(* Title: HOL/IntDiv.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
The division operators div, mod and the divides relation "dvd"
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));
*)
Goal "[| #0 $< k; k: int |] ==> 0 < zmagnitude(k)";
by (dtac zero_zless_imp_znegative_zminus 1);
by (dtac zneg_int_of 2);
by (auto_tac (claset(), simpset() addsimps [inst "x" "k" zminus_equation]));
by (subgoal_tac "0 < zmagnitude($# succ(x))" 1);
by (Asm_full_simp_tac 1);
by (asm_full_simp_tac (simpset_of Arith.thy addsimps [zmagnitude_int_of]) 1);
qed "zero_lt_zmagnitude";
(*** Inequality lemmas involving $#succ(m) ***)
Goal "(w $< z $+ $# succ(m)) <-> (w $< z $+ $#m | intify(w) = z $+ $#m)";
by (auto_tac (claset(),
simpset() addsimps [zless_iff_succ_zadd, zadd_assoc,
int_of_add RS sym]));
by (res_inst_tac [("x","0")] bexI 3);
by (TRYALL (dtac sym));
by (cut_inst_tac [("m","m")] int_succ_int_1 1);
by (cut_inst_tac [("m","n")] int_succ_int_1 1);
by (Asm_full_simp_tac 1);
by (eres_inst_tac [("n","x")] natE 1);
by Auto_tac;
by (res_inst_tac [("x","succ(x)")] bexI 1);
by Auto_tac;
qed "zless_add_succ_iff";
Goal "z : 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],
simpset() addsimps [zless_imp_zle, not_zless_iff_zle]));
qed "lemma";
Goal "(w $+ $# succ(m) $<= z) <-> (w $+ $#m $< z)";
by (cut_inst_tac [("z","intify(z)")] lemma 1);
by Auto_tac;
qed "zadd_succ_zle_iff";
(** Inequality reasoning **)
Goal "(w $< z $+ #1) <-> (w$<=z)";
by (subgoal_tac "#1 = $# 1" 1);
by (asm_simp_tac (simpset_of Int.thy
addsimps [zless_add_succ_iff, zle_def]) 1);
by Auto_tac;
qed "zless_add1_iff_zle";
Goal "(w $+ #1 $<= z) <-> (w $< z)";
by (subgoal_tac "#1 = $# 1" 1);
by (asm_simp_tac (simpset_of Int.thy addsimps [zadd_succ_zle_iff]) 1);
by Auto_tac;
qed "add1_zle_iff";
Goal "(#1 $+ w $<= z) <-> (w $< z)";
by (stac zadd_commute 1);
by (rtac add1_zle_iff 1);
qed "add1_left_zle_iff";
(*** Monotonicity results ***)
Goal "(v$+z $< w$+z) <-> (v $< w)";
by (Simp_tac 1);
qed "zadd_right_cancel_zless";
Goal "(z$+v $< z$+w) <-> (v $< w)";
by (Simp_tac 1);
qed "zadd_left_cancel_zless";
Addsimps [zadd_right_cancel_zless, zadd_left_cancel_zless];
Goal "(v$+z $<= w$+z) <-> (v $<= w)";
by (Simp_tac 1);
qed "zadd_right_cancel_zle";
Goal "(z$+v $<= z$+w) <-> (v $<= w)";
by (Simp_tac 1);
qed "zadd_left_cancel_zle";
Addsimps [zadd_right_cancel_zle, zadd_left_cancel_zle];
(*"v $<= w ==> v$+z $<= w$+z"*)
bind_thm ("zadd_zless_mono1", zadd_right_cancel_zless RS iffD2);
(*"v $<= w ==> z$+v $<= z$+w"*)
bind_thm ("zadd_zless_mono2", zadd_left_cancel_zless RS iffD2);
(*"v $<= w ==> v$+z $<= w$+z"*)
bind_thm ("zadd_zle_mono1", zadd_right_cancel_zle RS iffD2);
(*"v $<= w ==> z$+v $<= z$+w"*)
bind_thm ("zadd_zle_mono2", zadd_left_cancel_zle RS iffD2);
Goal "[| w' $<= w; z' $<= z |] ==> w' $+ z' $<= w $+ z";
by (etac (zadd_zle_mono1 RS zle_trans) 1);
by (Simp_tac 1);
qed "zadd_zle_mono";
Goal "[| w' $< w; z' $<= z |] ==> w' $+ z' $< w $+ z";
by (etac (zadd_zless_mono1 RS zless_zle_trans) 1);
by (Simp_tac 1);
qed "zadd_zless_mono";
(*** Monotonicity of Multiplication ***)
Goal "k : nat ==> i $<= j ==> i $* $#k $<= j $* $#k";
by (induct_tac "k" 1);
by (stac int_succ_int_1 2);
by (ALLGOALS
(asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2, zadd_zle_mono])));
val lemma = result();
Goal "[| i $<= j; #0 $<= k |] ==> i$*k $<= j$*k";
by (subgoal_tac "i $* intify(k) $<= j $* intify(k)" 1);
by (Full_simp_tac 1);
by (res_inst_tac [("b", "intify(k)")] (not_zneg_mag RS subst) 1);
by (rtac lemma 3);
by Auto_tac;
by (dtac znegative_imp_zless_0 1);
by (dtac zless_zle_trans 2);
by Auto_tac;
qed "zmult_zle_mono1";
Goal "[| i $<= j; k $<= #0 |] ==> j$*k $<= i$*k";
by (rtac (zminus_zle_zminus RS iffD1) 1);
by (asm_simp_tac (simpset() delsimps [zmult_zminus_right]
addsimps [zmult_zminus_right RS sym,
zmult_zle_mono1, zle_zminus]) 1);
qed "zmult_zle_mono1_neg";
Goal "[| i $<= j; #0 $<= k |] ==> k$*i $<= k$*j";
by (dtac zmult_zle_mono1 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
qed "zmult_zle_mono2";
Goal "[| i $<= j; k $<= #0 |] ==> k$*j $<= k$*i";
by (dtac zmult_zle_mono1_neg 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
qed "zmult_zle_mono2_neg";
(* $<= monotonicity, BOTH arguments*)
Goal "[| i $<= j; k $<= l; #0 $<= j; #0 $<= k |] ==> i$*k $<= j$*l";
by (etac (zmult_zle_mono1 RS zle_trans) 1);
by (assume_tac 1);
by (etac zmult_zle_mono2 1);
by (assume_tac 1);
qed "zmult_zle_mono";
