(* Title: ZF/IntDiv.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
Here is the division algorithm in ML:
fun posDivAlg (a,b) =
if a<b then (0,a)
else let val (q,r) = posDivAlg(a, 2*b)
in if 0<=r-b then (2*q+1, r-b) else (2*q, r)
end
fun negDivAlg (a,b) =
if 0<=a+b then (~1,a+b)
else let val (q,r) = negDivAlg(a, 2*b)
in if 0<=r-b then (2*q+1, r-b) else (2*q, r)
end;
fun negateSnd (q,r:int) = (q,~r);
fun divAlg (a,b) = if 0<=a then
if b>0 then posDivAlg (a,b)
else if a=0 then (0,0)
else negateSnd (negDivAlg (~a,~b))
else
if 0<b then negDivAlg (a,b)
else negateSnd (posDivAlg (~a,~b));
*)
section\<open>The Division Operators Div and Mod\<close>
theory IntDiv
imports Bin OrderArith
begin
definition
quorem :: "[i,i] => o" where
"quorem == % .
a = b$*q $+ r &
(#0$<b & #0$\<le>r & r$<b | ~(#0$<b) & b$<r & r $\<le> #0)"
definition
adjust :: "[i,i] => i" where
"adjust(b) == %. if #0 $\ r$-b then <#2$*q $+ #1,r$-b>
else <#2$*q,r>"
(** the division algorithm **)
definition
posDivAlg :: "i => i" where
(*for the case a>=0, b>0*)
(*recdef posDivAlg "inv_image less_than (%(a,b). nat_of(a $- b $+ #1))"*)
"posDivAlg(ab) ==
wfrec(measure(int*int, %<a,b>. nat_of (a $- b $+ #1)),
ab,
%<a,b> f. if (a$<b | b$\<le>#0) then <#0,a>
else adjust(b, f ` <a,#2$*b>))"
(*for the case a<0, b>0*)
definition
negDivAlg :: "i => i" where
(*recdef negDivAlg "inv_image less_than (%(a,b). nat_of(- a $- b))"*)
"negDivAlg(ab) ==
wfrec(measure(int*int, %<a,b>. nat_of ($- a $- b)),
ab,
%<a,b> f. if (#0 $\<le> a$+b | b$\<le>#0) then <#-1,a$+b>
else adjust(b, f ` <a,#2$*b>))"
(*for the general case @{term"b\<noteq>0"}*)
definition
negateSnd :: "i => i" where
"negateSnd == %. "
(*The full division algorithm considers all possible signs for a, b
including the special case a=0, b<0, because negDivAlg requires a<0*)
definition
divAlg :: "i => i" where
"divAlg ==
%<a,b>. if #0 $\<le> a then
if #0 $\<le> b then posDivAlg (<a,b>)
else if a=#0 then <#0,#0>
else negateSnd (negDivAlg (<$-a,$-b>))
else
if #0$<b then negDivAlg (<a,b>)
else negateSnd (posDivAlg (<$-a,$-b>))"
definition
zdiv :: "[i,i]=>i" (infixl \<open>zdiv\<close> 70) where
"a zdiv b == fst (divAlg ())"
definition
zmod :: "[i,i]=>i" (infixl \<open>zmod\<close> 70) where
"a zmod b == snd (divAlg ())"
(** Some basic laws by Sidi Ehmety (need linear arithmetic!) **)
lemma zspos_add_zspos_imp_zspos: "[| #0 $< x; #0 $< y |] ==> #0 $< x $+ y"
apply (rule_tac y = "y" in zless_trans)
apply (rule_tac [2] zdiff_zless_iff [THEN iffD1])
apply auto
done
lemma zpos_add_zpos_imp_zpos: "[| #0 $\ x; #0 $\ y |] ==> #0 $\ x $+ y"
apply (rule_tac y = "y" in zle_trans)
apply (rule_tac [2] zdiff_zle_iff [THEN iffD1])
apply auto
done
lemma zneg_add_zneg_imp_zneg: "[| x $< #0; y $< #0 |] ==> x $+ y $< #0"
apply (rule_tac y = "y" in zless_trans)
apply (rule zless_zdiff_iff [THEN iffD1])
apply auto
done
(* this theorem is used below *)
lemma zneg_or_0_add_zneg_or_0_imp_zneg_or_0:
"[| x $\ #0; y $\ #0 |] ==> x $+ y $\ #0"
apply (rule_tac y = "y" in zle_trans)
apply (rule zle_zdiff_iff [THEN iffD1])
apply auto
done
lemma zero_lt_zmagnitude: "[| #0 $< k; k \ int |] ==> 0 < zmagnitude(k)"
apply (drule zero_zless_imp_znegative_zminus)
apply (drule_tac [2] zneg_int_of)
apply (auto simp add: zminus_equation [of k])
apply (subgoal_tac "0 < zmagnitude ($# succ (n))")
apply simp
apply (simp only: zmagnitude_int_of)
apply simp
done
(*** Inequality lemmas involving $#succ(m) ***)
lemma zless_add_succ_iff:
"(w $< z $+ $# succ(m)) \ (w $< z $+ $#m | intify(w) = z $+ $#m)"
apply (auto simp add: zless_iff_succ_zadd zadd_assoc int_of_add [symmetric])
apply (rule_tac [3] x = "0" in bexI)
apply (cut_tac m = "m" in int_succ_int_1)
apply (cut_tac m = "n" in int_succ_int_1)
apply simp
apply (erule natE)
apply auto
apply (rule_tac x = "succ (n) " in bexI)
apply auto
done
lemma zadd_succ_lemma:
"z \ int ==> (w $+ $# succ(m) $\ z) \ (w $+ $#m $< z)"
apply (simp only: not_zless_iff_zle [THEN iff_sym] zless_add_succ_iff)
apply (auto intro: zle_anti_sym elim: zless_asym
simp add: zless_imp_zle not_zless_iff_zle)
done
lemma zadd_succ_zle_iff: "(w $+ $# succ(m) $\ z) \ (w $+ $#m $< z)"
apply (cut_tac z = "intify (z)" in zadd_succ_lemma)
apply auto
done
(** Inequality reasoning **)
lemma zless_add1_iff_zle: "(w $< z $+ #1) \ (w$\z)"
apply (subgoal_tac "#1 = $# 1")
apply (simp only: zless_add_succ_iff zle_def)
apply auto
done
lemma add1_zle_iff: "(w $+ #1 $\ z) \ (w $< z)"
apply (subgoal_tac "#1 = $# 1")
apply (simp only: zadd_succ_zle_iff)
apply auto
done
lemma add1_left_zle_iff: "(#1 $+ w $\ z) \ (w $< z)"
apply (subst zadd_commute)
apply (rule add1_zle_iff)
done
(*** Monotonicity of Multiplication ***)
lemma zmult_mono_lemma: "k \ nat ==> i $\ j ==> i $* $#k $\ j $* $#k"
apply (induct_tac "k")
prefer 2 apply (subst int_succ_int_1)
apply (simp_all (no_asm_simp) add: zadd_zmult_distrib2 zadd_zle_mono)
done
lemma zmult_zle_mono1: "[| i $\ j; #0 $\ k |] ==> i$*k $\ j$*k"
apply (subgoal_tac "i $* intify (k) $\ j $* intify (k) ")
apply (simp (no_asm_use))
apply (rule_tac b = "intify (k)" in not_zneg_mag [THEN subst])
apply (rule_tac [3] zmult_mono_lemma)
apply auto
apply (simp add: znegative_iff_zless_0 not_zless_iff_zle [THEN iff_sym])
done
lemma zmult_zle_mono1_neg: "[| i $\ j; k $\ #0 |] ==> j$*k $\ i$*k"
apply (rule zminus_zle_zminus [THEN iffD1])
apply (simp del: zmult_zminus_right
add: zmult_zminus_right [symmetric] zmult_zle_mono1 zle_zminus)
done
lemma zmult_zle_mono2: "[| i $\ j; #0 $\ k |] ==> k$*i $\ k$*j"
apply (drule zmult_zle_mono1)
apply (simp_all add: zmult_commute)
done
lemma zmult_zle_mono2_neg: "[| i $\ j; k $\ #0 |] ==> k$*j $\ k$*i"
apply (drule zmult_zle_mono1_neg)
apply (simp_all add: zmult_commute)
done
(* $\<le> monotonicity, BOTH arguments*)
lemma zmult_zle_mono:
"[| i $\ j; k $\ l; #0 $\ j; #0 $\ k |] ==> i$*k $\ j$*l"
apply (erule zmult_zle_mono1 [THEN zle_trans])
apply assumption
apply (erule zmult_zle_mono2)
apply assumption
done
(** strict, in 1st argument; proof is by induction on k>0 **)
lemma zmult_zless_mono2_lemma [rule_format]:
"[| i$ nat |] ==> 0 $#k $* i $< $#k $* j"
apply (induct_tac "k")
prefer 2
apply (subst int_succ_int_1)
apply (erule natE)
apply (simp_all add: zadd_zmult_distrib zadd_zless_mono zle_def)
apply (frule nat_0_le)
apply (subgoal_tac "i $+ (i $+ $# xa $* i) $< j $+ (j $+ $# xa $* j) ")
apply (simp (no_asm_use))
apply (rule zadd_zless_mono)
apply (simp_all (no_asm_simp) add: zle_def)
done
lemma zmult_zless_mono2: "[| i$ k$*i $< k$*j"
apply (subgoal_tac "intify (k) $* i $< intify (k) $* j")
apply (simp (no_asm_use))
