src/HOL/Divides.ML
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(*  Title:      HOL/Divides.ML
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

The division operators div, mod and the divides relation "dvd"
*)

(** Less-then properties **)

val wf_less_trans = [eq_reflection, wf_pred_nat RS wf_trancl] MRS
def_wfrec RS trans;

Goal "(%m. m mod n) = wfrec (trancl pred_nat) \
\                           (%f j. if j<n | n=0 then j else f (j-n))";
by (simp_tac (simpset() addsimps [mod_def]) 1);
qed "mod_eq";

Goal "(%m. m div n) = wfrec (trancl pred_nat) \
\            (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))";
by (simp_tac (simpset() addsimps [div_def]) 1);
qed "div_eq";

(** Aribtrary definitions for division by zero.  Useful to simplify
certain equations **)

Goal "a div 0 = 0";
by (rtac (div_eq RS wf_less_trans) 1);
by (Asm_simp_tac 1);
qed "DIVISION_BY_ZERO_DIV";  (*NOT for adding to default simpset*)

Goal "a mod 0 = a";
by (rtac (mod_eq RS wf_less_trans) 1);
by (Asm_simp_tac 1);
qed "DIVISION_BY_ZERO_MOD";  (*NOT for adding to default simpset*)

fun div_undefined_case_tac s i =
case_tac s i THEN
Full_simp_tac (i+1) THEN
DIVISION_BY_ZERO_MOD]) i;

(*** Remainder ***)

Goal "m<n ==> m mod n = (m::nat)";
by (rtac (mod_eq RS wf_less_trans) 1);
by (Asm_simp_tac 1);
qed "mod_less";

Goal "~ m < (n::nat) ==> m mod n = (m-n) mod n";
by (div_undefined_case_tac "n=0" 1);
by (rtac (mod_eq RS wf_less_trans) 1);
by (asm_simp_tac (simpset() addsimps [diff_less, cut_apply, less_eq]) 1);
qed "mod_geq";

(*Avoids the ugly ~m<n above*)
Goal "(n::nat) <= m ==> m mod n = (m-n) mod n";
by (asm_simp_tac (simpset() addsimps [mod_geq, not_less_iff_le]) 1);
qed "le_mod_geq";

Goal "m mod (n::nat) = (if m<n then m else (m-n) mod n)";
by (asm_simp_tac (simpset() addsimps [mod_less, mod_geq]) 1);
qed "mod_if";

Goal "m mod 1 = 0";
by (induct_tac "m" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_less, mod_geq])));
qed "mod_1";

Goal "n mod n = 0";
by (div_undefined_case_tac "n=0" 1);
by (asm_simp_tac (simpset() addsimps [mod_less, mod_geq]) 1);
qed "mod_self";

Goal "(m+n) mod n = m mod (n::nat)";
by (subgoal_tac "(n + m) mod n = (n+m-n) mod n" 1);
by (stac (mod_geq RS sym) 2);

Goal "(n+m) mod n = m mod (n::nat)";

Goal "(m + k*n) mod n = m mod (n::nat)";
by (induct_tac "k" 1);
by (ALLGOALS
(asm_simp_tac
qed "mod_mult_self1";

Goal "(m + n*k) mod n = m mod (n::nat)";
by (asm_simp_tac (simpset() addsimps [mult_commute, mod_mult_self1]) 1);
qed "mod_mult_self2";

Goal "(m mod n) * (k::nat) = (m*k) mod (n*k)";
by (div_undefined_case_tac "n=0" 1);
by (div_undefined_case_tac "k=0" 1);
by (res_inst_tac [("n","m")] less_induct 1);
by (stac mod_if 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [mod_less, mod_geq,
diff_less, diff_mult_distrib]) 1);
qed "mod_mult_distrib";

Goal "(k::nat) * (m mod n) = (k*m) mod (k*n)";
by (asm_simp_tac
mod_mult_distrib]) 1);
qed "mod_mult_distrib2";

Goal "(m*n) mod n = 0";
by (div_undefined_case_tac "n=0" 1);
by (induct_tac "m" 1);
by (asm_simp_tac (simpset() addsimps [mod_less]) 1);
by (rename_tac "k" 1);
qed "mod_mult_self_is_0";

Goal "(n*m) mod n = 0";
by (simp_tac (simpset() addsimps [mult_commute, mod_mult_self_is_0]) 1);
qed "mod_mult_self1_is_0";

(*** Quotient ***)

Goal "m<n ==> m div n = 0";
by (rtac (div_eq RS wf_less_trans) 1);
by (Asm_simp_tac 1);
qed "div_less";

