(*  Title:      HOL/Nat.ML

     ID:         $Id$

     Author:     Lawrence C Paulson and Tobias Nipkow

 Proofs about natural numbers and elementary arithmetic: addition,

 multiplication, etc.  Some from the Hoare example from Norbert Galm.

 *)

 (** conversion rules for nat_rec **)

 val [nat_rec_0, nat_rec_Suc] = nat.recs;

 bind_thm ("nat_rec_0", nat_rec_0);

 bind_thm ("nat_rec_Suc", nat_rec_Suc);

 (*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)

 val prems = Goal

     "[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c";

 by (simp_tac (simpset() addsimps prems) 1);

 qed "def_nat_rec_0";

 val prems = Goal

     "[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)";

 by (simp_tac (simpset() addsimps prems) 1);

 qed "def_nat_rec_Suc";

 val [nat_case_0, nat_case_Suc] = nat.cases;

 bind_thm ("nat_case_0", nat_case_0);

 bind_thm ("nat_case_Suc", nat_case_Suc);

 Goal "n ~= 0 ==> EX m. n = Suc m";

 by (case_tac "n" 1);

 by (REPEAT (Blast_tac 1));

 qed "not0_implies_Suc";

 Goal "!!n::nat. m<n ==> n ~= 0";

 by (case_tac "n" 1);

 by (ALLGOALS Asm_full_simp_tac);

 qed "gr_implies_not0";

 Goal "!!n::nat. (n ~= 0) = (0 < n)";

 by (case_tac "n" 1);

 by Auto_tac;

 qed "neq0_conv";

 AddIffs [neq0_conv];

 (*This theorem is useful with blast_tac: (n=0 ==> False) ==> 0<n *)

 bind_thm ("gr0I", [neq0_conv, notI] MRS iffD1);

 Goal "(0<n) = (EX m. n = Suc m)";

 by(fast_tac (claset() addIs [not0_implies_Suc]) 1);

 qed "gr0_conv_Suc";

 Goal "!!n::nat. (~(0 < n)) = (n=0)";

 by (rtac iffI 1);

  by (rtac ccontr 1);

  by (ALLGOALS Asm_full_simp_tac);

 qed "not_gr0";

 AddIffs [not_gr0];

 Goal "(Suc n <= m') --> (? m. m' = Suc m)";

 by (induct_tac "m'" 1);

 by  Auto_tac;

 qed_spec_mp "Suc_le_D";

 (*Useful in certain inductive arguments*)

 Goal "(m < Suc n) = (m=0 | (EX j. m = Suc j & j < n))";

 by (case_tac "m" 1);

 by Auto_tac;

 qed "less_Suc_eq_0_disj";

 val prems = Goal "[| P 0; P(Suc 0); !!k. P k ==> P (Suc (Suc k)) |] ==> P n";

 by (rtac nat_less_induct 1);

 by (case_tac "n" 1);

 by (case_tac "nat" 2);

 by (ALLGOALS (blast_tac (claset() addIs prems@[less_trans])));

 qed "nat_induct2";

 (** LEAST theorems for type "nat" by specialization **)

 bind_thm("LeastI", wellorder_LeastI);

 bind_thm("Least_le", wellorder_Least_le);

 bind_thm("not_less_Least", wellorder_not_less_Least);

 Goal "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))";

 by (case_tac "n" 1);

 by Auto_tac;  

 by (ftac LeastI 1); 

 by (dres_inst_tac [("P","%x. P (Suc x)")] LeastI 1);

 by (subgoal_tac "(LEAST x. P x) <= Suc (LEAST x. P (Suc x))" 1); 

 by (etac Least_le 2); 

 by (case_tac "LEAST x. P x" 1);

 by Auto_tac;  

 by (dres_inst_tac [("P","%x. P (Suc x)")] Least_le 1);

 by (blast_tac (claset() addIs [order_antisym]) 1); 

 qed "Least_Suc";

 Goal "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)";

 by (eatac (Least_Suc RS ssubst) 1 1);

 by (Asm_simp_tac 1);

 qed "Least_Suc2";