apply (rule_tac b = "intify (k)" in not_zneg_mag [THEN subst])
apply (rule_tac [3] zmult_zless_mono2_lemma)
apply auto
apply (simp add: znegative_iff_zless_0)
apply (drule zless_trans, assumption)
apply (auto simp add: zero_lt_zmagnitude)
done
lemma zmult_zless_mono1: "[| i$ 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 $< j; #0 $< k |] ==> i$*k $< j$*l"
apply (erule zmult_zless_mono1 [THEN zless_trans])
apply assumption
apply (erule zmult_zless_mono2)
apply assumption
done
lemma zmult_zless_mono1_neg: "[| i $< j; k $< #0 |] ==> j$*k $< i$*k"
apply (rule zminus_zless_zminus [THEN iffD1])
apply (simp del: zmult_zminus_right
add: zmult_zminus_right [symmetric] zmult_zless_mono1 zless_zminus)
done
lemma zmult_zless_mono2_neg: "[| i $< j; k $< #0 |] ==> k$*j $< k$*i"
apply (rule zminus_zless_zminus [THEN iffD1])
apply (simp del: zmult_zminus
add: zmult_zminus [symmetric] zmult_zless_mono2 zless_zminus)
done
(** Products of zeroes **)
lemma zmult_eq_lemma:
"[| m \ int; n \ int |] ==> (m = #0 | n = #0) \ (m$*n = #0)"
apply (case_tac "m $< #0")
apply (auto simp add: not_zless_iff_zle zle_def neq_iff_zless)
apply (force dest: zmult_zless_mono1_neg zmult_zless_mono1)+
done
lemma zmult_eq_0_iff [iff]: "(m$*n = #0) \ (intify(m) = #0 | intify(n) = #0)"
apply (simp add: zmult_eq_lemma)
done
(** Cancellation laws for k*m < k*n and m*k < n*k, also for @{text"\<le>"} and =,
but not (yet?) for k*m < n*k. **)
lemma zmult_zless_lemma:
"[| k \ int; m \ int; n \ int |]
==> (m$*k $< n$*k) \<longleftrightarrow> ((#0 $< k & m$<n) | (k $< #0 & n$<m))"
apply (case_tac "k = #0")
apply (auto simp add: neq_iff_zless zmult_zless_mono1 zmult_zless_mono1_neg)
apply (auto simp add: not_zless_iff_zle
not_zle_iff_zless [THEN iff_sym, of "m$*k"]
not_zle_iff_zless [THEN iff_sym, of m])
apply (auto elim: notE
simp add: zless_imp_zle zmult_zle_mono1 zmult_zle_mono1_neg)
done
lemma zmult_zless_cancel2:
"(m$*k $< n$*k) \ ((#0 $< k & m$
apply (cut_tac k = "intify (k)" and m = "intify (m)" and n = "intify (n)"
in zmult_zless_lemma)
apply auto
done
lemma zmult_zless_cancel1:
"(k$*m $< k$*n) \ ((#0 $< k & m$
by (simp add: zmult_commute [of k] zmult_zless_cancel2)
lemma zmult_zle_cancel2:
"(m$*k $\ n$*k) \ ((#0 $< k \ m$\n) & (k $< #0 \ n$\m))"
by (auto simp add: not_zless_iff_zle [THEN iff_sym] zmult_zless_cancel2)
lemma zmult_zle_cancel1:
"(k$*m $\ k$*n) \ ((#0 $< k \ m$\n) & (k $< #0 \ n$\m))"
by (auto simp add: not_zless_iff_zle [THEN iff_sym] zmult_zless_cancel1)
lemma int_eq_iff_zle: "[| m \ int; n \ int |] ==> m=n \ (m $\ n & n $\ m)"
apply (blast intro: zle_refl zle_anti_sym)
done
lemma zmult_cancel2_lemma:
"[| k \ int; m \ int; n \ int |] ==> (m$*k = n$*k) \ (k=#0 | m=n)"
apply (simp add: int_eq_iff_zle [of "m$*k"] int_eq_iff_zle [of m])
apply (auto simp add: zmult_zle_cancel2 neq_iff_zless)
done
lemma zmult_cancel2 [simp]:
"(m$*k = n$*k) \ (intify(k) = #0 | intify(m) = intify(n))"
apply (rule iff_trans)
apply (rule_tac [2] zmult_cancel2_lemma)
apply auto
done
lemma zmult_cancel1 [simp]:
"(k$*m = k$*n) \ (intify(k) = #0 | intify(m) = intify(n))"
by (simp add: zmult_commute [of k] zmult_cancel2)
subsection\<open>Uniqueness and monotonicity of quotients and remainders\<close>
lemma unique_quotient_lemma:
"[| b$*q' $+ r' $\ b$*q $+ r; #0 $\ r'; #0 $< b; r $< b |]
==> q' $\ q"
apply (subgoal_tac "r' $+ b $* (q'$-q) $\ r")
prefer 2 apply (simp add: zdiff_zmult_distrib2 zadd_ac zcompare_rls)
apply (subgoal_tac "#0 $< b $* (#1 $+ q $- q') ")
prefer 2
apply (erule zle_zless_trans)
apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2 zadd_ac zcompare_rls)
apply (erule zle_zless_trans)
apply simp
apply (subgoal_tac "b $* q' $< b $* (#1 $+ q)")
prefer 2
apply (simp add: zdiff_zmult_distrib2 zadd_zmult_distrib2 zadd_ac zcompare_rls)
apply (auto elim: zless_asym
simp add: zmult_zless_cancel1 zless_add1_iff_zle zadd_ac zcompare_rls)
done
lemma unique_quotient_lemma_neg:
"[| b$*q' $+ r' $\ b$*q $+ r; r $\ #0; b $< #0; b $< r' |]
==> q $\<le> q'"
apply (rule_tac b = "$-b" and r = "$-r'" and r' = "$-r"
in unique_quotient_lemma)
apply (auto simp del: zminus_zadd_distrib
simp add: zminus_zadd_distrib [symmetric] zle_zminus zless_zminus)
done
lemma unique_quotient:
"[| quorem (, ); quorem (, ); b \ int; b \ #0;
q \<in> int; q' \<in> int |] ==> q = q'"
apply (simp add: split_ifs quorem_def neq_iff_zless)
apply safe
apply simp_all
apply (blast intro: zle_anti_sym
dest: zle_eq_refl [THEN unique_quotient_lemma]
zle_eq_refl [THEN unique_quotient_lemma_neg] sym)+
done
lemma unique_remainder:
"[| quorem (, ); quorem (, ); b \ int; b \ #0;
q \<in> int; q' \<in> int;
r \<in> int; r' \<in> int |] ==> r = r'"
apply (subgoal_tac "q = q'")
prefer 2 apply (blast intro: unique_quotient)
apply (simp add: quorem_def)
done
subsection\<open>Correctness of posDivAlg,
the Division Algorithm for \<open>a\<ge>0\<close> and \<open>b>0\<close>\<close>
lemma adjust_eq [simp]:
"adjust(b, ) = (let diff = r$-b in
if #0 $\<le> diff then <#2$*q $+ #1,diff>
else <#2$*q,r>)"
by (simp add: Let_def adjust_def)
lemma posDivAlg_termination:
"[| #0 $< b; ~ a $< b |]
==> nat_of(a $- #2 $* b $+ #1) < nat_of(a $- b $+ #1)"
apply (simp (no_asm) add: zless_nat_conj)
apply (simp add: not_zless_iff_zle zless_add1_iff_zle zcompare_rls)
done
lemmas posDivAlg_unfold = def_wfrec [OF posDivAlg_def wf_measure]
lemma posDivAlg_eqn:
"[| #0 $< b; a \ int; b \ int |] ==>
posDivAlg(<a,b>) =
(if a$<b then <#0,a> else adjust(b, posDivAlg (<a, #2$*b>)))"
apply (rule posDivAlg_unfold [THEN trans])
apply (simp add: vimage_iff not_zless_iff_zle [THEN iff_sym])
apply (blast intro: posDivAlg_termination)
done
lemma posDivAlg_induct_lemma [rule_format]:
assumes prem:
"!!a b. [| a \ int; b \ int;
~ (a $< b | b $\<le> #0) \<longrightarrow> P(<a, #2 $* b>) |] ==> P(<a,b>)"
shows " \ int*int \ P()"
using wf_measure [where A = "int*int" and f = "%.nat_of (a $- b $+ #1)"]
proof (induct "" arbitrary: u v rule: wf_induct)
case (step x)
hence uv: "u \ int" "v \ int" by auto
thus ?case
apply (rule prem)
apply (rule impI)
apply (rule step)
apply (auto simp add: step uv not_zle_iff_zless posDivAlg_termination)
done
qed
lemma posDivAlg_induct [consumes 2]:
assumes u_int: "u \ int"
and v_int: "v \ int"
and ih: "!!a b. [| a \ int; b \ int;
~ (a $< b | b $\<le> #0) \<longrightarrow> P(a, #2 $* b) |] ==> P(a,b)"
shows "P(u,v)"
apply (subgoal_tac "(%. P (x,y)) ()")
apply simp
apply (rule posDivAlg_induct_lemma)
apply (simp (no_asm_use))
apply (rule ih)
apply (auto simp add: u_int v_int)
done
(*FIXME: use intify in integ_of so that we always have @{term"integ_of w \<in> int"}.