Goal "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)";
by (rtac (div_eq RS wf_less_trans) 1);
by (asm_simp_tac (simpset() addsimps [diff_less, cut_apply, less_eq]) 1);
qed "div_geq";

(*Avoids the ugly ~m<n above*)
Goal "[| 0<n;  n<=m |] ==> m div n = Suc((m-n) div n)";
by (asm_simp_tac (simpset() addsimps [div_geq, not_less_iff_le]) 1);
qed "le_div_geq";

Goal "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))";
by (asm_simp_tac (simpset() addsimps [div_less, div_geq]) 1);
qed "div_if";

(*Main Result about quotient and remainder.*)
Goal "(m div n)*n + m mod n = (m::nat)";
by (div_undefined_case_tac "n=0" 1);
by (res_inst_tac [("n","m")] less_induct 1);
by (stac mod_if 1);
by (ALLGOALS (asm_simp_tac
qed "mod_div_equality";

(* a simple rearrangement of mod_div_equality: *)
Goal "(n::nat) * (m div n) = m - (m mod n)";
by (cut_inst_tac [("m","m"),("n","n")] mod_div_equality 1);
by (EVERY1[etac subst, simp_tac (simpset() addsimps mult_ac),
K(IF_UNSOLVED no_tac)]);
qed "mult_div_cancel";

Goal "m div 1 = m";
by (induct_tac "m" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [div_less, div_geq])));
qed "div_1";

Goal "0<n ==> n div n = 1";
by (asm_simp_tac (simpset() addsimps [div_less, div_geq]) 1);
qed "div_self";

Goal "0<n ==> (m+n) div n = Suc (m div n)";
by (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n)" 1);
by (stac (div_geq RS sym) 2);

Goal "0<n ==> (n+m) div n = Suc (m div n)";

Goal "!!n. 0<n ==> (m + k*n) div n = k + m div n";
by (induct_tac "k" 1);
qed "div_mult_self1";

Goal "0<n ==> (m + n*k) div n = k + m div n";
by (asm_simp_tac (simpset() addsimps [mult_commute, div_mult_self1]) 1);
qed "div_mult_self2";

(** A dividend of zero **)

Goal "0 div m = 0";
by (div_undefined_case_tac "m=0" 1);
by (asm_simp_tac (simpset() addsimps [div_less]) 1);
qed "div_0";

Goal "0 mod m = 0";
by (div_undefined_case_tac "m=0" 1);
by (asm_simp_tac (simpset() addsimps [mod_less]) 1);
qed "mod_0";

(* Monotonicity of div in first argument *)
Goal "ALL m::nat. m <= n --> (m div k) <= (n div k)";
by (div_undefined_case_tac "k=0" 1);
by (res_inst_tac [("n","n")] less_induct 1);
by (Clarify_tac 1);
by (case_tac "n<k" 1);
(* 1  case n<k *)
by (asm_simp_tac (simpset() addsimps [div_less]) 1);
(* 2  case n >= k *)
by (case_tac "m<k" 1);
(* 2.1  case m<k *)
by (asm_simp_tac (simpset() addsimps [div_less]) 1);
(* 2.2  case m>=k *)
by (asm_simp_tac (simpset() addsimps [div_geq, diff_less, diff_le_mono]) 1);
qed_spec_mp "div_le_mono";

(* Antimonotonicity of div in second argument *)
Goal "[| 0<m; m<=n |] ==> (k div n) <= (k div m)";
by (subgoal_tac "0<n" 1);
by (Asm_simp_tac 2);
by (res_inst_tac [("n","k")] less_induct 1);
by (rename_tac "k" 1);
by (case_tac "k<n" 1);
by (asm_simp_tac (simpset() addsimps [div_less]) 1);
by (subgoal_tac "~(k<m)" 1);
by (Asm_simp_tac 2);
by (asm_simp_tac (simpset() addsimps [div_geq]) 1);
by (subgoal_tac "(k-n) div n <= (k-m) div n" 1);
by (REPEAT (ares_tac [div_le_mono,diff_le_mono2] 2));
by (rtac le_trans 1);
by (Asm_simp_tac 1);
by (asm_simp_tac (simpset() addsimps [diff_less]) 1);
qed "div_le_mono2";

Goal "m div n <= (m::nat)";
by (div_undefined_case_tac "n=0" 1);
by (subgoal_tac "m div n <= m div 1" 1);
by (Asm_full_simp_tac 1);
by (rtac div_le_mono2 1);
by (ALLGOALS Asm_simp_tac);
qed "div_le_dividend";