 (** min and max **)

 Goal "min 0 n = (0::nat)";

 by (rtac min_leastL 1);

 by (Simp_tac 1);

 qed "min_0L";

 Goal "min n 0 = (0::nat)";

 by (rtac min_leastR 1);

 by (Simp_tac 1);

 qed "min_0R";

 Goal "min (Suc m) (Suc n) = Suc (min m n)";

 by (simp_tac (simpset() addsimps [min_of_mono]) 1);

 qed "min_Suc_Suc";

 Addsimps [min_0L,min_0R,min_Suc_Suc];

 Goal "max 0 n = (n::nat)";

 by (rtac max_leastL 1);

 by (Simp_tac 1);

 qed "max_0L";

 Goal "max n 0 = (n::nat)";

 by (rtac max_leastR 1);

 by (Simp_tac 1);

 qed "max_0R";

 Goal "max (Suc m) (Suc n) = Suc(max m n)";

 by (simp_tac (simpset() addsimps [max_of_mono]) 1);

 qed "max_Suc_Suc";

 Addsimps [max_0L,max_0R,max_Suc_Suc];

 (*** Basic rewrite rules for the arithmetic operators ***)

 (** Difference **)

 Goal "0 - n = (0::nat)";

 by (induct_tac "n" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_0_eq_0";

 (*Must simplify BEFORE the induction!  (Else we get a critical pair)

   Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)

 Goal "Suc(m) - Suc(n) = m - n";

 by (Simp_tac 1);

 by (induct_tac "n" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_Suc_Suc";

 Addsimps [diff_0_eq_0, diff_Suc_Suc];

 (* Could be (and is, below) generalized in various ways;

    However, none of the generalizations are currently in the simpset,

    and I dread to think what happens if I put them in *)

 Goal "0 < n ==> Suc(n - Suc 0) = n";

 by (asm_simp_tac (simpset() addsplits [nat.split]) 1);

 qed "Suc_pred";

 Addsimps [Suc_pred];

 Delsimps [diff_Suc];

 (**** Inductive properties of the operators ****)

 (*** Addition ***)

 Goal "m + 0 = (m::nat)";

 by (induct_tac "m" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "add_0_right";

 Goal "m + Suc(n) = Suc(m+n)";

 by (induct_tac "m" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "add_Suc_right";

 Addsimps [add_0_right,add_Suc_right];

 (*Associative law for addition*)

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

 by (induct_tac "m" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "add_assoc";

 (*Commutative law for addition*)

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

 by (induct_tac "m" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "add_commute";

 Goal "x+(y+z)=y+((x+z)::nat)";

 by (rtac (add_commute RS trans) 1);

 by (rtac (add_assoc RS trans) 1);

 by (rtac (add_commute RS arg_cong) 1);

 qed "add_left_commute";

 (*Addition is an AC-operator*)

 bind_thms ("add_ac", [add_assoc, add_commute, add_left_commute]);

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

 by (induct_tac "k" 1);

 by (Simp_tac 1);

 by (Asm_simp_tac 1);

 qed "add_left_cancel";

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

 by (induct_tac "k" 1);

 by (Simp_tac 1);

 by (Asm_simp_tac 1);

 qed "add_right_cancel";

 Goal "(k + m <= k + n) = (m<=(n::nat))";

 by (induct_tac "k" 1);

 by (Simp_tac 1);

 by (Asm_simp_tac 1);

 qed "add_left_cancel_le";

 Goal "(k + m < k + n) = (m<(n::nat))";

 by (induct_tac "k" 1);

 by (Simp_tac 1);

 by (Asm_simp_tac 1);

 qed "add_left_cancel_less";

 Addsimps [add_left_cancel, add_right_cancel,

           add_left_cancel_le, add_left_cancel_less];

 (** Reasoning about m+0=0, etc. **)

 Goal "!!m::nat. (m+n = 0) = (m=0 & n=0)";

 by (case_tac "m" 1);

 by (Auto_tac);

 qed "add_is_0";

