--- a/src/HOL/Nat.ML Mon Aug 05 14:27:42 2002 +0200
+++ b/src/HOL/Nat.ML Mon Aug 05 14:27:55 2002 +0200
@@ -1,719 +1,243 @@
(* 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 **)
+(** legacy ML bindings **)
-val [nat_rec_0, nat_rec_Suc] = nat.recs;
+structure nat =
+struct
+ val distinct = thms "nat.distinct";
+ val inject = thms "nat.inject";
+ val exhaust = thm "nat.exhaust";
+ val cases = thms "nat.cases";
+ val split = thm "nat.split";
+ val split_asm = thm "nat.split_asm";
+ val induct = thm "nat.induct";
+ val recs = thms "nat.recs";
+ val simps = thms "nat.simps";
+end;
+
+val [nat_rec_0, nat_rec_Suc] = thms "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;
+val [nat_case_0, nat_case_Suc] = thms "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_assoc,add_commute] MRS
- read_instantiate[("f","op +")](thm"mk_left_commute")) 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 ([mult_assoc,mult_commute] MRS
- read_instantiate[("f","op *")](thm"mk_left_commute")) 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, diff_is_0_eq RS iffD2];
-
-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";
+val LeastI = thm "LeastI";
+val Least_Suc = thm "Least_Suc";
+val Least_Suc2 = thm "Least_Suc2";
+val Least_le = thm "Least_le";
+val One_nat_def = thm "One_nat_def";
+val Suc_Suc_eq = thm "Suc_Suc_eq";
+val Suc_def = thm "Suc_def";
+val Suc_diff_diff = thm "Suc_diff_diff";
+val Suc_diff_le = thm "Suc_diff_le";
+val Suc_inject = thm "Suc_inject";
+val Suc_leD = thm "Suc_leD";
+val Suc_leI = thm "Suc_leI";
+val Suc_le_D = thm "Suc_le_D";
+val Suc_le_eq = thm "Suc_le_eq";
+val Suc_le_lessD = thm "Suc_le_lessD";
+val Suc_le_mono = thm "Suc_le_mono";
+val Suc_lessD = thm "Suc_lessD";
+val Suc_lessE = thm "Suc_lessE";
+val Suc_lessI = thm "Suc_lessI";
+val Suc_less_SucD = thm "Suc_less_SucD";
+val Suc_less_eq = thm "Suc_less_eq";
+val Suc_mono = thm "Suc_mono";
+val Suc_mult_cancel1 = thm "Suc_mult_cancel1";
+val Suc_mult_le_cancel1 = thm "Suc_mult_le_cancel1";
+val Suc_mult_less_cancel1 = thm "Suc_mult_less_cancel1";
+val Suc_n_not_le_n = thm "Suc_n_not_le_n";
+val Suc_n_not_n = thm "Suc_n_not_n";
+val Suc_neq_Zero = thm "Suc_neq_Zero";
+val Suc_not_Zero = thm "Suc_not_Zero";
+val Suc_pred = thm "Suc_pred";
+val Zero_nat_def = thm "Zero_nat_def";
+val Zero_neq_Suc = thm "Zero_neq_Suc";
+val Zero_not_Suc = thm "Zero_not_Suc";
+val add_0 = thm "add_0";
+val add_0_right = thm "add_0_right";
+val add_Suc = thm "add_Suc";
+val add_Suc_right = thm "add_Suc_right";
+val add_ac = thms "add_ac";
+val add_assoc = thm "add_assoc";
+val add_commute = thm "add_commute";
+val add_diff_inverse = thm "add_diff_inverse";
+val add_eq_self_zero = thm "add_eq_self_zero";
+val add_gr_0 = thm "add_gr_0";
+val add_is_0 = thm "add_is_0";
+val add_is_1 = thm "add_is_1";
+val add_leD1 = thm "add_leD1";
+val add_leD2 = thm "add_leD2";
+val add_leE = thm "add_leE";
+val add_le_mono = thm "add_le_mono";
+val add_le_mono1 = thm "add_le_mono1";
+val add_left_cancel = thm "add_left_cancel";
+val add_left_cancel_le = thm "add_left_cancel_le";
+val add_left_cancel_less = thm "add_left_cancel_less";
+val add_left_commute = thm "add_left_commute";
+val add_lessD1 = thm "add_lessD1";