then this rewrite can work for all constants!!*)
lemma intify_eq_0_iff_zle: "intify(m) = #0 \ (m $\ #0 & #0 $\ m)"
by (simp add: int_eq_iff_zle)
subsection\<open>Some convenient biconditionals for products of signs\<close>
lemma zmult_pos: "[| #0 $< i; #0 $< j |] ==> #0 $< i $* j"
by (drule zmult_zless_mono1, auto)
lemma zmult_neg: "[| i $< #0; j $< #0 |] ==> #0 $< i $* j"
by (drule zmult_zless_mono1_neg, auto)
lemma zmult_pos_neg: "[| #0 $< i; j $< #0 |] ==> i $* j $< #0"
by (drule zmult_zless_mono1_neg, auto)
(** Inequality reasoning **)
lemma int_0_less_lemma:
"[| x \ int; y \ int |]
==> (#0 $< x $* y) \<longleftrightarrow> (#0 $< x & #0 $< y | x $< #0 & y $< #0)"
apply (auto simp add: zle_def not_zless_iff_zle zmult_pos zmult_neg)
apply (rule ccontr)
apply (rule_tac [2] ccontr)
apply (auto simp add: zle_def not_zless_iff_zle)
apply (erule_tac P = "#0$< x$* y" in rev_mp)
apply (erule_tac [2] P = "#0$< x$* y" in rev_mp)
apply (drule zmult_pos_neg, assumption)
prefer 2
apply (drule zmult_pos_neg, assumption)
apply (auto dest: zless_not_sym simp add: zmult_commute)
done
lemma int_0_less_mult_iff:
"(#0 $< x $* y) \ (#0 $< x & #0 $< y | x $< #0 & y $< #0)"
apply (cut_tac x = "intify (x)" and y = "intify (y)" in int_0_less_lemma)
apply auto
done
lemma int_0_le_lemma:
"[| x \ int; y \ int |]
==> (#0 $\<le> x $* y) \<longleftrightarrow> (#0 $\<le> x & #0 $\<le> y | x $\<le> #0 & y $\<le> #0)"
by (auto simp add: zle_def not_zless_iff_zle int_0_less_mult_iff)
lemma int_0_le_mult_iff:
"(#0 $\ x $* y) \ ((#0 $\ x & #0 $\ y) | (x $\ #0 & y $\ #0))"
apply (cut_tac x = "intify (x)" and y = "intify (y)" in int_0_le_lemma)
apply auto
done
lemma zmult_less_0_iff:
"(x $* y $< #0) \ (#0 $< x & y $< #0 | x $< #0 & #0 $< y)"
apply (auto simp add: int_0_le_mult_iff not_zle_iff_zless [THEN iff_sym])
apply (auto dest: zless_not_sym simp add: not_zle_iff_zless)
done
lemma zmult_le_0_iff:
"(x $* y $\ #0) \ (#0 $\ x & y $\ #0 | x $\ #0 & #0 $\ y)"
by (auto dest: zless_not_sym
simp add: int_0_less_mult_iff not_zless_iff_zle [THEN iff_sym])
(*Typechecking for posDivAlg*)
lemma posDivAlg_type [rule_format]:
"[| a \ int; b \ int |] ==> posDivAlg() \ int * int"
apply (rule_tac u = "a" and v = "b" in posDivAlg_induct)
apply assumption+
apply (case_tac "#0 $< ba")
apply (simp add: posDivAlg_eqn adjust_def integ_of_type
split: split_if_asm)
apply clarify
apply (simp add: int_0_less_mult_iff not_zle_iff_zless)
apply (simp add: not_zless_iff_zle)
apply (subst posDivAlg_unfold)
apply simp
done
(*Correctness of posDivAlg: it computes quotients correctly*)
lemma posDivAlg_correct [rule_format]:
"[| a \ int; b \ int |]
==> #0 $\<le> a \<longrightarrow> #0 $< b \<longrightarrow> quorem (<a,b>, posDivAlg(<a,b>))"
apply (rule_tac u = "a" and v = "b" in posDivAlg_induct)
apply auto
apply (simp_all add: quorem_def)
txt\<open>base case: a<b\<close>
apply (simp add: posDivAlg_eqn)
apply (simp add: not_zless_iff_zle [THEN iff_sym])
apply (simp add: int_0_less_mult_iff)
txt\<open>main argument\<close>
apply (subst posDivAlg_eqn)
apply (simp_all (no_asm_simp))
apply (erule splitE)
apply (rule posDivAlg_type)
apply (simp_all add: int_0_less_mult_iff)
apply (auto simp add: zadd_zmult_distrib2 Let_def)
txt\<open>now just linear arithmetic\<close>
apply (simp add: not_zle_iff_zless zdiff_zless_iff)
done
subsection\<open>Correctness of negDivAlg, the division algorithm for a<0 and b>0\<close>
lemma negDivAlg_termination:
"[| #0 $< b; a $+ b $< #0 |]
==> nat_of($- a $- #2 $* b) < nat_of($- a $- b)"
apply (simp (no_asm) add: zless_nat_conj)
apply (simp add: zcompare_rls not_zle_iff_zless zless_zdiff_iff [THEN iff_sym]
zless_zminus)
done
lemmas negDivAlg_unfold = def_wfrec [OF negDivAlg_def wf_measure]
lemma negDivAlg_eqn:
"[| #0 $< b; a \ int; b \ int |] ==>
negDivAlg(<a,b>) =
(if #0 $\<le> a$+b then <#-1,a$+b>
else adjust(b, negDivAlg (<a, #2$*b>)))"
apply (rule negDivAlg_unfold [THEN trans])
apply (simp (no_asm_simp) add: vimage_iff not_zless_iff_zle [THEN iff_sym])
apply (blast intro: negDivAlg_termination)
done
lemma negDivAlg_induct_lemma [rule_format]:
assumes prem:
"!!a b. [| a \ int; b \ int;
~ (#0 $\<le> a $+ b | b $\<le> #0) \<longrightarrow> P(<a, #2 $* b>) |]
==> P(<a,b>)"
shows " \ int*int \ P()"
using wf_measure [where A = "int*int" and f = "%.nat_of ($- a $- b)"]
proof (induct "" arbitrary: u v rule: wf_induct)
case (step x)
hence uv: "u \ int" "v \ int" by auto
thus ?case
apply (rule prem)
apply (rule impI)
apply (rule step)
apply (auto simp add: step uv not_zle_iff_zless negDivAlg_termination)
done
qed
lemma negDivAlg_induct [consumes 2]:
assumes u_int: "u \ int"
and v_int: "v \ int"
and ih: "!!a b. [| a \ int; b \ int;
~ (#0 $\<le> a $+ b | b $\<le> #0) \<longrightarrow> P(a, #2 $* b) |]
==> P(a,b)"
shows "P(u,v)"
apply (subgoal_tac " (%. P (x,y)) ()")
apply simp
apply (rule negDivAlg_induct_lemma)
apply (simp (no_asm_use))
apply (rule ih)
apply (auto simp add: u_int v_int)
done
(*Typechecking for negDivAlg*)
lemma negDivAlg_type:
"[| a \ int; b \ int |] ==> negDivAlg() \ int * int"
apply (rule_tac u = "a" and v = "b" in negDivAlg_induct)
apply assumption+
apply (case_tac "#0 $< ba")
apply (simp add: negDivAlg_eqn adjust_def integ_of_type
split: split_if_asm)
apply clarify
apply (simp add: int_0_less_mult_iff not_zle_iff_zless)
apply (simp add: not_zless_iff_zle)
apply (subst negDivAlg_unfold)
apply simp
done
(*Correctness of negDivAlg: it computes quotients correctly
It doesn't work if a=0 because the 0/b=0 rather than -1*)
lemma negDivAlg_correct [rule_format]:
"[| a \ int; b \ int |]
==> a $< #0 \<longrightarrow> #0 $< b \<longrightarrow> quorem (<a,b>, negDivAlg(<a,b>))"
apply (rule_tac u = "a" and v = "b" in negDivAlg_induct)
apply auto
apply (simp_all add: quorem_def)
txt\<open>base case: \<^term>\<open>0$\<le>a$+b\<close>\<close>
apply (simp add: negDivAlg_eqn)
apply (simp add: not_zless_iff_zle [THEN iff_sym])
apply (simp add: int_0_less_mult_iff)
txt\<open>main argument\<close>
apply (subst negDivAlg_eqn)
apply (simp_all (no_asm_simp))
apply (erule splitE)
apply (rule negDivAlg_type)
apply (simp_all add: int_0_less_mult_iff)
apply (auto simp add: zadd_zmult_distrib2 Let_def)
txt\<open>now just linear arithmetic\<close>
apply (simp add: not_zle_iff_zless zdiff_zless_iff)
done
subsection\<open>Existence shown by proving the division algorithm to be correct\<close>
(*the case a=0*)
lemma quorem_0: "[|b \ #0; b \ int|] ==> quorem (<#0,b>, <#0,#0>)"
by (force simp add: quorem_def neq_iff_zless)
lemma posDivAlg_zero_divisor: "posDivAlg() = <#0,a>"
apply (subst posDivAlg_unfold)
apply simp
done
lemma posDivAlg_0 [simp]: "posDivAlg (<#0,b>) = <#0,#0>"
apply (subst posDivAlg_unfold)
apply (simp add: not_zle_iff_zless)
done
(*Needed below. Actually it's an equivalence.*)