 AddIffs [add_is_0];

 Goal "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)";

 by (case_tac "m" 1);

 by (Auto_tac);

 qed "add_is_1";

 Goal "(Suc 0 = m+n) = (m = Suc 0 & n=0 | m=0 & n = Suc 0)";

 by (rtac ([eq_commute, add_is_1] MRS trans) 1);

 qed "one_is_add";

 Goal "!!m::nat. (0<m+n) = (0<m | 0<n)";

 by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);

 qed "add_gr_0";

 AddIffs [add_gr_0];

 Goal "!!m::nat. m + n = m ==> n = 0";

 by (dtac (add_0_right RS ssubst) 1);

 by (asm_full_simp_tac (simpset() addsimps [add_assoc]

                                  delsimps [add_0_right]) 1);

 qed "add_eq_self_zero";

 (**** Additional theorems about "less than" ****)

 (*Deleted less_natE; instead use less_imp_Suc_add RS exE*)

 Goal "m<n --> (EX k. n=Suc(m+k))";

 by (induct_tac "n" 1);

 by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less])));

 by (blast_tac (claset() addSEs [less_SucE]

                         addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);

 qed_spec_mp "less_imp_Suc_add";

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

 by (induct_tac "m" 1);

 by (ALLGOALS Simp_tac);

 by (etac le_SucI 1);

 qed "le_add2";

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

 by (simp_tac (simpset() addsimps add_ac) 1);

 by (rtac le_add2 1);

 qed "le_add1";

 bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));

 bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));

 Goal "(m<n) = (EX k. n=Suc(m+k))";

 by (blast_tac (claset() addSIs [less_add_Suc1, less_imp_Suc_add]) 1);

 qed "less_iff_Suc_add";

 (*"i <= j ==> i <= j+m"*)

 bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));

 (*"i <= j ==> i <= m+j"*)

 bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));

 (*"i < j ==> i < j+m"*)

 bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));

 (*"i < j ==> i < m+j"*)

 bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));

 Goal "i+j < (k::nat) --> i<k";

 by (induct_tac "j" 1);

 by (ALLGOALS Asm_simp_tac);

 by (blast_tac (claset() addDs [Suc_lessD]) 1);

 qed_spec_mp "add_lessD1";

 Goal "~ (i+j < (i::nat))";

 by (rtac notI 1);

 by (etac (add_lessD1 RS less_irrefl) 1);

 qed "not_add_less1";

 Goal "~ (j+i < (i::nat))";

 by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);

 qed "not_add_less2";

 AddIffs [not_add_less1, not_add_less2];

 Goal "m+k<=n --> m<=(n::nat)";

 by (induct_tac "k" 1);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));

 qed_spec_mp "add_leD1";

 Goal "m+k<=n ==> k<=(n::nat)";

 by (full_simp_tac (simpset() addsimps [add_commute]) 1);

 by (etac add_leD1 1);

 qed_spec_mp "add_leD2";

 Goal "m+k<=n ==> m<=n & k<=(n::nat)";

 by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);

 bind_thm ("add_leE", result() RS conjE);

 (*needs !!k for add_ac to work*)

 Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";

 by (force_tac (claset(),

               simpset() delsimps [add_Suc_right]

                         addsimps [less_iff_Suc_add,

                                   add_Suc_right RS sym] @ add_ac) 1);

 qed "less_add_eq_less";

 (*** Monotonicity of Addition ***)

 (*strict, in 1st argument*)

 Goal "i < j ==> i + k < j + (k::nat)";

 by (induct_tac "k" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "add_less_mono1";

 (*strict, in both arguments*)

 Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";

 by (rtac (add_less_mono1 RS less_trans) 1);

 by (REPEAT (assume_tac 1));

 by (induct_tac "j" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "add_less_mono";

 (*A [clumsy] way of lifting < monotonicity to <= monotonicity *)

 val [lt_mono,le] = Goal

      "[| !!i j::nat. i<j ==> f(i) < f(j);       \

 \        i <= j                                 \

 \     |] ==> f(i) <= (f(j)::nat)";

 by (cut_facts_tac [le] 1);

 by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1);

 by (blast_tac (claset() addSIs [lt_mono]) 1);

 qed "less_mono_imp_le_mono";