(* Similar for "less than" *)
Goal "1<n ==> (0 < m) --> (m div n < m)";
by (res_inst_tac [("n","m")] less_induct 1);
by (rename_tac "m" 1);
by (case_tac "m<n" 1);
by (asm_full_simp_tac (simpset() addsimps [div_less]) 1);
by (subgoal_tac "0<n" 1);
by (Asm_simp_tac 2);
by (asm_full_simp_tac (simpset() addsimps [div_geq]) 1);
by (case_tac "n<m" 1);
by (subgoal_tac "(m-n) div n < (m-n)" 1);
by (REPEAT (ares_tac [impI,less_trans_Suc] 1));
by (asm_full_simp_tac (simpset() addsimps [diff_less]) 1);
by (asm_full_simp_tac (simpset() addsimps [diff_less]) 1);
(* case n=m *)
by (subgoal_tac "m=n" 1);
by (Asm_simp_tac 2);
by (asm_simp_tac (simpset() addsimps [div_less]) 1);
qed_spec_mp "div_less_dividend";

(*** Further facts about mod (mainly for the mutilated chess board ***)

Goal "0<n ==> Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))";
by (res_inst_tac [("n","m")] less_induct 1);
by (excluded_middle_tac "Suc(na)<n" 1);
(* case Suc(na) < n *)
by (forward_tac [lessI RS less_trans] 2);
by (asm_simp_tac (simpset() addsimps [mod_less, less_not_refl3]) 2);
(* case n <= Suc(na) *)
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le, le_Suc_eq,
mod_geq]) 1);
by (etac disjE 1);
by (asm_simp_tac (simpset() addsimps [mod_less]) 2);
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, diff_less,
le_mod_geq]) 1);
qed "mod_Suc";

Goal "0<n ==> m mod n < n";
by (res_inst_tac [("n","m")] less_induct 1);
by (case_tac "na<n" 1);
(*case n le na*)
by (asm_full_simp_tac (simpset() addsimps [mod_geq, diff_less]) 2);
(*case na<n*)
by (asm_simp_tac (simpset() addsimps [mod_less]) 1);
qed "mod_less_divisor";

(** Evens and Odds **)

(*With less_zeroE, causes case analysis on b<2*)

Goal "b<2 ==> (k mod 2 = b) | (k mod 2 = (if b=1 then 0 else 1))";
by (subgoal_tac "k mod 2 < 2" 1);
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2);
by (Asm_simp_tac 1);
by Safe_tac;
qed "mod2_cases";

Goal "Suc(Suc(m)) mod 2 = m mod 2";
by (subgoal_tac "m mod 2 < 2" 1);
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2);
by Safe_tac;
by (ALLGOALS (asm_simp_tac (simpset() addsimps [mod_Suc])));
qed "mod2_Suc_Suc";

Goal "(0 < m mod 2) = (m mod 2 = 1)";
by (subgoal_tac "m mod 2 < 2" 1);
by (asm_simp_tac (simpset() addsimps [mod_less_divisor]) 2);
by Auto_tac;
qed "mod2_gr_0";

Goal "(m+m) mod 2 = 0";
by (induct_tac "m" 1);
by (simp_tac (simpset() addsimps [mod_less]) 1);
by (Asm_simp_tac 1);

Goal "((m+m)+n) mod 2 = n mod 2";
by (induct_tac "m" 1);
by (simp_tac (simpset() addsimps [mod_less]) 1);
by (Asm_simp_tac 1);

(*Restore the default*)
Delrules [less_SucE];

(*** More division laws ***)

Goal "0<n ==> (m*n) div n = m";
by (cut_inst_tac [("m", "m*n"),("n","n")] mod_div_equality 1);
by (asm_full_simp_tac (simpset() addsimps [mod_mult_self_is_0]) 1);
qed "div_mult_self_is_m";

Goal "0<n ==> (n*m) div n = m";
by (asm_simp_tac (simpset() addsimps [mult_commute, div_mult_self_is_m]) 1);
qed "div_mult_self1_is_m";

(*Cancellation law for division*)
Goal "0<k ==> (k*m) div (k*n) = m div n";
by (div_undefined_case_tac "n=0" 1);
by (res_inst_tac [("n","m")] less_induct 1);
by (case_tac "na<n" 1);
by (asm_simp_tac (simpset() addsimps [div_less, zero_less_mult_iff,
mult_less_mono2]) 1);
by (subgoal_tac "~ k*na < k*n" 1);
by (asm_simp_tac
diff_mult_distrib2 RS sym, diff_less]) 1);
le_refl RS mult_le_mono]) 1);
qed "div_cancel";

(*mod_mult_distrib2 above is the counterpart for remainder*)

(************************************************)
(** Divides Relation                           **)
(************************************************)