lemma linear_arith_lemma: "~ (#0 $\ #-1 $+ b) ==> (b $\ #0)"
apply (simp add: not_zle_iff_zless)
apply (drule zminus_zless_zminus [THEN iffD2])
apply (simp add: zadd_commute zless_add1_iff_zle zle_zminus)
done
lemma negDivAlg_minus1 [simp]: "negDivAlg (<#-1,b>) = <#-1, b$-#1>"
apply (subst negDivAlg_unfold)
apply (simp add: linear_arith_lemma integ_of_type vimage_iff)
done
lemma negateSnd_eq [simp]: "negateSnd () = "
apply (unfold negateSnd_def)
apply auto
done
lemma negateSnd_type: "qr \ int * int ==> negateSnd (qr) \ int * int"
apply (unfold negateSnd_def)
apply auto
done
lemma quorem_neg:
"[|quorem (<$-a,$-b>, qr); a \ int; b \ int; qr \ int * int|]
==> quorem (<a,b>, negateSnd(qr))"
apply clarify
apply (auto elim: zless_asym simp add: quorem_def zless_zminus)
txt\<open>linear arithmetic from here on\<close>
apply (simp_all add: zminus_equation [of a] zminus_zless)
apply (cut_tac [2] z = "b" and w = "#0" in zless_linear)
apply (cut_tac [1] z = "b" and w = "#0" in zless_linear)
apply auto
apply (blast dest: zle_zless_trans)+
done
lemma divAlg_correct:
"[|b \ #0; a \ int; b \ int|] ==> quorem (, divAlg())"
apply (auto simp add: quorem_0 divAlg_def)
apply (safe intro!: quorem_neg posDivAlg_correct negDivAlg_correct
posDivAlg_type negDivAlg_type)
apply (auto simp add: quorem_def neq_iff_zless)
txt\<open>linear arithmetic from here on\<close>
apply (auto simp add: zle_def)
done
lemma divAlg_type: "[|a \ int; b \ int|] ==> divAlg() \ int * int"
apply (auto simp add: divAlg_def)
apply (auto simp add: posDivAlg_type negDivAlg_type negateSnd_type)
done
(** intify cancellation **)
lemma zdiv_intify1 [simp]: "intify(x) zdiv y = x zdiv y"
by (simp add: zdiv_def)
lemma zdiv_intify2 [simp]: "x zdiv intify(y) = x zdiv y"
by (simp add: zdiv_def)
lemma zdiv_type [iff,TC]: "z zdiv w \ int"
apply (unfold zdiv_def)
apply (blast intro: fst_type divAlg_type)
done
lemma zmod_intify1 [simp]: "intify(x) zmod y = x zmod y"
by (simp add: zmod_def)
lemma zmod_intify2 [simp]: "x zmod intify(y) = x zmod y"
by (simp add: zmod_def)
lemma zmod_type [iff,TC]: "z zmod w \ int"
apply (unfold zmod_def)
apply (rule snd_type)
apply (blast intro: divAlg_type)
done
(** Arbitrary definitions for division by zero. Useful to simplify
certain equations **)
lemma DIVISION_BY_ZERO_ZDIV: "a zdiv #0 = #0"
by (simp add: zdiv_def divAlg_def posDivAlg_zero_divisor)
lemma DIVISION_BY_ZERO_ZMOD: "a zmod #0 = intify(a)"
by (simp add: zmod_def divAlg_def posDivAlg_zero_divisor)
(** Basic laws about division and remainder **)
lemma raw_zmod_zdiv_equality:
"[| a \ int; b \ int |] ==> a = b $* (a zdiv b) $+ (a zmod b)"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (cut_tac a = "a" and b = "b" in divAlg_correct)
apply (auto simp add: quorem_def zdiv_def zmod_def split_def)
done
lemma zmod_zdiv_equality: "intify(a) = b $* (a zdiv b) $+ (a zmod b)"
apply (rule trans)
apply (rule_tac b = "intify (b)" in raw_zmod_zdiv_equality)
apply auto
done
lemma pos_mod: "#0 $< b ==> #0 $\ a zmod b & a zmod b $< b"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in divAlg_correct)
apply (auto simp add: intify_eq_0_iff_zle quorem_def zmod_def split_def)
apply (blast dest: zle_zless_trans)+
done
lemmas pos_mod_sign = pos_mod [THEN conjunct1]
and pos_mod_bound = pos_mod [THEN conjunct2]
lemma neg_mod: "b $< #0 ==> a zmod b $\ #0 & b $< a zmod b"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in divAlg_correct)
apply (auto simp add: intify_eq_0_iff_zle quorem_def zmod_def split_def)
apply (blast dest: zle_zless_trans)
apply (blast dest: zless_trans)+
done
lemmas neg_mod_sign = neg_mod [THEN conjunct1]
and neg_mod_bound = neg_mod [THEN conjunct2]
(** proving general properties of zdiv and zmod **)
lemma quorem_div_mod:
"[|b \ #0; a \ int; b \ int |]
==> quorem (<a,b>, <a zdiv b, a zmod b>)"
apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
apply (auto simp add: quorem_def neq_iff_zless pos_mod_sign pos_mod_bound
neg_mod_sign neg_mod_bound)
done
(*Surely quorem(<a,b>,<q,r>) implies @{term"a \<in> int"}, but it doesn't matter*)
lemma quorem_div:
"[| quorem(,); b \ #0; a \ int; b \ int; q \ int |]
==> a zdiv b = q"
by (blast intro: quorem_div_mod [THEN unique_quotient])
lemma quorem_mod:
"[| quorem(,); b \ #0; a \ int; b \ int; q \ int; r \ int |]
==> a zmod b = r"
by (blast intro: quorem_div_mod [THEN unique_remainder])
lemma zdiv_pos_pos_trivial_raw:
"[| a \ int; b \ int; #0 $\ a; a $< b |] ==> a zdiv b = #0"
apply (rule quorem_div)
apply (auto simp add: quorem_def)
(*linear arithmetic*)
apply (blast dest: zle_zless_trans)+
done
lemma zdiv_pos_pos_trivial: "[| #0 $\ a; a $< b |] ==> a zdiv b = #0"
apply (cut_tac a = "intify (a)" and b = "intify (b)"
in zdiv_pos_pos_trivial_raw)
apply auto
done
lemma zdiv_neg_neg_trivial_raw:
"[| a \ int; b \ int; a $\ #0; b $< a |] ==> a zdiv b = #0"
apply (rule_tac r = "a" in quorem_div)
apply (auto simp add: quorem_def)
(*linear arithmetic*)
apply (blast dest: zle_zless_trans zless_trans)+
done
lemma zdiv_neg_neg_trivial: "[| a $\ #0; b $< a |] ==> a zdiv b = #0"
apply (cut_tac a = "intify (a)" and b = "intify (b)"
in zdiv_neg_neg_trivial_raw)
apply auto
done
lemma zadd_le_0_lemma: "[| a$+b $\ #0; #0 $< a; #0 $< b |] ==> False"
apply (drule_tac z' = "#0" and z = "b" in zadd_zless_mono)
apply (auto simp add: zle_def)
apply (blast dest: zless_trans)
done
lemma zdiv_pos_neg_trivial_raw:
"[| a \ int; b \ int; #0 $< a; a$+b $\ #0 |] ==> a zdiv b = #-1"
apply (rule_tac r = "a $+ b" in quorem_div)
apply (auto simp add: quorem_def)
(*linear arithmetic*)
apply (blast dest: zadd_le_0_lemma zle_zless_trans)+
done
lemma zdiv_pos_neg_trivial: "[| #0 $< a; a$+b $\ #0 |] ==> a zdiv b = #-1"
apply (cut_tac a = "intify (a)" and b = "intify (b)"
in zdiv_pos_neg_trivial_raw)
apply auto
done
(*There is no zdiv_neg_pos_trivial because #0 zdiv b = #0 would supersede it*)
lemma zmod_pos_pos_trivial_raw:
"[| a \ int; b \ int; #0 $\ a; a $< b |] ==> a zmod b = a"
apply (rule_tac q = "#0" in quorem_mod)
apply (auto simp add: quorem_def)
(*linear arithmetic*)
apply (blast dest: zle_zless_trans)+
done
lemma zmod_pos_pos_trivial: "[| #0 $\ a; a $< b |] ==> a zmod b = intify(a)"
apply (cut_tac a = "intify (a)" and b = "intify (b)"
in zmod_pos_pos_trivial_raw)
apply auto
done
lemma zmod_neg_neg_trivial_raw:
"[| a \ int; b \ int; a $\ #0; b $< a |] ==> a zmod b = a"
apply (rule_tac q = "#0" in quorem_mod)
apply (auto simp add: quorem_def)
(*linear arithmetic*)
apply (blast dest: zle_zless_trans zless_trans)+
done
lemma zmod_neg_neg_trivial: "[| a $\ #0; b $< a |] ==> a zmod b = intify(a)"
apply (cut_tac a = "intify (a)" and b = "intify (b)"
in zmod_neg_neg_trivial_raw)
apply auto
done
lemma zmod_pos_neg_trivial_raw:
"[| a \ int; b \ int; #0 $< a; a$+b $\ #0 |] ==> a zmod b = a$+b"
apply (rule_tac q = "#-1" in quorem_mod)