 (*non-strict, in 1st argument*)

 Goal "i<=j ==> i + k <= j + (k::nat)";

 by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);

 by (etac add_less_mono1 1);

 by (assume_tac 1);

 qed "add_le_mono1";

 (*non-strict, in both arguments*)

 Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";

 by (etac (add_le_mono1 RS le_trans) 1);

 by (simp_tac (simpset() addsimps [add_commute]) 1);

 qed "add_le_mono";

 (*** Multiplication ***)

 (*right annihilation in product*)

 Goal "!!m::nat. m * 0 = 0";

 by (induct_tac "m" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "mult_0_right";

 (*right successor law for multiplication*)

 Goal  "m * Suc(n) = m + (m * n)";

 by (induct_tac "m" 1);

 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));

 qed "mult_Suc_right";

 Addsimps [mult_0_right, mult_Suc_right];

 Goal "(1::nat) * n = n";

 by (Asm_simp_tac 1);

 qed "mult_1";

 Goal "n * (1::nat) = n";

 by (Asm_simp_tac 1);

 qed "mult_1_right";

 (*Commutative law for multiplication*)

 Goal "m * n = n * (m::nat)";

 by (induct_tac "m" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "mult_commute";

 (*addition distributes over multiplication*)

 Goal "(m + n)*k = (m*k) + ((n*k)::nat)";

 by (induct_tac "m" 1);

 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));

 qed "add_mult_distrib";

 Goal "k*(m + n) = (k*m) + ((k*n)::nat)";

 by (induct_tac "m" 1);

 by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));

 qed "add_mult_distrib2";

 (*Associative law for multiplication*)

 Goal "(m * n) * k = m * ((n * k)::nat)";

 by (induct_tac "m" 1);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib])));

 qed "mult_assoc";

 Goal "x*(y*z) = y*((x*z)::nat)";

 by (rtac trans 1);

 by (rtac mult_commute 1);

 by (rtac trans 1);

 by (rtac mult_assoc 1);

 by (rtac (mult_commute RS arg_cong) 1);

 qed "mult_left_commute";

 bind_thms ("mult_ac", [mult_assoc,mult_commute,mult_left_commute]);

 Goal "!!m::nat. (m*n = 0) = (m=0 | n=0)";

 by (induct_tac "m" 1);

 by (induct_tac "n" 2);

 by (ALLGOALS Asm_simp_tac);

 qed "mult_is_0";

 Addsimps [mult_is_0];

 (*** Difference ***)

 Goal "!!m::nat. m - m = 0";

 by (induct_tac "m" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_self_eq_0";

 Addsimps [diff_self_eq_0];

 (*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)

 Goal "~ m<n --> n+(m-n) = (m::nat)";

 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);

 by (ALLGOALS Asm_simp_tac);

 qed_spec_mp "add_diff_inverse";

 Goal "n<=m ==> n+(m-n) = (m::nat)";

 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);

 qed "le_add_diff_inverse";

 Goal "n<=m ==> (m-n)+n = (m::nat)";

 by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);

 qed "le_add_diff_inverse2";

 Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];

 (*** More results about difference ***)

 Goal "n <= m ==> Suc(m)-n = Suc(m-n)";

 by (etac rev_mp 1);

 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);

 by (ALLGOALS Asm_simp_tac);

 qed "Suc_diff_le";

 Goal "m - n < Suc(m)";

 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);

 by (etac less_SucE 3);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));

 qed "diff_less_Suc";

 Goal "m - n <= (m::nat)";

 by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_SucI])));

 qed "diff_le_self";

 Addsimps [diff_le_self];

 (* j<k ==> j-n < k *)

 bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);

 Goal "!!i::nat. i-j-k = i - (j+k)";

 by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_diff_left";

 Goal "(Suc m - n) - Suc k = m - n - k";

 by (simp_tac (simpset() addsimps [diff_diff_left]) 1);

 qed "Suc_diff_diff";