(** strict, in 1st argument; proof is by induction on k>0 **)
Goal "[| i$<j; k : nat |] ==> 0<k --> $#k $* i $< $#k $* j";
by (induct_tac "k" 1);
by (stac int_succ_int_1 2);
by (etac natE 2);
by (ALLGOALS (asm_full_simp_tac
(simpset() addsimps [zadd_zmult_distrib, zadd_zless_mono,
zle_def])));
by (ftac nat_0_le 1);
by (mp_tac 1);
by (subgoal_tac "i $+ (i $+ $# xa $* i) $< j $+ (j $+ $# xa $* j)" 1);
by (Full_simp_tac 1);
by (rtac zadd_zless_mono 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [zle_def])));
val lemma = result() RS mp;
Goal "[| i$<j; #0 $< k |] ==> k$*i $< k$*j";
by (subgoal_tac "intify(k) $* i $< intify(k) $* j" 1);
by (Full_simp_tac 1);
by (res_inst_tac [("b", "intify(k)")] (not_zneg_mag RS subst) 1);
by (rtac lemma 3);
by Auto_tac;
by (dtac znegative_imp_zless_0 1);
by (dtac zless_trans 2 THEN assume_tac 2);
by (auto_tac (claset(), simpset() addsimps [zero_lt_zmagnitude]));
qed "zmult_zless_mono2";
Goal "[| i$<j; #0 $< k |] ==> i$*k $< j$*k";
by (dtac zmult_zless_mono2 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [zmult_commute])));
qed "zmult_zless_mono1";
(* < monotonicity, BOTH arguments*)
Goal "[| i $< j; k $< l; #0 $< j; #0 $< k |] ==> i$*k $< j$*l";
by (etac (zmult_zless_mono1 RS zless_trans) 1);
by (assume_tac 1);
by (etac zmult_zless_mono2 1);
by (assume_tac 1);
qed "zmult_zless_mono";
Goal "[| i $< j; k $< #0 |] ==> j$*k $< i$*k";
by (rtac (zminus_zless_zminus RS iffD1) 1);
by (asm_simp_tac (simpset() delsimps [zmult_zminus_right]
addsimps [zmult_zminus_right RS sym,
zmult_zless_mono1, zless_zminus]) 1);
qed "zmult_zless_mono1_neg";
Goal "[| i $< j; k $< #0 |] ==> k$*j $< k$*i";
by (rtac (zminus_zless_zminus RS iffD1) 1);
by (asm_simp_tac (simpset() delsimps [zmult_zminus]
addsimps [zmult_zminus RS sym,
zmult_zless_mono2, zless_zminus]) 1);
qed "zmult_zless_mono2_neg";
Goal "[| m: int; n: int |] ==> (m$*n = #0) <-> (m = #0 | n = #0)";
by (case_tac "m $< #0" 1);
by (auto_tac (claset(),
simpset() addsimps [not_zless_iff_zle, zle_def, neq_iff_zless]));
by (REPEAT
(force_tac (claset() addDs [zmult_zless_mono1_neg, zmult_zless_mono1],
simpset()) 1));
val lemma = result();
Goal "(m$*n = #0) <-> (intify(m) = #0 | intify(n) = #0)";
by (asm_full_simp_tac (simpset() addsimps [lemma RS iff_sym]) 1);
qed "zmult_eq_0_iff";
(** 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 |] \
\ ==> (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,
zmult_zless_mono1, zmult_zless_mono1_neg]));
by (auto_tac (claset(),
simpset() addsimps [not_zless_iff_zle,
inst "w1" "m$*k" (not_zle_iff_zless RS iff_sym),
inst "w1" "m" (not_zle_iff_zless RS iff_sym)]));
by (ALLGOALS (etac notE));
by (auto_tac (claset(), simpset() addsimps [zless_imp_zle, zmult_zle_mono1,
zmult_zle_mono1_neg]));
val lemma = result();
Goal "(m$*k $< n$*k) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))";
by (cut_inst_tac [("k","intify(k)"),("m","intify(m)"),("n","intify(n)")]
lemma 1);
by Auto_tac;
qed "zmult_zless_cancel2";
Goal "(k$*m $< k$*n) <-> ((#0 $< k & m$<n) | (k $< #0 & n$<m))";
by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute,
zmult_zless_cancel2]) 1);
qed "zmult_zless_cancel1";
Goal "(m$*k $<= n$*k) <-> ((#0 $< k --> m$<=n) & (k $< #0 --> n$<=m))";
by (simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym,
zmult_zless_cancel2]) 1);
by Auto_tac;
qed "zmult_zle_cancel2";
Goal "(k$*m $<= k$*n) <-> ((#0 $< k --> m$<=n) & (k $< #0 --> n$<=m))";
by (simp_tac (simpset() addsimps [not_zless_iff_zle RS iff_sym,
zmult_zless_cancel1]) 1);
by Auto_tac;
qed "zmult_zle_cancel1";
Goal "[| m: int; n: 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$*k) <-> (k=#0 | m=n)";
by (asm_simp_tac (simpset() addsimps [inst "m" "m$*k" int_eq_iff_zle,
inst "m" "m" int_eq_iff_zle]) 1);
by (auto_tac (claset(),
simpset() addsimps [zmult_zle_cancel2, neq_iff_zless]));
val lemma = result();
Goal "(m$*k = n$*k) <-> (intify(k) = #0 | intify(m) = intify(n))";
by (rtac iff_trans 1);
by (rtac lemma 2);
by Auto_tac;
qed "zmult_cancel2";
Goal "(k$*m = k$*n) <-> (intify(k) = #0 | intify(m) = intify(n))";
by (simp_tac (simpset() addsimps [inst "z" "k" zmult_commute,
zmult_cancel2]) 1);
qed "zmult_cancel1";
Addsimps [zmult_cancel1, zmult_cancel2];
(*** Uniqueness and monotonicity of quotients and remainders ***)
Goal "[| b$*q' $+ r' $<= b$*q $+ r; #0 $<= r'; #0 $< b; r $< b |] \
\ ==> q' $<= q";
by (subgoal_tac "r' $+ b $* (q'$-q) $<= r" 1);
by (full_simp_tac