apply (auto simp add: quorem_def)
(*linear arithmetic*)
apply (blast dest: zadd_le_0_lemma zle_zless_trans)+
done
lemma zmod_pos_neg_trivial: "[| #0 $< a; a$+b $\ #0 |] ==> a zmod b = a$+b"
apply (cut_tac a = "intify (a)" and b = "intify (b)"
in zmod_pos_neg_trivial_raw)
apply auto
done
(*There is no zmod_neg_pos_trivial...*)
(*Simpler laws such as -a zdiv b = -(a zdiv b) FAIL*)
lemma zdiv_zminus_zminus_raw:
"[|a \ int; b \ int|] ==> ($-a) zdiv ($-b) = a zdiv b"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_div])
apply auto
done
lemma zdiv_zminus_zminus [simp]: "($-a) zdiv ($-b) = a zdiv b"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in zdiv_zminus_zminus_raw)
apply auto
done
(*Simpler laws such as -a zmod b = -(a zmod b) FAIL*)
lemma zmod_zminus_zminus_raw:
"[|a \ int; b \ int|] ==> ($-a) zmod ($-b) = $- (a zmod b)"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (subst quorem_div_mod [THEN quorem_neg, simplified, THEN quorem_mod])
apply auto
done
lemma zmod_zminus_zminus [simp]: "($-a) zmod ($-b) = $- (a zmod b)"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_zminus_zminus_raw)
apply auto
done
subsection\<open>division of a number by itself\<close>
lemma self_quotient_aux1: "[| #0 $< a; a = r $+ a$*q; r $< a |] ==> #1 $\ q"
apply (subgoal_tac "#0 $< a$*q")
apply (cut_tac w = "#0" and z = "q" in add1_zle_iff)
apply (simp add: int_0_less_mult_iff)
apply (blast dest: zless_trans)
(*linear arithmetic...*)
apply (drule_tac t = "%x. x $- r" in subst_context)
apply (drule sym)
apply (simp add: zcompare_rls)
done
lemma self_quotient_aux2: "[| #0 $< a; a = r $+ a$*q; #0 $\ r |] ==> q $\ #1"
apply (subgoal_tac "#0 $\ a$* (#1$-q)")
apply (simp add: int_0_le_mult_iff zcompare_rls)
apply (blast dest: zle_zless_trans)
apply (simp add: zdiff_zmult_distrib2)
apply (drule_tac t = "%x. x $- a $* q" in subst_context)
apply (simp add: zcompare_rls)
done
lemma self_quotient:
"[| quorem(,); a \ int; q \ int; a \ #0|] ==> q = #1"
apply (simp add: split_ifs quorem_def neq_iff_zless)
apply (rule zle_anti_sym)
apply safe
apply auto
prefer 4 apply (blast dest: zless_trans)
apply (blast dest: zless_trans)
apply (rule_tac [3] a = "$-a" and r = "$-r" in self_quotient_aux1)
apply (rule_tac a = "$-a" and r = "$-r" in self_quotient_aux2)
apply (rule_tac [6] zminus_equation [THEN iffD1])
apply (rule_tac [2] zminus_equation [THEN iffD1])
apply (force intro: self_quotient_aux1 self_quotient_aux2
simp add: zadd_commute zmult_zminus)+
done
lemma self_remainder:
"[|quorem(,); a \ int; q \ int; r \ int; a \ #0|] ==> r = #0"
apply (frule self_quotient)
apply (auto simp add: quorem_def)
done
lemma zdiv_self_raw: "[|a \ #0; a \ int|] ==> a zdiv a = #1"
apply (blast intro: quorem_div_mod [THEN self_quotient])
done
lemma zdiv_self [simp]: "intify(a) \ #0 ==> a zdiv a = #1"
apply (drule zdiv_self_raw)
apply auto
done
(*Here we have 0 zmod 0 = 0, also assumed by Knuth (who puts m zmod 0 = 0) *)
lemma zmod_self_raw: "a \ int ==> a zmod a = #0"
apply (case_tac "a = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (blast intro: quorem_div_mod [THEN self_remainder])
done
lemma zmod_self [simp]: "a zmod a = #0"
apply (cut_tac a = "intify (a)" in zmod_self_raw)
apply auto
done
subsection\<open>Computation of division and remainder\<close>
lemma zdiv_zero [simp]: "#0 zdiv b = #0"
by (simp add: zdiv_def divAlg_def)
lemma zdiv_eq_minus1: "#0 $< b ==> #-1 zdiv b = #-1"
by (simp (no_asm_simp) add: zdiv_def divAlg_def)
lemma zmod_zero [simp]: "#0 zmod b = #0"
by (simp add: zmod_def divAlg_def)
lemma zdiv_minus1: "#0 $< b ==> #-1 zdiv b = #-1"
by (simp add: zdiv_def divAlg_def)
lemma zmod_minus1: "#0 $< b ==> #-1 zmod b = b $- #1"
by (simp add: zmod_def divAlg_def)
(** a positive, b positive **)
lemma zdiv_pos_pos: "[| #0 $< a; #0 $\ b |]
==> a zdiv b = fst (posDivAlg(<intify(a), intify(b)>))"
apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
apply (auto simp add: zle_def)
done
lemma zmod_pos_pos:
"[| #0 $< a; #0 $\ b |]
==> a zmod b = snd (posDivAlg(<intify(a), intify(b)>))"
apply (simp (no_asm_simp) add: zmod_def divAlg_def)
apply (auto simp add: zle_def)
done
(** a negative, b positive **)
lemma zdiv_neg_pos:
"[| a $< #0; #0 $< b |]
==> a zdiv b = fst (negDivAlg(<intify(a), intify(b)>))"
apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
apply (blast dest: zle_zless_trans)
done
lemma zmod_neg_pos:
"[| a $< #0; #0 $< b |]
==> a zmod b = snd (negDivAlg(<intify(a), intify(b)>))"
apply (simp (no_asm_simp) add: zmod_def divAlg_def)
apply (blast dest: zle_zless_trans)
done
(** a positive, b negative **)
lemma zdiv_pos_neg:
"[| #0 $< a; b $< #0 |]
==> a zdiv b = fst (negateSnd(negDivAlg (<$-a, $-b>)))"
apply (simp (no_asm_simp) add: zdiv_def divAlg_def intify_eq_0_iff_zle)
apply auto
apply (blast dest: zle_zless_trans)+
apply (blast dest: zless_trans)
apply (blast intro: zless_imp_zle)
done
lemma zmod_pos_neg:
"[| #0 $< a; b $< #0 |]
==> a zmod b = snd (negateSnd(negDivAlg (<$-a, $-b>)))"
apply (simp (no_asm_simp) add: zmod_def divAlg_def intify_eq_0_iff_zle)
apply auto
apply (blast dest: zle_zless_trans)+
apply (blast dest: zless_trans)
apply (blast intro: zless_imp_zle)
done
(** a negative, b negative **)
lemma zdiv_neg_neg:
"[| a $< #0; b $\ #0 |]
==> a zdiv b = fst (negateSnd(posDivAlg(<$-a, $-b>)))"
apply (simp (no_asm_simp) add: zdiv_def divAlg_def)
apply auto
apply (blast dest!: zle_zless_trans)+
done
lemma zmod_neg_neg:
"[| a $< #0; b $\ #0 |]
==> a zmod b = snd (negateSnd(posDivAlg(<$-a, $-b>)))"
apply (simp (no_asm_simp) add: zmod_def divAlg_def)
apply auto
apply (blast dest!: zle_zless_trans)+
done
declare zdiv_pos_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
declare zdiv_neg_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
declare zdiv_pos_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
declare zdiv_neg_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
declare zmod_pos_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
declare zmod_neg_pos [of "integ_of (v)" "integ_of (w)", simp] for v w
declare zmod_pos_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
declare zmod_neg_neg [of "integ_of (v)" "integ_of (w)", simp] for v w
declare posDivAlg_eqn [of concl: "integ_of (v)" "integ_of (w)", simp] for v w
declare negDivAlg_eqn [of concl: "integ_of (v)" "integ_of (w)", simp] for v w
(** Special-case simplification **)
lemma zmod_1 [simp]: "a zmod #1 = #0"
apply (cut_tac a = "a" and b = "#1" in pos_mod_sign)
apply (cut_tac [2] a = "a" and b = "#1" in pos_mod_bound)
apply auto
(*arithmetic*)
apply (drule add1_zle_iff [THEN iffD2])
apply (rule zle_anti_sym)
apply auto
done
lemma zdiv_1 [simp]: "a zdiv #1 = intify(a)"
apply (cut_tac a = "a" and b = "#1" in zmod_zdiv_equality)
apply auto
done
lemma zmod_minus1_right [simp]: "a zmod #-1 = #0"