 Addsimps [Suc_diff_diff];

 Goal "0<n ==> n - Suc i < n";

 by (case_tac "n" 1);

 by Safe_tac;

 by (asm_simp_tac (simpset() addsimps le_simps) 1);

 qed "diff_Suc_less";

 Addsimps [diff_Suc_less];

 (*This and the next few suggested by Florian Kammueller*)

 Goal "!!i::nat. i-j-k = i-k-j";

 by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);

 qed "diff_commute";

 Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";

 by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);

 by (ALLGOALS Asm_simp_tac);

 qed_spec_mp "diff_add_assoc";

 Goal "k <= (j::nat) --> (j + i) - k = (j - k) + i";

 by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);

 qed_spec_mp "diff_add_assoc2";

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

 by (induct_tac "n" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_add_inverse";

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

 by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);

 qed "diff_add_inverse2";

 Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";

 by Safe_tac;

 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_add_inverse2])));

 qed "le_imp_diff_is_add";

 Goal "!!m::nat. (m-n = 0) = (m <= n)";

 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_is_0_eq";

 Addsimps [diff_is_0_eq];

 Goal "!!m::nat. (0<n-m) = (m<n)";

 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);

 by (ALLGOALS Asm_simp_tac);

 qed "zero_less_diff";

 Addsimps [zero_less_diff];

 Goal "i < j  ==> EX k::nat. 0<k & i+k = j";

 by (res_inst_tac [("x","j - i")] exI 1);

 by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);

 qed "less_imp_add_positive";

 Goal "P(k) --> (ALL n. P(Suc(n))--> P(n)) --> P(k-i)";

 by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);

 by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));

 qed "zero_induct_lemma";

 val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";

 by (rtac (diff_self_eq_0 RS subst) 1);

 by (rtac (zero_induct_lemma RS mp RS mp) 1);

 by (REPEAT (ares_tac ([impI,allI]@prems) 1));

 qed "zero_induct";

 Goal "(k+m) - (k+n) = m - (n::nat)";

 by (induct_tac "k" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_cancel";

 Goal "(m+k) - (n+k) = m - (n::nat)";

 by (asm_simp_tac

     (simpset() addsimps [diff_cancel, inst "n" "k" add_commute]) 1);

 qed "diff_cancel2";

 Goal "n - (n+m) = (0::nat)";

 by (induct_tac "n" 1);

 by (ALLGOALS Asm_simp_tac);

 qed "diff_add_0";

 (** Difference distributes over multiplication **)

 Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";

 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_cancel])));

 qed "diff_mult_distrib" ;

 Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";

 val mult_commute_k = read_instantiate [("m","k")] mult_commute;

 by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);

 qed "diff_mult_distrib2" ;

 (*NOT added as rewrites, since sometimes they are used from right-to-left*)

 bind_thms ("nat_distrib",

   [add_mult_distrib, add_mult_distrib2, diff_mult_distrib, diff_mult_distrib2]);

 (*** Monotonicity of Multiplication ***)

 Goal "i <= (j::nat) ==> i*k<=j*k";

 by (induct_tac "k" 1);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));

 qed "mult_le_mono1";

 Goal "i <= (j::nat) ==> k*i <= k*j";

 by (dtac mult_le_mono1 1);

 by (asm_simp_tac (simpset() addsimps [mult_commute]) 1);

 qed "mult_le_mono2";

 (* <= monotonicity, BOTH arguments*)

 Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";

 by (etac (mult_le_mono1 RS le_trans) 1);

 by (etac mult_le_mono2 1);

 qed "mult_le_mono";

 (*strict, in 1st argument; proof is by induction on k>0*)

 Goal "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";

 by (eres_inst_tac [("m1","0")] (less_imp_Suc_add RS exE) 1);

 by (Asm_simp_tac 1);

 by (induct_tac "x" 1);

 by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));

 qed "mult_less_mono2";