(simpset() addsimps [zdiff_zmult_distrib2]@zadd_ac@zcompare_rls) 2);
by (subgoal_tac "#0 $< b $* (#1 $+ q $- q')" 1);
by (etac zle_zless_trans 2);
by (full_simp_tac
(simpset() addsimps [zdiff_zmult_distrib2,
zadd_zmult_distrib2]@zadd_ac@zcompare_rls) 2);
by (etac zle_zless_trans 2);
by (Asm_simp_tac 2);
by (subgoal_tac "b $* q' $< b $* (#1 $+ q)" 1);
by (full_simp_tac
(simpset() addsimps [zdiff_zmult_distrib2,
zadd_zmult_distrib2]@zadd_ac@zcompare_rls) 2);
by (auto_tac (claset() addEs [zless_asym],
simpset() addsimps [zmult_zless_cancel1, zless_add1_iff_zle]@
zadd_ac@zcompare_rls));
qed "unique_quotient_lemma";
Goal "[| b$*q' $+ r' $<= b$*q $+ r; r $<= #0; b $< #0; b $< r' |] \
\ ==> q $<= q'";
by (res_inst_tac [("b", "$-b"), ("r", "$-r'"), ("r'", "$-r")]
unique_quotient_lemma 1);
by (auto_tac (claset(),
simpset() delsimps [zminus_zadd_distrib]
addsimps [zminus_zadd_distrib RS sym,
zle_zminus, zless_zminus]));
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'";
by (asm_full_simp_tac
(simpset() addsimps split_ifs@
[quorem_def, neq_iff_zless]) 1);
by Safe_tac;
by (ALLGOALS Asm_full_simp_tac);
by (REPEAT
(blast_tac (claset() addIs [zle_anti_sym]
addDs [zle_eq_refl RS unique_quotient_lemma,
zle_eq_refl RS unique_quotient_lemma_neg,
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'";
by (subgoal_tac "q = q'" 1);
by (blast_tac (claset() addIs [unique_quotient]) 2);
by (asm_full_simp_tac (simpset() addsimps [quorem_def]) 1);
by Auto_tac;
qed "unique_remainder";
(*** THE REST TO PORT LATER. The division algorithm can wait; most properties
of division flow from the uniqueness properties proved above...
(*** 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>)";
by (simp_tac (simpset() addsimps [Let_def,adjust_def]) 1);
qed "adjust_eq";
Addsimps [adjust_eq];
(*Proving posDivAlg's termination condition*)
val [tc] = posDivAlg.tcs;
goalw_cterm [] (cterm_of (sign_of (the_context ())) (HOLogic.mk_Trueprop tc));
by Auto_tac;
val lemma = result();
(* removing the termination condition from the generated theorems *)
bind_thm ("posDivAlg_raw_eqn", lemma RS hd posDivAlg.simps);
(**use with simproc to avoid re-proving the premise*)
Goal "#0 $< b ==> \
\ posDivAlg <a,b> = \
\ (if a$<b then <#0,a> else adjust a b (posDivAlg<a,#2$*b>))";
by (rtac (posDivAlg_raw_eqn RS trans) 1);
by (Asm_simp_tac 1);
qed "posDivAlg_eqn";
bind_thm ("posDivAlg_induct", lemma RS posDivAlg.induct);
(*Correctness of posDivAlg: it computes quotients correctly*)
Goal "#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 [posDivAlg_eqn]) 1);
(*main argument*)
by (stac posDivAlg_eqn 1);
by (ALLGOALS Asm_simp_tac);
by (etac splitE 1);
by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
qed_spec_mp "posDivAlg_correct";
(*** Correctness of negDivAlg, the division algorithm for a<0 and b>0 ***)
(*Proving negDivAlg's termination condition*)
val [tc] = negDivAlg.tcs;
goalw_cterm [] (cterm_of (sign_of (the_context ())) (HOLogic.mk_Trueprop tc));
by Auto_tac;
val lemma = result();
(* removing the termination condition from the generated theorems *)
bind_thm ("negDivAlg_raw_eqn", lemma RS hd negDivAlg.simps);
(**use with simproc to avoid re-proving the premise*)
Goal "#0 $< b ==> \
\ negDivAlg <a,b> = \
\ (if #0$<=a$+b then <#-1,a$+b> else adjust a b (negDivAlg<a,#2$*b>))";
by (rtac (negDivAlg_raw_eqn RS trans) 1);
by (Asm_simp_tac 1);
qed "negDivAlg_eqn";
bind_thm ("negDivAlg_induct", lemma RS negDivAlg.induct);
(*Correctness of negDivAlg: it computes quotients correctly
It doesn't work if a=0 because the 0/b=0 rather than -1*)
Goal "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 [negDivAlg_eqn]) 1);
(*main argument*)
by (stac negDivAlg_eqn 1);
by (ALLGOALS Asm_simp_tac);
by (etac splitE 1);
by (auto_tac (claset(), simpset() addsimps [zadd_zmult_distrib2, Let_def]));
qed_spec_mp "negDivAlg_correct";
(*** Existence shown by proving the division algorithm to be correct ***)
(*the case a=0*)
Goal "b ~= #0 ==> quorem (<#0,b>, <#0,#0>)";
by (auto_tac (claset(),
simpset() addsimps [quorem_def, neq_iff_zless]));
qed "quorem_0";
Goal "posDivAlg <#0,b> = <#0,#0>";
by (stac posDivAlg_raw_eqn 1);
by Auto_tac;
qed "posDivAlg_0";
Addsimps [posDivAlg_0];
Goal "negDivAlg <#-1,b> = <#-1,b-#1>";
by (stac negDivAlg_raw_eqn 1);
by Auto_tac;
qed "negDivAlg_minus1";
Addsimps [negDivAlg_minus1];
Goalw [negateSnd_def] "negateSnd<q,r> = <q,-r>";
by Auto_tac;
qed "negateSnd_eq";
Addsimps [negateSnd_eq];