apply (cut_tac a = "a" and b = "#-1" in neg_mod_sign)
apply (cut_tac [2] a = "a" and b = "#-1" in neg_mod_bound)
apply auto
(*arithmetic*)
apply (drule add1_zle_iff [THEN iffD2])
apply (rule zle_anti_sym)
apply auto
done
lemma zdiv_minus1_right_raw: "a \ int ==> a zdiv #-1 = $-a"
apply (cut_tac a = "a" and b = "#-1" in zmod_zdiv_equality)
apply auto
apply (rule equation_zminus [THEN iffD2])
apply auto
done
lemma zdiv_minus1_right: "a zdiv #-1 = $-a"
apply (cut_tac a = "intify (a)" in zdiv_minus1_right_raw)
apply auto
done
declare zdiv_minus1_right [simp]
subsection\<open>Monotonicity in the first argument (divisor)\<close>
lemma zdiv_mono1: "[| a $\ a'; #0 $< b |] ==> a zdiv b $\ a' zdiv b"
apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
apply (cut_tac a = "a'" and b = "b" in zmod_zdiv_equality)
apply (rule unique_quotient_lemma)
apply (erule subst)
apply (erule subst)
apply (simp_all (no_asm_simp) add: pos_mod_sign pos_mod_bound)
done
lemma zdiv_mono1_neg: "[| a $\ a'; b $< #0 |] ==> a' zdiv b $\ a zdiv b"
apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
apply (cut_tac a = "a'" and b = "b" in zmod_zdiv_equality)
apply (rule unique_quotient_lemma_neg)
apply (erule subst)
apply (erule subst)
apply (simp_all (no_asm_simp) add: neg_mod_sign neg_mod_bound)
done
subsection\<open>Monotonicity in the second argument (dividend)\<close>
lemma q_pos_lemma:
"[| #0 $\ b'$*q' $+ r'; r' $< b'; #0 $< b' |] ==> #0 $\ q'"
apply (subgoal_tac "#0 $< b'$* (q' $+ #1)")
apply (simp add: int_0_less_mult_iff)
apply (blast dest: zless_trans intro: zless_add1_iff_zle [THEN iffD1])
apply (simp add: zadd_zmult_distrib2)
apply (erule zle_zless_trans)
apply (erule zadd_zless_mono2)
done
lemma zdiv_mono2_lemma:
"[| b$*q $+ r = b'$*q' $+ r'; #0 $\ b'$*q' $+ r';
r' $< b'; #0 $\<le> r; #0 $< b'; b' $\<le> b |]
==> q $\<le> q'"
apply (frule q_pos_lemma, assumption+)
apply (subgoal_tac "b$*q $< b$* (q' $+ #1)")
apply (simp add: zmult_zless_cancel1)
apply (force dest: zless_add1_iff_zle [THEN iffD1] zless_trans zless_zle_trans)
apply (subgoal_tac "b$*q = r' $- r $+ b'$*q'")
prefer 2 apply (simp add: zcompare_rls)
apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
apply (subst zadd_commute [of "b $* q'"], rule zadd_zless_mono)
prefer 2 apply (blast intro: zmult_zle_mono1)
apply (subgoal_tac "r' $+ #0 $< b $+ r")
apply (simp add: zcompare_rls)
apply (rule zadd_zless_mono)
apply auto
apply (blast dest: zless_zle_trans)
done
lemma zdiv_mono2_raw:
"[| #0 $\ a; #0 $< b'; b' $\ b; a \ int |]
==> a zdiv b $\<le> a zdiv b'"
apply (subgoal_tac "#0 $< b")
prefer 2 apply (blast dest: zless_zle_trans)
apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
apply (cut_tac a = "a" and b = "b'" in zmod_zdiv_equality)
apply (rule zdiv_mono2_lemma)
apply (erule subst)
apply (erule subst)
apply (simp_all add: pos_mod_sign pos_mod_bound)
done
lemma zdiv_mono2:
"[| #0 $\ a; #0 $< b'; b' $\ b |]
==> a zdiv b $\<le> a zdiv b'"
apply (cut_tac a = "intify (a)" in zdiv_mono2_raw)
apply auto
done
lemma q_neg_lemma:
"[| b'$*q' $+ r' $< #0; #0 $\ r'; #0 $< b' |] ==> q' $< #0"
apply (subgoal_tac "b'$*q' $< #0")
prefer 2 apply (force intro: zle_zless_trans)
apply (simp add: zmult_less_0_iff)
apply (blast dest: zless_trans)
done
lemma zdiv_mono2_neg_lemma:
"[| b$*q $+ r = b'$*q' $+ r'; b'$*q' $+ r' $< #0;
r $< b; #0 $\<le> r'; #0 $< b'; b' $\<le> b |]
==> q' $\ q"
apply (subgoal_tac "#0 $< b")
prefer 2 apply (blast dest: zless_zle_trans)
apply (frule q_neg_lemma, assumption+)
apply (subgoal_tac "b$*q' $< b$* (q $+ #1)")
apply (simp add: zmult_zless_cancel1)
apply (blast dest: zless_trans zless_add1_iff_zle [THEN iffD1])
apply (simp (no_asm_simp) add: zadd_zmult_distrib2)
apply (subgoal_tac "b$*q' $\ b'$*q'")
prefer 2
apply (simp add: zmult_zle_cancel2)
apply (blast dest: zless_trans)
apply (subgoal_tac "b'$*q' $+ r $< b $+ (b$*q $+ r)")
prefer 2
apply (erule ssubst)
apply simp
apply (drule_tac w' = "r" and z' = "#0" in zadd_zless_mono)
apply (assumption)
apply simp
apply (simp (no_asm_use) add: zadd_commute)
apply (rule zle_zless_trans)
prefer 2 apply (assumption)
apply (simp (no_asm_simp) add: zmult_zle_cancel2)
apply (blast dest: zless_trans)
done
lemma zdiv_mono2_neg_raw:
"[| a $< #0; #0 $< b'; b' $\ b; a \ int |]
==> a zdiv b' $\ a zdiv b"
apply (subgoal_tac "#0 $< b")
prefer 2 apply (blast dest: zless_zle_trans)
apply (cut_tac a = "a" and b = "b" in zmod_zdiv_equality)
apply (cut_tac a = "a" and b = "b'" in zmod_zdiv_equality)
apply (rule zdiv_mono2_neg_lemma)
apply (erule subst)
apply (erule subst)
apply (simp_all add: pos_mod_sign pos_mod_bound)
done
lemma zdiv_mono2_neg: "[| a $< #0; #0 $< b'; b' $\ b |]
==> a zdiv b' $\ a zdiv b"
apply (cut_tac a = "intify (a)" in zdiv_mono2_neg_raw)
apply auto
done
subsection\<open>More algebraic laws for zdiv and zmod\<close>
(** proving (a*b) zdiv c = a $* (b zdiv c) $+ a * (b zmod c) **)
lemma zmult1_lemma:
"[| quorem(, ); c \ int; c \ #0 |]
==> quorem (<a$*b, c>, <a$*q $+ (a$*r) zdiv c, (a$*r) zmod c>)"
apply (auto simp add: split_ifs quorem_def neq_iff_zless zadd_zmult_distrib2
pos_mod_sign pos_mod_bound neg_mod_sign neg_mod_bound)
apply (auto intro: raw_zmod_zdiv_equality)
done
lemma zdiv_zmult1_eq_raw:
"[|b \ int; c \ int|]
==> (a$*b) zdiv c = a$*(b zdiv c) $+ a$*(b zmod c) zdiv c"
apply (case_tac "c = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (rule quorem_div_mod [THEN zmult1_lemma, THEN quorem_div])
apply auto
done
lemma zdiv_zmult1_eq: "(a$*b) zdiv c = a$*(b zdiv c) $+ a$*(b zmod c) zdiv c"
apply (cut_tac b = "intify (b)" and c = "intify (c)" in zdiv_zmult1_eq_raw)
apply auto
done
lemma zmod_zmult1_eq_raw:
"[|b \ int; c \ int|] ==> (a$*b) zmod c = a$*(b zmod c) zmod c"
apply (case_tac "c = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (rule quorem_div_mod [THEN zmult1_lemma, THEN quorem_mod])
apply auto
done
lemma zmod_zmult1_eq: "(a$*b) zmod c = a$*(b zmod c) zmod c"
apply (cut_tac b = "intify (b)" and c = "intify (c)" in zmod_zmult1_eq_raw)
apply auto
done
lemma zmod_zmult1_eq': "(a$*b) zmod c = ((a zmod c) $* b) zmod c"
apply (rule trans)
apply (rule_tac b = " (b $* a) zmod c" in trans)
apply (rule_tac [2] zmod_zmult1_eq)
apply (simp_all (no_asm) add: zmult_commute)
done
lemma zmod_zmult_distrib: "(a$*b) zmod c = ((a zmod c) $* (b zmod c)) zmod c"
apply (rule zmod_zmult1_eq' [THEN trans])
apply (rule zmod_zmult1_eq)
done
lemma zdiv_zmult_self1 [simp]: "intify(b) \ #0 ==> (a$*b) zdiv b = intify(a)"
by (simp add: zdiv_zmult1_eq)
lemma zdiv_zmult_self2 [simp]: "intify(b) \ #0 ==> (b$*a) zdiv b = intify(a)"
by (simp add: zmult_commute)
lemma zmod_zmult_self1 [simp]: "(a$*b) zmod b = #0"
by (simp add: zmod_zmult1_eq)
lemma zmod_zmult_self2 [simp]: "(b$*a) zmod b = #0"
by (simp add: zmult_commute zmod_zmult1_eq)