Goalw [dvd_def] "m dvd 0";
by (blast_tac (claset() addIs [mult_0_right RS sym]) 1);
qed "dvd_0_right";

Goalw [dvd_def] "0 dvd m ==> m = 0";
by Auto_tac;
qed "dvd_0_left";

Goalw [dvd_def] "1 dvd k";
by (Simp_tac 1);
qed "dvd_1_left";

Goalw [dvd_def] "m dvd (m::nat)";
by (blast_tac (claset() addIs [mult_1_right RS sym]) 1);
qed "dvd_refl";

Goalw [dvd_def] "[| m dvd n; n dvd p |] ==> m dvd (p::nat)";
by (blast_tac (claset() addIs [mult_assoc] ) 1);
qed "dvd_trans";

Goalw [dvd_def] "[| m dvd n; n dvd m |] ==> m = (n::nat)";
qed "dvd_anti_sym";

Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)";

Goalw [dvd_def] "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)";
by (blast_tac (claset() addIs [diff_mult_distrib2 RS sym]) 1);
qed "dvd_diff";

Goal "[| k dvd (m-n); k dvd n; n<=m |] ==> k dvd (m::nat)";
by (etac (not_less_iff_le RS iffD2 RS add_diff_inverse RS subst) 1);
qed "dvd_diffD";

Goalw [dvd_def] "k dvd n ==> k dvd (m*n :: nat)";
by (blast_tac (claset() addIs [mult_left_commute]) 1);
qed "dvd_mult";

Goal "k dvd m ==> k dvd (m*n :: nat)";
by (stac mult_commute 1);
by (etac dvd_mult 1);
qed "dvd_mult2";

(* k dvd (m*k) *)
AddIffs [dvd_refl RS dvd_mult, dvd_refl RS dvd_mult2];

Goal "k dvd (n + k) = k dvd (n::nat)";
br iffI 1;
br  dvd_refl 2;
by (subgoal_tac "n = (n+k)-k" 1);
by  (Simp_tac 2);
be ssubst 1;
be dvd_diff 1;
br dvd_refl 1;
qed "dvd_reduce";

Goalw [dvd_def] "[| f dvd m; f dvd n; 0<n |] ==> f dvd (m mod n)";
by (Clarify_tac 1);
by (Full_simp_tac 1);
by (res_inst_tac
[("x", "(((k div ka)*ka + k mod ka) - ((f*k) div (f*ka)) * ka)")]
exI 1);
by (asm_simp_tac
(simpset() addsimps [diff_mult_distrib2, mod_mult_distrib2 RS sym,
qed "dvd_mod";

Goal "[| (k::nat) dvd (m mod n);  k dvd n |] ==> k dvd m";
by (subgoal_tac "k dvd ((m div n)*n + m mod n)" 1);
by (asm_full_simp_tac (simpset() addsimps [mod_div_equality]) 1);
qed "dvd_mod_imp_dvd";

Goalw [dvd_def]  "!!k::nat. [| (k*m) dvd (k*n); 0<k |] ==> m dvd n";
by (etac exE 1);
by (asm_full_simp_tac (simpset() addsimps mult_ac) 1);
by (Blast_tac 1);
qed "dvd_mult_cancel";

Goalw [dvd_def] "[| i dvd m; j dvd n|] ==> (i*j) dvd (m*n :: nat)";
by (Clarify_tac 1);
by (res_inst_tac [("x","k*ka")] exI 1);
by (asm_simp_tac (simpset() addsimps mult_ac) 1);
qed "mult_dvd_mono";

Goalw [dvd_def] "(i*j :: nat) dvd k ==> i dvd k";
by (full_simp_tac (simpset() addsimps [mult_assoc]) 1);
by (Blast_tac 1);
qed "dvd_mult_left";

Goalw [dvd_def] "[| k dvd n; 0 < n |] ==> k <= n";
by (Clarify_tac 1);
by (ALLGOALS (full_simp_tac (simpset() addsimps [zero_less_mult_iff])));
by (etac conjE 1);
by (rtac le_trans 1);
by (rtac (le_refl RS mult_le_mono) 2);
by (etac Suc_leI 2);
by (Simp_tac 1);
qed "dvd_imp_le";

Goalw [dvd_def] "(k dvd n) = (n mod k = 0)";
by (div_undefined_case_tac "k=0" 1);
by Safe_tac;
by (asm_simp_tac (simpset() addsimps [mult_commute]) 1);
by (res_inst_tac [("t","n"),("n1","k")] (mod_div_equality RS subst) 1);
by (stac mult_commute 1);
by (Asm_simp_tac 1);
qed "dvd_eq_mod_eq_0";
```