Goal "quorem (<-a,-b>, qr) ==> quorem (<a,b>, negateSnd qr)";
by (auto_tac (claset(), simpset() addsimps split_ifs@[quorem_def]));
qed "quorem_neg";
Goal "b ~= #0 ==> quorem (<a,b>, divAlg<a,b>)";
by (auto_tac (claset(),
simpset() addsimps [quorem_0, divAlg_def]));
by (REPEAT_FIRST (resolve_tac [quorem_neg, posDivAlg_correct,
negDivAlg_correct]));
by (auto_tac (claset(),
simpset() addsimps [quorem_def, neq_iff_zless]));
qed "divAlg_correct";
(** Aribtrary definitions for division by zero. Useful to simplify
certain equations **)
Goal "a div (#0::int) = #0";
by (simp_tac (simpset() addsimps [div_def, divAlg_def, posDivAlg_raw_eqn]) 1);
qed "DIVISION_BY_ZERO_ZDIV"; (*NOT for adding to default simpset*)
Goal "a mod (#0::int) = a";
by (simp_tac (simpset() addsimps [mod_def, divAlg_def, posDivAlg_raw_eqn]) 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 = b $* (a div b) $+ (a mod 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, div_def, mod_def]));
qed "zmod_zdiv_equality";
Goal "(#0::int) $< b ==> #0 $<= a mod b & a mod b $< b";
by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
by (auto_tac (claset(),
simpset() addsimps [quorem_def, mod_def]));
bind_thm ("pos_mod_sign", result() RS conjunct1);
bind_thm ("pos_mod_bound", result() RS conjunct2);
Goal "b $< (#0::int) ==> a mod b $<= #0 & b $< a mod b";
by (cut_inst_tac [("a","a"),("b","b")] divAlg_correct 1);
by (auto_tac (claset(),
simpset() addsimps [quorem_def, div_def, mod_def]));
bind_thm ("neg_mod_sign", result() RS conjunct1);
bind_thm ("neg_mod_bound", result() RS conjunct2);
(** proving general properties of div and mod **)
Goal "b ~= #0 ==> quorem (<a,b>, <a div b,a mod b>)";
by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 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";
Goal "[| quorem(<a,b>,<q,r>); b ~= #0 |] ==> a div b = q";
by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_quotient]) 1);
qed "quorem_div";
Goal "[| quorem(<a,b>,<q,r>); b ~= #0 |] ==> a mod b = r";
by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS unique_remainder]) 1);
qed "quorem_mod";
Goal "[| (#0::int) $<= a; a $< b |] ==> a div b = #0";
by (rtac quorem_div 1);
by (auto_tac (claset(), simpset() addsimps [quorem_def]));
qed "div_pos_pos_trivial";
Goal "[| a $<= (#0::int); b $< a |] ==> a div b = #0";
by (rtac quorem_div 1);
by (auto_tac (claset(), simpset() addsimps [quorem_def]));
qed "div_neg_neg_trivial";
Goal "[| (#0::int) $< a; a$+b $<= #0 |] ==> a div b = #-1";
by (rtac quorem_div 1);
by (auto_tac (claset(), simpset() addsimps [quorem_def]));
qed "div_pos_neg_trivial";
(*There is no div_neg_pos_trivial because #0 div b = #0 would supersede it*)
Goal "[| (#0::int) $<= a; a $< b |] ==> a mod b = a";
by (res_inst_tac [("q","#0")] quorem_mod 1);
by (auto_tac (claset(), simpset() addsimps [quorem_def]));
qed "mod_pos_pos_trivial";
Goal "[| a $<= (#0::int); b $< a |] ==> a mod b = a";
by (res_inst_tac [("q","#0")] quorem_mod 1);
by (auto_tac (claset(), simpset() addsimps [quorem_def]));
qed "mod_neg_neg_trivial";
Goal "[| (#0::int) $< a; a$+b $<= #0 |] ==> a mod b = a$+b";
by (res_inst_tac [("q","#-1")] quorem_mod 1);
by (auto_tac (claset(), simpset() addsimps [quorem_def]));
qed "mod_pos_neg_trivial";
(*There is no mod_neg_pos_trivial...*)
(*Simpler laws such as -a div b = -(a div b) FAIL*)
Goal "(-a) div (-b) = a div 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";
Addsimps [zdiv_zminus_zminus];
(*Simpler laws such as -a mod b = -(a mod b) FAIL*)
Goal "(-a) mod (-b) = - (a mod 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";
Addsimps [zmod_zminus_zminus];
(*** division of a number by itself ***)
Goal "[| (#0::int) $< a; a = r $+ a$*q; r $< a |] ==> #1 $<= q";
by (subgoal_tac "#0 $< a$*q" 1);
by (arith_tac 2);
by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 1);
val lemma1 = result();
Goal "[| (#0::int) $< 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 (asm_full_simp_tac (simpset() addsimps [int_0_le_mult_iff]) 1);
val lemma2 = result();
Goal "[| quorem(<a,a>,<q,r>); a ~= (#0::int) |] ==> q = #1";
by (asm_full_simp_tac
(simpset() addsimps split_ifs@[quorem_def, neq_iff_zless]) 1);
by (rtac order_antisym 1);
by Safe_tac;
by Auto_tac;
by (res_inst_tac [("a", "-a"),("r", "-r")] lemma1 3);
by (res_inst_tac [("a", "-a"),("r", "-r")] lemma2 1);
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 ~= (#0::int) |] ==> r = #0";
by (ftac self_quotient 1);
by (assume_tac 1);