(** proving (a$+b) zdiv c =
a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c) **)
lemma zadd1_lemma:
"[| quorem(, ); quorem(, );
c \<in> int; c \<noteq> #0 |]
==> quorem (<a$+b, c>, <aq $+ bq $+ (ar$+br) zdiv c, (ar$+br) zmod c>)"
apply (auto simp add: split_ifs quorem_def neq_iff_zless zadd_zmult_distrib2
pos_mod_sign pos_mod_bound neg_mod_sign neg_mod_bound)
apply (auto intro: raw_zmod_zdiv_equality)
done
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
lemma zdiv_zadd1_eq_raw:
"[|a \ int; b \ int; c \ int|] ==>
(a$+b) zdiv c = a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c)"
apply (case_tac "c = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod,
THEN quorem_div])
done
lemma zdiv_zadd1_eq:
"(a$+b) zdiv c = a zdiv c $+ b zdiv c $+ ((a zmod c $+ b zmod c) zdiv c)"
apply (cut_tac a = "intify (a)" and b = "intify (b)" and c = "intify (c)"
in zdiv_zadd1_eq_raw)
apply auto
done
lemma zmod_zadd1_eq_raw:
"[|a \ int; b \ int; c \ int|]
==> (a$+b) zmod c = (a zmod c $+ b zmod c) zmod c"
apply (case_tac "c = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (blast intro: zadd1_lemma [OF quorem_div_mod quorem_div_mod,
THEN quorem_mod])
done
lemma zmod_zadd1_eq: "(a$+b) zmod c = (a zmod c $+ b zmod c) zmod c"
apply (cut_tac a = "intify (a)" and b = "intify (b)" and c = "intify (c)"
in zmod_zadd1_eq_raw)
apply auto
done
lemma zmod_div_trivial_raw:
"[|a \ int; b \ int|] ==> (a zmod b) zdiv b = #0"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (auto simp add: neq_iff_zless pos_mod_sign pos_mod_bound
zdiv_pos_pos_trivial neg_mod_sign neg_mod_bound zdiv_neg_neg_trivial)
done
lemma zmod_div_trivial [simp]: "(a zmod b) zdiv b = #0"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_div_trivial_raw)
apply auto
done
lemma zmod_mod_trivial_raw:
"[|a \ int; b \ int|] ==> (a zmod b) zmod b = a zmod b"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (auto simp add: neq_iff_zless pos_mod_sign pos_mod_bound
zmod_pos_pos_trivial neg_mod_sign neg_mod_bound zmod_neg_neg_trivial)
done
lemma zmod_mod_trivial [simp]: "(a zmod b) zmod b = a zmod b"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_mod_trivial_raw)
apply auto
done
lemma zmod_zadd_left_eq: "(a$+b) zmod c = ((a zmod c) $+ b) zmod c"
apply (rule trans [symmetric])
apply (rule zmod_zadd1_eq)
apply (simp (no_asm))
apply (rule zmod_zadd1_eq [symmetric])
done
lemma zmod_zadd_right_eq: "(a$+b) zmod c = (a $+ (b zmod c)) zmod c"
apply (rule trans [symmetric])
apply (rule zmod_zadd1_eq)
apply (simp (no_asm))
apply (rule zmod_zadd1_eq [symmetric])
done
lemma zdiv_zadd_self1 [simp]:
"intify(a) \ #0 ==> (a$+b) zdiv a = b zdiv a $+ #1"
by (simp (no_asm_simp) add: zdiv_zadd1_eq)
lemma zdiv_zadd_self2 [simp]:
"intify(a) \ #0 ==> (b$+a) zdiv a = b zdiv a $+ #1"
by (simp (no_asm_simp) add: zdiv_zadd1_eq)
lemma zmod_zadd_self1 [simp]: "(a$+b) zmod a = b zmod a"
apply (case_tac "a = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (simp (no_asm_simp) add: zmod_zadd1_eq)
done
lemma zmod_zadd_self2 [simp]: "(b$+a) zmod a = b zmod a"
apply (case_tac "a = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (simp (no_asm_simp) add: zmod_zadd1_eq)
done
subsection\<open>proving a zdiv (b*c) = (a zdiv b) zdiv c\<close>
(*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 **)
lemma zdiv_zmult2_aux1:
"[| #0 $< c; b $< r; r $\ #0 |] ==> b$*c $< b$*(q zmod c) $+ r"
apply (subgoal_tac "b $* (c $- q zmod c) $< r $* #1")
apply (simp add: zdiff_zmult_distrib2 zadd_commute zcompare_rls)
apply (rule zle_zless_trans)
apply (erule_tac [2] zmult_zless_mono1)
apply (rule zmult_zle_mono2_neg)
apply (auto simp add: zcompare_rls zadd_commute add1_zle_iff pos_mod_bound)
apply (blast intro: zless_imp_zle dest: zless_zle_trans)
done
lemma zdiv_zmult2_aux2:
"[| #0 $< c; b $< r; r $\ #0 |] ==> b $* (q zmod c) $+ r $\ #0"
apply (subgoal_tac "b $* (q zmod c) $\ #0")
prefer 2
apply (simp add: zmult_le_0_iff pos_mod_sign)
apply (blast intro: zless_imp_zle dest: zless_zle_trans)
(*arithmetic*)
apply (drule zadd_zle_mono)
apply assumption
apply (simp add: zadd_commute)
done
lemma zdiv_zmult2_aux3:
"[| #0 $< c; #0 $\ r; r $< b |] ==> #0 $\ b $* (q zmod c) $+ r"
apply (subgoal_tac "#0 $\ b $* (q zmod c)")
prefer 2
apply (simp add: int_0_le_mult_iff pos_mod_sign)
apply (blast intro: zless_imp_zle dest: zle_zless_trans)
(*arithmetic*)
apply (drule zadd_zle_mono)
apply assumption
apply (simp add: zadd_commute)
done
lemma zdiv_zmult2_aux4:
"[| #0 $< c; #0 $\ r; r $< b |] ==> b $* (q zmod c) $+ r $< b $* c"
apply (subgoal_tac "r $* #1 $< b $* (c $- q zmod c)")
apply (simp add: zdiff_zmult_distrib2 zadd_commute zcompare_rls)
apply (rule zless_zle_trans)
apply (erule zmult_zless_mono1)
apply (rule_tac [2] zmult_zle_mono2)
apply (auto simp add: zcompare_rls zadd_commute add1_zle_iff pos_mod_bound)
apply (blast intro: zless_imp_zle dest: zle_zless_trans)
done
lemma zdiv_zmult2_lemma:
"[| quorem (, ); a \ int; b \ int; b \ #0; #0 $< c |]
==> quorem (<a,b$*c>, <q zdiv c, b$*(q zmod c) $+ r>)"
apply (auto simp add: zmult_ac zmod_zdiv_equality [symmetric] quorem_def
neq_iff_zless int_0_less_mult_iff
zadd_zmult_distrib2 [symmetric] zdiv_zmult2_aux1 zdiv_zmult2_aux2
zdiv_zmult2_aux3 zdiv_zmult2_aux4)
apply (blast dest: zless_trans)+
done
lemma zdiv_zmult2_eq_raw:
"[|#0 $< c; a \ int; b \ int|] ==> a zdiv (b$*c) = (a zdiv b) zdiv c"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (rule quorem_div_mod [THEN zdiv_zmult2_lemma, THEN quorem_div])
apply (auto simp add: intify_eq_0_iff_zle)
apply (blast dest: zle_zless_trans)
done
lemma zdiv_zmult2_eq: "#0 $< c ==> a zdiv (b$*c) = (a zdiv b) zdiv c"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in zdiv_zmult2_eq_raw)
apply auto
done
lemma zmod_zmult2_eq_raw:
"[|#0 $< c; a \ int; b \ int|]
==> a zmod (b$*c) = b$*(a zdiv b zmod c) $+ a zmod b"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (rule quorem_div_mod [THEN zdiv_zmult2_lemma, THEN quorem_mod])
apply (auto simp add: intify_eq_0_iff_zle)
apply (blast dest: zle_zless_trans)
done
lemma zmod_zmult2_eq:
"#0 $< c ==> a zmod (b$*c) = b$*(a zdiv b zmod c) $+ a zmod b"
apply (cut_tac a = "intify (a)" and b = "intify (b)" in zmod_zmult2_eq_raw)
apply auto
done
subsection\<open>Cancellation of common factors in "zdiv"\<close>
lemma zdiv_zmult_zmult1_aux1:
"[| #0 $< b; intify(c) \ #0 |] ==> (c$*a) zdiv (c$*b) = a zdiv b"
apply (subst zdiv_zmult2_eq)
apply auto
done
lemma zdiv_zmult_zmult1_aux2:
"[| b $< #0; intify(c) \ #0 |] ==> (c$*a) zdiv (c$*b) = a zdiv b"
apply (subgoal_tac " (c $* ($-a)) zdiv (c $* ($-b)) = ($-a) zdiv ($-b)")
apply (rule_tac [2] zdiv_zmult_zmult1_aux1)
apply auto
done
lemma zdiv_zmult_zmult1_raw:
"[|intify(c) \ #0; b \ int|] ==> (c$*a) zdiv (c$*b) = a zdiv b"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (auto simp add: neq_iff_zless [of b]