 Goal "!!i::nat. [| i<j; 0<k |] ==> i*k < j*k";

 by (dtac mult_less_mono2 1);

 by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));

 qed "mult_less_mono1";

 Goal "!!m::nat. (0 < m*n) = (0<m & 0<n)";

 by (induct_tac "m" 1);

 by (case_tac "n" 2);

 by (ALLGOALS Asm_simp_tac);

 qed "zero_less_mult_iff";

 Addsimps [zero_less_mult_iff];

 Goal "(Suc 0 <= m*n) = (1<=m & 1<=n)";

 by (induct_tac "m" 1);

 by (case_tac "n" 2);

 by (ALLGOALS Asm_simp_tac);

 qed "one_le_mult_iff";

 Addsimps [one_le_mult_iff];

 Goal "(m*n = Suc 0) = (m=1 & n=1)";

 by (induct_tac "m" 1);

 by (Simp_tac 1);

 by (induct_tac "n" 1);

 by (Simp_tac 1);

 by (fast_tac (claset() addss simpset()) 1);

 qed "mult_eq_1_iff";

 Addsimps [mult_eq_1_iff];

 Goal "(Suc 0 = m*n) = (m=1 & n=1)";

 by(rtac (mult_eq_1_iff RSN (2,trans)) 1);

 by (fast_tac (claset() addss simpset()) 1);

 qed "one_eq_mult_iff";

 Addsimps [one_eq_mult_iff];

 Goal "!!m::nat. (m*k < n*k) = (0<k & m<n)";

 by (safe_tac (claset() addSIs [mult_less_mono1]));

 by (case_tac "k" 1);

 by Auto_tac;  

 by (full_simp_tac (simpset() delsimps [le_0_eq]

 			     addsimps [linorder_not_le RS sym]) 1);

 by (blast_tac (claset() addIs [mult_le_mono1]) 1); 

 qed "mult_less_cancel2";

 Goal "!!m::nat. (k*m < k*n) = (0<k & m<n)";

 by (simp_tac (simpset() addsimps [mult_less_cancel2, 

                                   inst "m" "k" mult_commute]) 1);

 qed "mult_less_cancel1";

 Addsimps [mult_less_cancel1, mult_less_cancel2];

 Goal "!!m::nat. (m*k <= n*k) = (0<k --> m<=n)";

 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);

 by Auto_tac;  

 qed "mult_le_cancel2";

 Goal "!!m::nat. (k*m <= k*n) = (0<k --> m<=n)";

 by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);

 by Auto_tac;  

 qed "mult_le_cancel1";

 Addsimps [mult_le_cancel1, mult_le_cancel2];

 Goal "(m*k = n*k) = (m=n | (k = (0::nat)))";

 by (cut_facts_tac [less_linear] 1);

 by Safe_tac;

 by Auto_tac; 	

 by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));

 by (ALLGOALS Asm_full_simp_tac);

 qed "mult_cancel2";

 Goal "(k*m = k*n) = (m=n | (k = (0::nat)))";

 by (simp_tac (simpset() addsimps [mult_cancel2, inst "m" "k" mult_commute]) 1);

 qed "mult_cancel1";

 Addsimps [mult_cancel1, mult_cancel2];

 Goal "(Suc k * m < Suc k * n) = (m < n)";

 by (stac mult_less_cancel1 1);

 by (Simp_tac 1);

 qed "Suc_mult_less_cancel1";

 Goal "(Suc k * m <= Suc k * n) = (m <= n)";

 by (stac mult_le_cancel1 1);

 by (Simp_tac 1);

 qed "Suc_mult_le_cancel1";

 Goal "(Suc k * m = Suc k * n) = (m = n)";

 by (stac mult_cancel1 1);

 by (Simp_tac 1);

 qed "Suc_mult_cancel1";

 (*Lemma for gcd*)

 Goal "!!m::nat. m = m*n ==> n=1 | m=0";

 by (dtac sym 1);

 by (rtac disjCI 1);

 by (rtac nat_less_cases 1 THEN assume_tac 2);

 by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);

 by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);

 qed "mult_eq_self_implies_10";