by (asm_full_simp_tac (simpset() addsimps [quorem_def]) 1);
qed "self_remainder";
Goal "a ~= #0 ==> a div a = (#1::int)";
by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_quotient]) 1);
qed "zdiv_self";
Addsimps [zdiv_self];
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
Goal "a mod a = (#0::int)";
by (zdiv_undefined_case_tac "a = #0" 1);
by (asm_simp_tac (simpset() addsimps [quorem_div_mod RS self_remainder]) 1);
qed "zmod_self";
Addsimps [zmod_self];
(*** Computation of division and remainder ***)
Goal "(#0::int) div b = #0";
by (simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
qed "zdiv_zero";
Goal "(#0::int) $< b ==> #-1 div b = #-1";
by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
qed "div_eq_minus1";
Goal "(#0::int) mod b = #0";
by (simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
qed "zmod_zero";
Addsimps [zdiv_zero, zmod_zero];
Goal "(#0::int) $< b ==> #-1 div b = #-1";
by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
qed "zdiv_minus1";
Goal "(#0::int) $< b ==> #-1 mod b = b-#1";
by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
qed "zmod_minus1";
(** a positive, b positive **)
Goal "[| #0 $< a; #0 $<= b |] ==> a div b = fst (posDivAlg<a,b>)";
by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
qed "div_pos_pos";
Goal "[| #0 $< a; #0 $<= b |] ==> a mod b = snd (posDivAlg<a,b>)";
by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
qed "mod_pos_pos";
(** a negative, b positive **)
Goal "[| a $< #0; #0 $< b |] ==> a div b = fst (negDivAlg<a,b>)";
by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
qed "div_neg_pos";
Goal "[| a $< #0; #0 $< b |] ==> a mod b = snd (negDivAlg<a,b>)";
by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
qed "mod_neg_pos";
(** a positive, b negative **)
Goal "[| #0 $< a; b $< #0 |] ==> a div b = fst (negateSnd(negDivAlg<-a,-b>))";
by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
qed "div_pos_neg";
Goal "[| #0 $< a; b $< #0 |] ==> a mod b = snd (negateSnd(negDivAlg<-a,-b>))";
by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
qed "mod_pos_neg";
(** a negative, b negative **)
Goal "[| a $< #0; b $<= #0 |] ==> a div b = fst (negateSnd(posDivAlg<-a,-b>))";
by (asm_simp_tac (simpset() addsimps [div_def, divAlg_def]) 1);
qed "div_neg_neg";
Goal "[| a $< #0; b $<= #0 |] ==> a mod b = snd (negateSnd(posDivAlg<-a,-b>))";
by (asm_simp_tac (simpset() addsimps [mod_def, divAlg_def]) 1);
qed "mod_neg_neg";
Addsimps (map (read_instantiate_sg (sign_of (the_context ()))
[("a", "number_of ?v"), ("b", "number_of ?w")])
[div_pos_pos, div_neg_pos, div_pos_neg, div_neg_neg,
mod_pos_pos, mod_neg_pos, mod_pos_neg, mod_neg_neg,
posDivAlg_eqn, negDivAlg_eqn]);
(** Special-case simplification **)
Goal "a mod (#1::int) = #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;
qed "zmod_1";
Addsimps [zmod_1];
Goal "a div (#1::int) = a";
by (cut_inst_tac [("a","a"),("b","#1")] zmod_zdiv_equality 1);
by Auto_tac;
qed "zdiv_1";
Addsimps [zdiv_1];
Goal "a mod (#-1::int) = #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;
qed "zmod_minus1_right";
Addsimps [zmod_minus1_right];
Goal "a div (#-1::int) = -a";
by (cut_inst_tac [("a","a"),("b","#-1")] zmod_zdiv_equality 1);
by Auto_tac;
qed "zdiv_minus1_right";
Addsimps [zdiv_minus1_right];
(*** Monotonicity in the first argument (divisor) ***)
Goal "[| a $<= a'; #0 $< b |] ==> a div b $<= a' div 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' div b $<= a div b";
by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
by (cut_inst_tac [("a","a'"),("b","b")] zmod_zdiv_equality 1);
by (rtac unique_quotient_lemma_neg 1);
by (etac subst 1);
by (etac subst 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [neg_mod_sign,neg_mod_bound])));
qed "zdiv_mono1_neg";
(*** Monotonicity in the second argument (dividend) ***)
Goal "[| b$*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' $+ #1)" 2);
by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 3);
by (asm_full_simp_tac (simpset() addsimps [int_0_less_mult_iff]) 2);
by (subgoal_tac "b$*q $< b$*(q' $+ #1)" 1);
by (Asm_full_simp_tac 1);
by (subgoal_tac "b$*q = r' $- r $+ b'$*q'" 1);
by (Simp_tac 2);
by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
by (stac zadd_commute 1 THEN rtac zadd_zless_mono 1 THEN arith_tac 1);
by (rtac zmult_zle_mono1 1);
by Auto_tac;
qed "zdiv_mono2_lemma";
Goal "[| (#0::int) $<= a; #0 $< b'; b' $<= b |] \
\ ==> a div b $<= a div b'";