+val add_less_mono = thm "add_less_mono";
+val add_less_mono1 = thm "add_less_mono1";
+val add_mult_distrib = thm "add_mult_distrib";
+val add_mult_distrib2 = thm "add_mult_distrib2";
+val add_right_cancel = thm "add_right_cancel";
+val def_nat_rec_0 = thm "def_nat_rec_0";
+val def_nat_rec_Suc = thm "def_nat_rec_Suc";
+val diff_0 = thm "diff_0";
+val diff_0_eq_0 = thm "diff_0_eq_0";
+val diff_Suc = thm "diff_Suc";
+val diff_Suc_Suc = thm "diff_Suc_Suc";
+val diff_Suc_less = thm "diff_Suc_less";
+val diff_add_0 = thm "diff_add_0";
+val diff_add_assoc = thm "diff_add_assoc";
+val diff_add_assoc2 = thm "diff_add_assoc2";
+val diff_add_inverse = thm "diff_add_inverse";
+val diff_add_inverse2 = thm "diff_add_inverse2";
+val diff_cancel = thm "diff_cancel";
+val diff_cancel2 = thm "diff_cancel2";
+val diff_commute = thm "diff_commute";
+val diff_diff_left = thm "diff_diff_left";
+val diff_induct = thm "diff_induct";
+val diff_is_0_eq = thm "diff_is_0_eq";
+val diff_le_self = thm "diff_le_self";
+val diff_less_Suc = thm "diff_less_Suc";
+val diff_mult_distrib = thm "diff_mult_distrib";
+val diff_mult_distrib2 = thm "diff_mult_distrib2";
+val diff_self_eq_0 = thm "diff_self_eq_0";
+val eq_imp_le = thm "eq_imp_le";
+val gr0I = thm "gr0I";
+val gr0_conv_Suc = thm "gr0_conv_Suc";
+val gr_implies_not0 = thm "gr_implies_not0";
+val inj_Rep_Nat = thm "inj_Rep_Nat";
+val inj_Suc = thm "inj_Suc";
+val inj_on_Abs_Nat = thm "inj_on_Abs_Nat";
+val le0 = thm "le0";
+val leD = thm "leD";
+val leE = thm "leE";
+val leI = thm "leI";
+val le_0_eq = thm "le_0_eq";
+val le_SucE = thm "le_SucE";
+val le_SucI = thm "le_SucI";
+val le_Suc_eq = thm "le_Suc_eq";
+val le_add1 = thm "le_add1";
+val le_add2 = thm "le_add2";
+val le_add_diff_inverse = thm "le_add_diff_inverse";
+val le_add_diff_inverse2 = thm "le_add_diff_inverse2";
+val le_anti_sym = thm "le_anti_sym";
+val le_def = thm "le_def";
+val le_eq_less_or_eq = thm "le_eq_less_or_eq";
+val le_imp_diff_is_add = thm "le_imp_diff_is_add";
+val le_imp_less_Suc = thm "le_imp_less_Suc";
+val le_imp_less_or_eq = thm "le_imp_less_or_eq";
+val le_less_trans = thm "le_less_trans";
+val le_neq_implies_less = thm "le_neq_implies_less";
+val le_refl = thm "le_refl";
+val le_simps = thms "le_simps";
+val le_trans = thm "le_trans";
+val lessE = thm "lessE";
+val lessI = thm "lessI";
+val less_Suc0 = thm "less_Suc0";
+val less_SucE = thm "less_SucE";
+val less_SucI = thm "less_SucI";
+val less_Suc_eq = thm "less_Suc_eq";
+val less_Suc_eq_0_disj = thm "less_Suc_eq_0_disj";
+val less_Suc_eq_le = thm "less_Suc_eq_le";
+val less_add_Suc1 = thm "less_add_Suc1";
+val less_add_Suc2 = thm "less_add_Suc2";
+val less_add_eq_less = thm "less_add_eq_less";
+val less_asym = thm "less_asym";
+val less_def = thm "less_def";
+val less_eq = thm "less_eq";
+val less_iff_Suc_add = thm "less_iff_Suc_add";
+val less_imp_Suc_add = thm "less_imp_Suc_add";
+val less_imp_add_positive = thm "less_imp_add_positive";
+val less_imp_diff_less = thm "less_imp_diff_less";
+val less_imp_le = thm "less_imp_le";
+val less_irrefl = thm "less_irrefl";
+val less_le_trans = thm "less_le_trans";
+val less_linear = thm "less_linear";
+val less_mono_imp_le_mono = thm "less_mono_imp_le_mono";