zdiv_zmult_zmult1_aux1 zdiv_zmult_zmult1_aux2)
done
lemma zdiv_zmult_zmult1: "intify(c) \ #0 ==> (c$*a) zdiv (c$*b) = a zdiv b"
apply (cut_tac b = "intify (b)" in zdiv_zmult_zmult1_raw)
apply auto
done
lemma zdiv_zmult_zmult2: "intify(c) \ #0 ==> (a$*c) zdiv (b$*c) = a zdiv b"
apply (drule zdiv_zmult_zmult1)
apply (auto simp add: zmult_commute)
done
subsection\<open>Distribution of factors over "zmod"\<close>
lemma zmod_zmult_zmult1_aux1:
"[| #0 $< b; intify(c) \ #0 |]
==> (c$*a) zmod (c$*b) = c $* (a zmod b)"
apply (subst zmod_zmult2_eq)
apply auto
done
lemma zmod_zmult_zmult1_aux2:
"[| b $< #0; intify(c) \ #0 |]
==> (c$*a) zmod (c$*b) = c $* (a zmod b)"
apply (subgoal_tac " (c $* ($-a)) zmod (c $* ($-b)) = c $* (($-a) zmod ($-b))")
apply (rule_tac [2] zmod_zmult_zmult1_aux1)
apply auto
done
lemma zmod_zmult_zmult1_raw:
"[|b \ int; c \ int|] ==> (c$*a) zmod (c$*b) = c $* (a zmod b)"
apply (case_tac "b = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (case_tac "c = #0")
apply (simp add: DIVISION_BY_ZERO_ZDIV DIVISION_BY_ZERO_ZMOD)
apply (auto simp add: neq_iff_zless [of b]
zmod_zmult_zmult1_aux1 zmod_zmult_zmult1_aux2)
done
lemma zmod_zmult_zmult1: "(c$*a) zmod (c$*b) = c $* (a zmod b)"
apply (cut_tac b = "intify (b)" and c = "intify (c)" in zmod_zmult_zmult1_raw)
apply auto
done
lemma zmod_zmult_zmult2: "(a$*c) zmod (b$*c) = (a zmod b) $* c"
apply (cut_tac c = "c" in zmod_zmult_zmult1)
apply (auto simp add: zmult_commute)
done
(** Quotients of signs **)
lemma zdiv_neg_pos_less0: "[| a $< #0; #0 $< b |] ==> a zdiv b $< #0"
apply (subgoal_tac "a zdiv b $\ #-1")
apply (erule zle_zless_trans)
apply (simp (no_asm))
apply (rule zle_trans)
apply (rule_tac a' = "#-1" in zdiv_mono1)
apply (rule zless_add1_iff_zle [THEN iffD1])
apply (simp (no_asm))
apply (auto simp add: zdiv_minus1)
done
lemma zdiv_nonneg_neg_le0: "[| #0 $\ a; b $< #0 |] ==> a zdiv b $\ #0"
apply (drule zdiv_mono1_neg)
apply auto
done
lemma pos_imp_zdiv_nonneg_iff: "#0 $< b ==> (#0 $\ a zdiv b) \ (#0 $\ a)"
apply auto
apply (drule_tac [2] zdiv_mono1)
apply (auto simp add: neq_iff_zless)
apply (simp (no_asm_use) add: not_zless_iff_zle [THEN iff_sym])
apply (blast intro: zdiv_neg_pos_less0)
done
lemma neg_imp_zdiv_nonneg_iff: "b $< #0 ==> (#0 $\ a zdiv b) \ (a $\ #0)"
apply (subst zdiv_zminus_zminus [symmetric])
apply (rule iff_trans)
apply (rule pos_imp_zdiv_nonneg_iff)
apply auto
done
(*But not (a zdiv b $\<le> 0 iff a$\<le>0); consider a=1, b=2 when a zdiv b = 0.*)
lemma pos_imp_zdiv_neg_iff: "#0 $< b ==> (a zdiv b $< #0) \ (a $< #0)"
apply (simp (no_asm_simp) add: not_zle_iff_zless [THEN iff_sym])
apply (erule pos_imp_zdiv_nonneg_iff)
done
(*Again the law fails for $\<le>: consider a = -1, b = -2 when a zdiv b = 0*)
lemma neg_imp_zdiv_neg_iff: "b $< #0 ==> (a zdiv b $< #0) \ (#0 $< a)"
apply (simp (no_asm_simp) add: not_zle_iff_zless [THEN iff_sym])
apply (erule neg_imp_zdiv_nonneg_iff)
done
(*
THESE REMAIN TO BE CONVERTED -- but aren't that useful!
subsection{* Speeding up the division algorithm with shifting *}
(** computing "zdiv" by shifting **)
lemma pos_zdiv_mult_2: "#0 $\ a ==> (#1 $+ #2$*b) zdiv (#2$*a) = b zdiv a"
apply (case_tac "a = #0")
apply (subgoal_tac "#1 $\ a")
apply (arith_tac 2)
apply (subgoal_tac "#1 $< a $* #2")
apply (arith_tac 2)
apply (subgoal_tac "#2$* (#1 $+ b zmod a) $\ #2$*a")
apply (rule_tac [2] zmult_zle_mono2)
apply (auto simp add: zadd_commute zmult_commute add1_zle_iff pos_mod_bound)
apply (subst zdiv_zadd1_eq)
apply (simp (no_asm_simp) add: zdiv_zmult_zmult2 zmod_zmult_zmult2 zdiv_pos_pos_trivial)
apply (subst zdiv_pos_pos_trivial)
apply (simp (no_asm_simp) add: [zmod_pos_pos_trivial pos_mod_sign [THEN zadd_zle_mono1] RSN (2,zle_trans) ])
apply (auto simp add: zmod_pos_pos_trivial)
apply (subgoal_tac "#0 $\ b zmod a")
apply (asm_simp_tac (simpset () add: pos_mod_sign) 2)
apply arith
done
lemma neg_zdiv_mult_2: "a $\ #0 ==> (#1 $+ #2$*b) zdiv (#2$*a) \ (b$+#1) zdiv a"
apply (subgoal_tac " (#1 $+ #2$* ($-b-#1)) zdiv (#2 $* ($-a)) \ ($-b-#1) zdiv ($-a)")
apply (rule_tac [2] pos_zdiv_mult_2)
apply (auto simp add: zmult_zminus_right)
apply (subgoal_tac " (#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))")
apply (Simp_tac 2)
apply (asm_full_simp_tac (HOL_ss add: zdiv_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric])
done
(*Not clear why this must be proved separately; probably integ_of causes
simplification problems*)
lemma lemma: "~ #0 $\ x ==> x $\ #0"
apply auto
done
lemma zdiv_integ_of_BIT: "integ_of (v BIT b) zdiv integ_of (w BIT False) =
(if ~b | #0 $\<le> integ_of w
then integ_of v zdiv (integ_of w)
else (integ_of v $+ #1) zdiv (integ_of w))"
apply (simp_tac (simpset_of @{theory_context Int} add: zadd_assoc integ_of_BIT)
apply (simp (no_asm_simp) del: bin_arith_extra_simps@bin_rel_simps add: zdiv_zmult_zmult1 pos_zdiv_mult_2 lemma neg_zdiv_mult_2)
done
declare zdiv_integ_of_BIT [simp]
(** computing "zmod" by shifting (proofs resemble those for "zdiv") **)
lemma pos_zmod_mult_2: "#0 $\ a ==> (#1 $+ #2$*b) zmod (#2$*a) = #1 $+ #2 $* (b zmod a)"
apply (case_tac "a = #0")
apply (subgoal_tac "#1 $\ a")
apply (arith_tac 2)
apply (subgoal_tac "#1 $< a $* #2")
apply (arith_tac 2)
apply (subgoal_tac "#2$* (#1 $+ b zmod a) $\ #2$*a")
apply (rule_tac [2] zmult_zle_mono2)
apply (auto simp add: zadd_commute zmult_commute add1_zle_iff pos_mod_bound)
apply (subst zmod_zadd1_eq)
apply (simp (no_asm_simp) add: zmod_zmult_zmult2 zmod_pos_pos_trivial)
apply (rule zmod_pos_pos_trivial)
apply (simp (no_asm_simp) # add: [zmod_pos_pos_trivial pos_mod_sign [THEN zadd_zle_mono1] RSN (2,zle_trans) ])
apply (auto simp add: zmod_pos_pos_trivial)
apply (subgoal_tac "#0 $\ b zmod a")
apply (asm_simp_tac (simpset () add: pos_mod_sign) 2)
apply arith
done
lemma neg_zmod_mult_2: "a $\ #0 ==> (#1 $+ #2$*b) zmod (#2$*a) = #2 $* ((b$+#1) zmod a) - #1"
apply (subgoal_tac " (#1 $+ #2$* ($-b-#1)) zmod (#2$* ($-a)) = #1 $+ #2$* (($-b-#1) zmod ($-a))")
apply (rule_tac [2] pos_zmod_mult_2)
apply (auto simp add: zmult_zminus_right)
apply (subgoal_tac " (#-1 - (#2 $* b)) = - (#1 $+ (#2 $* b))")
apply (Simp_tac 2)
apply (asm_full_simp_tac (HOL_ss add: zmod_zminus_zminus zdiff_def zminus_zadd_distrib [symmetric])
apply (dtac (zminus_equation [THEN iffD1, symmetric])
apply auto
done
lemma zmod_integ_of_BIT: "integ_of (v BIT b) zmod integ_of (w BIT False) =
(if b then
if #0 $\<le> 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))"
apply (simp_tac (simpset_of @{theory_context Int} add: zadd_assoc integ_of_BIT)
apply (simp (no_asm_simp) del: bin_arith_extra_simps@bin_rel_simps add: zmod_zmult_zmult1 pos_zmod_mult_2 lemma neg_zmod_mult_2)
done
declare zmod_integ_of_BIT [simp]
*)
end
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