by (subgoal_tac "b ~= #0" 1);
by (arith_tac 2);
by (cut_inst_tac [("a","a"),("b","b")] zmod_zdiv_equality 1);
by (cut_inst_tac [("a","a"),("b","b'")] zmod_zdiv_equality 1);
by (rtac zdiv_mono2_lemma 1);
by (etac subst 1);
by (etac subst 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [pos_mod_sign,pos_mod_bound])));
qed "zdiv_mono2";
Goal "[| b$*q $+ r = b'$*q' $+ r'; b'$*q' $+ r' $< #0; \
\ r $< b; #0 $<= r'; #0 $< b'; b' $<= b |] \
\ ==> q' $<= q";
by (subgoal_tac "q' $< #0" 1);
by (subgoal_tac "b'$*q' $< #0" 2);
by (arith_tac 3);
by (asm_full_simp_tac (simpset() addsimps [zmult_less_0_iff]) 2);
by (subgoal_tac "b$*q' $< b$*(q $+ #1)" 1);
by (Asm_full_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [zadd_zmult_distrib2]) 1);
by (subgoal_tac "b$*q' $<= b'$*q'" 1);
by (asm_simp_tac (simpset() addsimps [zmult_zle_mono1_neg]) 2);
by (subgoal_tac "b'$*q' $< b $+ b$*q" 1);
by (Asm_simp_tac 2);
by (arith_tac 1);
qed "zdiv_mono2_neg_lemma";
Goal "[| a $< (#0::int); #0 $< b'; b' $<= b |] \
\ ==> a div b' $<= a div 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 zdiv_mono2_neg_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_mono2_neg";
(*** More algebraic laws for div and mod ***)
(** proving (a*b) div c = a $* (b div c) $+ a * (b mod c) **)
Goal "[| quorem(<b,c>,<q,r>); c ~= #0 |] \
\ ==> quorem (<a$*b,c>, <a$*q $+ a$*r div c,a$*r mod c>)";
by (auto_tac
(claset(),
simpset() addsimps split_ifs@
[quorem_def, neq_iff_zless,
zadd_zmult_distrib2,
pos_mod_sign,pos_mod_bound,
neg_mod_sign, neg_mod_bound]));
by (ALLGOALS(rtac zmod_zdiv_equality));
val lemma = result();
Goal "(a$*b) div c = a$*(b div c) $+ a$*(b mod c) div c";
by (zdiv_undefined_case_tac "c = #0" 1);
by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_div]) 1);
qed "zdiv_zmult1_eq";
Goal "(a$*b) mod c = a$*(b mod c) mod c";
by (zdiv_undefined_case_tac "c = #0" 1);
by (blast_tac (claset() addIs [quorem_div_mod RS lemma RS quorem_mod]) 1);
qed "zmod_zmult1_eq";
Goal "(a$*b) mod c = ((a mod c) $* b) mod c";
by (rtac trans 1);
by (res_inst_tac [("s","b$*a mod c")] trans 1);
by (rtac zmod_zmult1_eq 2);
by (ALLGOALS (simp_tac (simpset() addsimps [zmult_commute])));
qed "zmod_zmult1_eq'";
Goal "(a$*b) mod c = ((a mod c) $* (b mod c)) mod c";
by (rtac (zmod_zmult1_eq' RS trans) 1);
by (rtac zmod_zmult1_eq 1);
qed "zmod_zmult_distrib";
Goal "b ~= (#0::int) ==> (a$*b) div b = a";
by (asm_simp_tac (simpset() addsimps [zdiv_zmult1_eq]) 1);
qed "zdiv_zmult_self1";
Goal "b ~= (#0::int) ==> (b$*a) div b = a";
by (stac zmult_commute 1 THEN etac zdiv_zmult_self1 1);
qed "zdiv_zmult_self2";
Addsimps [zdiv_zmult_self1, zdiv_zmult_self2];
Goal "(a$*b) mod b = (#0::int)";
by (simp_tac (simpset() addsimps [zmod_zmult1_eq]) 1);
qed "zmod_zmult_self1";
Goal "(b$*a) mod b = (#0::int)";
by (simp_tac (simpset() addsimps [zmult_commute, zmod_zmult1_eq]) 1);
qed "zmod_zmult_self2";
Addsimps [zmod_zmult_self1, zmod_zmult_self2];
(** proving (a$+b) div c = a div c $+ b div c $+ ((a mod c $+ b mod c) div c) **)
Goal "[| quorem(<a,c>,<aq,ar>); quorem(<b,c>,<bq,br>); c ~= #0 |] \
\ ==> quorem (<a$+b,c>, (aq $+ bq $+ (ar$+br) div c, (ar$+br) mod c))";
by (auto_tac
(claset(),
simpset() addsimps split_ifs@
[quorem_def, neq_iff_zless,
zadd_zmult_distrib2,
pos_mod_sign,pos_mod_bound,
neg_mod_sign, neg_mod_bound]));
by (ALLGOALS(rtac zmod_zdiv_equality));
val lemma = result();
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
Goal "(a$+b) div c = a div c $+ b div c $+ ((a mod c $+ b mod c) div c)";
by (zdiv_undefined_case_tac "c = #0" 1);
by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
MRS lemma RS quorem_div]) 1);
qed "zdiv_zadd1_eq";
Goal "(a$+b) mod c = (a mod c $+ b mod c) mod c";
by (zdiv_undefined_case_tac "c = #0" 1);
by (blast_tac (claset() addIs [[quorem_div_mod,quorem_div_mod]
MRS lemma RS quorem_mod]) 1);
qed "zmod_zadd1_eq";
Goal "(a mod b) div b = (#0::int)";
by (zdiv_undefined_case_tac "b = #0" 1);
by (auto_tac (claset(),
simpset() addsimps [neq_iff_zless,
pos_mod_sign, pos_mod_bound, div_pos_pos_trivial,
neg_mod_sign, neg_mod_bound, div_neg_neg_trivial]));
qed "mod_div_trivial";
Addsimps [mod_div_trivial];
Goal "(a mod b) mod b = a mod b";
by (zdiv_undefined_case_tac "b = #0" 1);
by (auto_tac (claset(),
simpset() addsimps [neq_iff_zless,
pos_mod_sign, pos_mod_bound, mod_pos_pos_trivial,
neg_mod_sign, neg_mod_bound, mod_neg_neg_trivial]));
qed "mod_mod_trivial";
Addsimps [mod_mod_trivial];
