #2 $* (integ_of v zmod integ_of w) $+ #1 \
+ \ else #2 $* ((integ_of v $+ #1) zmod integ_of w) - #1 \
+ \ else #2 $* (integ_of v zmod integ_of w))";
+ by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, integ_of_BIT]) 1);
+ by (asm_simp_tac (simpset()
+ delsimps bin_arith_extra_simps@bin_rel_simps
+ addsimps [zmod_zmult_zmult1,
+ pos_zmod_mult_2, lemma, neg_zmod_mult_2]) 1);
+ qed "zmod_integ_of_BIT";
+
+ Addsimps [zmod_integ_of_BIT];
+*)