+val less_not_refl = thm "less_not_refl";
+val less_not_refl2 = thm "less_not_refl2";
+val less_not_refl3 = thm "less_not_refl3";
+val less_not_sym = thm "less_not_sym";
+val less_one = thm "less_one";
+val less_or_eq_imp_le = thm "less_or_eq_imp_le";
+val less_trans = thm "less_trans";
+val less_trans_Suc = thm "less_trans_Suc";
+val less_zeroE = thm "less_zeroE";
+val max_0L = thm "max_0L";
+val max_0R = thm "max_0R";
+val max_Suc_Suc = thm "max_Suc_Suc";
+val min_0L = thm "min_0L";
+val min_0R = thm "min_0R";
+val min_Suc_Suc = thm "min_Suc_Suc";
+val mult_0 = thm "mult_0";
+val mult_0_right = thm "mult_0_right";
+val mult_1 = thm "mult_1";
+val mult_1_right = thm "mult_1_right";
+val mult_Suc = thm "mult_Suc";
+val mult_Suc_right = thm "mult_Suc_right";
+val mult_ac = thms "mult_ac";
+val mult_assoc = thm "mult_assoc";
+val mult_cancel1 = thm "mult_cancel1";
+val mult_cancel2 = thm "mult_cancel2";
+val mult_commute = thm "mult_commute";
+val mult_eq_1_iff = thm "mult_eq_1_iff";
+val mult_eq_self_implies_10 = thm "mult_eq_self_implies_10";
+val mult_is_0 = thm "mult_is_0";
+val mult_le_cancel1 = thm "mult_le_cancel1";
+val mult_le_cancel2 = thm "mult_le_cancel2";
+val mult_le_mono = thm "mult_le_mono";
+val mult_le_mono1 = thm "mult_le_mono1";
+val mult_le_mono2 = thm "mult_le_mono2";
+val mult_left_commute = thm "mult_left_commute";
+val mult_less_cancel1 = thm "mult_less_cancel1";
+val mult_less_cancel2 = thm "mult_less_cancel2";
+val mult_less_mono1 = thm "mult_less_mono1";
+val mult_less_mono2 = thm "mult_less_mono2";
+val n_not_Suc_n = thm "n_not_Suc_n";
+val nat_distrib = thms "nat_distrib";
+val nat_induct = thm "nat_induct";
+val nat_induct2 = thm "nat_induct2";
+val nat_le_linear = thm "nat_le_linear";
+val nat_less_cases = thm "nat_less_cases";
+val nat_less_induct = thm "nat_less_induct";
+val nat_less_le = thm "nat_less_le";
+val nat_neq_iff = thm "nat_neq_iff";
+val nat_not_singleton = thm "nat_not_singleton";
+val neq0_conv = thm "neq0_conv";
+val not0_implies_Suc = thm "not0_implies_Suc";
+val not_add_less1 = thm "not_add_less1";
+val not_add_less2 = thm "not_add_less2";
+val not_gr0 = thm "not_gr0";
+val not_leE = thm "not_leE";
+val not_le_iff_less = thm "not_le_iff_less";
+val not_less0 = thm "not_less0";
+val not_less_Least = thm "not_less_Least";
+val not_less_eq = thm "not_less_eq";
+val not_less_iff_le = thm "not_less_iff_le";
+val not_less_less_Suc_eq = thm "not_less_less_Suc_eq";
+val not_less_simps = thms "not_less_simps";
+val one_eq_mult_iff = thm "one_eq_mult_iff";
+val one_is_add = thm "one_is_add";
+val one_le_mult_iff = thm "one_le_mult_iff";
+val one_reorient = thm "one_reorient";
+val powerI = thm "powerI";
+val pred_nat_def = thm "pred_nat_def";
+val trans_le_add1 = thm "trans_le_add1";
+val trans_le_add2 = thm "trans_le_add2";
+val trans_less_add1 = thm "trans_less_add1";
+val trans_less_add2 = thm "trans_less_add2";
+val wf_less = thm "wf_less";
+val wf_pred_nat = thm "wf_pred_nat";
+val zero_induct = thm "zero_induct";
+val zero_induct_lemma = thm "zero_induct_lemma";
+val zero_less_Suc = thm "zero_less_Suc";
+val zero_less_diff = thm "zero_less_diff";
+val zero_less_mult_iff = thm "zero_less_mult_iff";
+val zero_reorient = thm "zero_reorient";