Goal "(a$+b) mod c = ((a mod c) $+ b) mod c";
by (rtac (trans RS sym) 1);
by (rtac zmod_zadd1_eq 1);
by (Simp_tac 1);
by (rtac (zmod_zadd1_eq RS sym) 1);
qed "zmod_zadd_left_eq";
Goal "(a$+b) mod c = (a $+ (b mod c)) mod c";
by (rtac (trans RS sym) 1);
by (rtac zmod_zadd1_eq 1);
by (Simp_tac 1);
by (rtac (zmod_zadd1_eq RS sym) 1);
qed "zmod_zadd_right_eq";
Goal "a ~= (#0::int) ==> (a$+b) div a = b div a $+ #1";
by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
qed "zdiv_zadd_self1";
Goal "a ~= (#0::int) ==> (b$+a) div a = b div a $+ #1";
by (asm_simp_tac (simpset() addsimps [zdiv_zadd1_eq]) 1);
qed "zdiv_zadd_self2";
Addsimps [zdiv_zadd_self1, zdiv_zadd_self2];
Goal "(a$+b) mod a = b mod 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) mod a = b mod 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 div (b*c) = (a div b) div c ***)
(*The condition c>0 seems necessary. Consider that 7 div ~6 = ~2 but
7 div 2 div ~3 = 3 div ~3 = ~1. The subcase (a div b) mod c = 0 seems
to cause particular problems.*)
(** first, four lemmas to bound the remainder for the cases b<0 and b>0 **)
Goal "[| (#0::int) $< c; b $< r; r $<= #0 |] ==> b$*c $< b$*(q mod c) $+ r";
by (subgoal_tac "b $* (c $- q mod c) $< r $* #1" 1);
by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
by (rtac order_le_less_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_eq, pos_mod_bound]));
val lemma1 = result();
Goal "[| (#0::int) $< c; b $< r; r $<= #0 |] ==> b $* (q mod c) $+ r $<= #0";
by (subgoal_tac "b $* (q mod c) $<= #0" 1);
by (arith_tac 1);
by (asm_simp_tac (simpset() addsimps [zmult_le_0_iff, pos_mod_sign]) 1);
val lemma2 = result();
Goal "[| (#0::int) $< c; #0 $<= r; r $< b |] ==> #0 $<= b $* (q mod c) $+ r";
by (subgoal_tac "#0 $<= b $* (q mod c)" 1);
by (arith_tac 1);
by (asm_simp_tac (simpset() addsimps [int_0_le_mult_iff, pos_mod_sign]) 1);
val lemma3 = result();
Goal "[| (#0::int) $< c; #0 $<= r; r $< b |] ==> b $* (q mod c) $+ r $< b $* c";
by (subgoal_tac "r $* #1 $< b $* (c $- q mod c)" 1);
by (asm_full_simp_tac (simpset() addsimps [zdiff_zmult_distrib2]) 1);
by (rtac order_less_le_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_eq, pos_mod_bound]));
val lemma4 = result();
Goal "[| quorem (<a,b>, <q,r>); b ~= #0; #0 $< c |] \
\ ==> quorem (<a,b$*c>, (q div c, b$*(q mod c) $+ r))";
by (auto_tac
(claset(),
simpset() addsimps zmult_ac@
[zmod_zdiv_equality, quorem_def, neq_iff_zless,
int_0_less_mult_iff,
zadd_zmult_distrib2 RS sym,
lemma1, lemma2, lemma3, lemma4]));
val lemma = result();
Goal "(#0::int) $< c ==> a div (b$*c) = (a div b) div c";
by (zdiv_undefined_case_tac "b = #0" 1);
by (force_tac (claset(),
simpset() addsimps [quorem_div_mod RS lemma RS quorem_div,
zmult_eq_0_iff]) 1);
qed "zdiv_zmult2_eq";
Goal "(#0::int) $< c ==> a mod (b$*c) = b$*(a div b mod c) $+ a mod b";
by (zdiv_undefined_case_tac "b = #0" 1);
by (force_tac (claset(),
simpset() addsimps [quorem_div_mod RS lemma RS quorem_mod,
zmult_eq_0_iff]) 1);
qed "zmod_zmult2_eq";
(*** Cancellation of common factors in "div" ***)
Goal "[| (#0::int) $< b; c ~= #0 |] ==> (c$*a) div (c$*b) = a div b";
by (stac zdiv_zmult2_eq 1);
by Auto_tac;
val lemma1 = result();
Goal "[| b $< (#0::int); c ~= #0 |] ==> (c$*a) div (c$*b) = a div b";
by (subgoal_tac "(c $* (-a)) div (c $* (-b)) = (-a) div (-b)" 1);
by (rtac lemma1 2);
by Auto_tac;
val lemma2 = result();
Goal "c ~= (#0::int) ==> (c$*a) div (c$*b) = a div b";
by (zdiv_undefined_case_tac "b = #0" 1);
by (auto_tac
(claset(),
simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless,
lemma1, lemma2]));
qed "zdiv_zmult_zmult1";
Goal "c ~= (#0::int) ==> (a$*c) div (b$*c) = a div b";
by (dtac zdiv_zmult_zmult1 1);
by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
qed "zdiv_zmult_zmult2";
(*** Distribution of factors over "mod" ***)
Goal "[| (#0::int) $< b; c ~= #0 |] ==> (c$*a) mod (c$*b) = c $* (a mod b)";
by (stac zmod_zmult2_eq 1);
by Auto_tac;
val lemma1 = result();
Goal "[| b $< (#0::int); c ~= #0 |] ==> (c$*a) mod (c$*b) = c $* (a mod b)";
by (subgoal_tac "(c $* (-a)) mod (c $* (-b)) = c $* ((-a) mod (-b))" 1);
by (rtac lemma1 2);
by Auto_tac;
val lemma2 = result();
Goal "(c$*a) mod (c$*b) = c $* (a mod b)";
by (zdiv_undefined_case_tac "b = #0" 1);
by (zdiv_undefined_case_tac "c = #0" 1);
by (auto_tac
(claset(),
simpset() addsimps [read_instantiate [("x", "b")] neq_iff_zless,
lemma1, lemma2]));
qed "zmod_zmult_zmult1";
Goal "(a$*c) mod (b$*c) = (a mod b) $* c";
by (cut_inst_tac [("c","c")] zmod_zmult_zmult1 1);
by (auto_tac (claset(), simpset() addsimps [zmult_commute]));
qed "zmod_zmult_zmult2";
(*** Speeding up the division algorithm with shifting ***)
(** computing "div" by shifting **)
Goal "(#0::int) $<= a ==> (#1 $+ #2$*b) div (#2$*a) = b div 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 mod a) $<= #2$*a" 1);
by (rtac zmult_zle_mono2 2);
by (auto_tac (claset(),
simpset() addsimps [zadd_commute, zmult_commute,
add1_zle_eq, pos_mod_bound]));
by (stac zdiv_zadd1_eq 1);
by (asm_simp_tac (simpset() addsimps [zdiv_zmult_zmult2, zmod_zmult_zmult2,
div_pos_pos_trivial]) 1);
by (stac div_pos_pos_trivial 1);
by (asm_simp_tac (simpset()
addsimps [mod_pos_pos_trivial,
pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
by (auto_tac (claset(),
simpset() addsimps [mod_pos_pos_trivial]));
by (subgoal_tac "#0 $<= b mod a" 1);
by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
by (arith_tac 1);
qed "pos_zdiv_mult_2";
Goal "a $<= (#0::int) ==> (#1 $+ #2$*b) div (#2$*a) = (b$+#1) div a";
by (subgoal_tac "(#1 $+ #2$*(-b-#1)) div (#2 $* (-a)) = (-b-#1) div (-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 number_of causes
simplification problems*)
Goal "~ #0 $<= x ==> x $<= (#0::int)";
by Auto_tac;
val lemma = result();
Goal "number_of (v BIT b) div number_of (w BIT False) = \
\ (if ~b | (#0::int) $<= number_of w \
\ then number_of v div (number_of w) \
\ else (number_of v $+ (#1::int)) div (number_of w))";
by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_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_number_of_BIT";
Addsimps [zdiv_number_of_BIT];
(** computing "mod" by shifting (proofs resemble those for "div") **)
Goal "(#0::int) $<= a ==> (#1 $+ #2$*b) mod (#2$*a) = #1 $+ #2 $* (b mod 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 mod a) $<= #2$*a" 1);
by (rtac zmult_zle_mono2 2);
by (auto_tac (claset(),
simpset() addsimps [zadd_commute, zmult_commute,
add1_zle_eq, pos_mod_bound]));
by (stac zmod_zadd1_eq 1);
by (asm_simp_tac (simpset() addsimps [zmod_zmult_zmult2,
mod_pos_pos_trivial]) 1);
by (rtac mod_pos_pos_trivial 1);
by (asm_simp_tac (simpset()
addsimps [mod_pos_pos_trivial,
pos_mod_sign RS zadd_zle_mono1 RSN (2,order_trans)]) 1);
by (auto_tac (claset(),
simpset() addsimps [mod_pos_pos_trivial]));
by (subgoal_tac "#0 $<= b mod a" 1);
by (asm_simp_tac (simpset() addsimps [pos_mod_sign]) 2);
by (arith_tac 1);
qed "pos_zmod_mult_2";
Goal "a $<= (#0::int) ==> (#1 $+ #2$*b) mod (#2$*a) = #2 $* ((b$+#1) mod a) - #1";
by (subgoal_tac
"(#1 $+ #2$*(-b-#1)) mod (#2$*(-a)) = #1 $+ #2$*((-b-#1) mod (-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 "number_of (v BIT b) mod number_of (w BIT False) = \
\ (if b then \
\ if (#0::int) $<= number_of w \
\ then #2 $* (number_of v mod number_of w) $+ #1 \
\ else #2 $* ((number_of v $+ (#1::int)) mod number_of w) - #1 \
\ else #2 $* (number_of v mod number_of w))";
by (simp_tac (simpset_of Int.thy addsimps [zadd_assoc, number_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_number_of_BIT";
Addsimps [zmod_number_of_BIT];
(** Quotients of signs **)
Goal "[| a $< (#0::int); #0 $< b |] ==> a div b $< #0";
by (subgoal_tac "a div b $<= #-1" 1);
by (Force_tac 1);
by (rtac order_trans 1);
by (res_inst_tac [("a'","#-1")] zdiv_mono1 1);
by (auto_tac (claset(), simpset() addsimps [zdiv_minus1]));
qed "div_neg_pos_less0";
Goal "[| (#0::int) $<= a; b $< #0 |] ==> a div b $<= #0";
by (dtac zdiv_mono1_neg 1);
by Auto_tac;
qed "div_nonneg_neg_le0";
Goal "(#0::int) $< b ==> (#0 $<= a div 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 sym]) 1);
by (blast_tac (claset() addIs [div_neg_pos_less0]) 1);
qed "pos_imp_zdiv_nonneg_iff";
Goal "b $< (#0::int) ==> (#0 $<= a div b) = (a $<= (#0::int))";
by (stac (zdiv_zminus_zminus RS sym) 1);
by (stac pos_imp_zdiv_nonneg_iff 1);
by Auto_tac;
qed "neg_imp_zdiv_nonneg_iff";
(*But not (a div b $<= 0 iff a$<=0); consider a=1, b=2 when a div b = 0.*)
Goal "(#0::int) $< b ==> (a div b $< #0) = (a $< #0)";
by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
pos_imp_zdiv_nonneg_iff]) 1);
qed "pos_imp_zdiv_neg_iff";
(*Again the law fails for $<=: consider a = -1, b = -2 when a div b = 0*)
Goal "b $< (#0::int) ==> (a div b $< #0) = (#0 $< a)";
by (asm_simp_tac (simpset() addsimps [linorder_not_le RS sym,
neg_imp_zdiv_nonneg_iff]) 1);
qed "neg_imp_zdiv_neg_iff";
*)