(* 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 1; !!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 -- the least number operator ***)
(*This version is polymorphic over type class "order"! *)
Goalw [Least_def]
"[| P(k::'a::order); ALL x. P x --> k <= x |] ==> (LEAST x. P(x)) = k";
by (blast_tac (claset() addIs [some_equality, order_antisym]) 1);
bind_thm ("Least_equality", allI RSN (2, result()));
(*LEAST and wellorderings*)
Goal "wf({(x,y::'a). x<y}) \
\ ==> P(k::'a::linorder) --> P(LEAST x. P(x)) & (LEAST x. P(x)) <= k";
by (eres_inst_tac [("a","k")] wf_induct 1);
by (rtac impI 1);
by (rtac classical 1);
by (res_inst_tac [("s","x")] (Least_equality RS ssubst) 1);
by Auto_tac;
by (auto_tac (claset(), simpset() addsimps [linorder_not_less RS sym]));
by (blast_tac (claset() addIs [order_less_trans]) 1);
bind_thm("wellorder_LeastI", result() RS mp RS conjunct1);
bind_thm("wellorder_Least_le", result() RS mp RS conjunct2);
Goal "[| wf({(x,y::'a). x<y}); k < (LEAST x. P(x)) |] \
\ ==> ~P(k::'a::linorder)";
by (rtac notI 1);
by (dtac wellorder_Least_le 1);
by (asm_full_simp_tac (simpset() addsimps [linorder_not_le RS sym]) 2);
by (fast_tac (claset() addIs []) 2);
by (assume_tac 1);
qed "wellorder_not_less_Least";
(** LEAST theorems for type "nat" by specialization **)
Goalw [less_def] "wf {(x,y::nat). x<y}";
by (rtac (wf_pred_nat RS wf_trancl RS wf_subset) 1);
by (Blast_tac 1);
qed "wf_less";
bind_thm("LeastI", wf_less RS wellorder_LeastI);
bind_thm("Least_le", wf_less RS wellorder_Least_le);
bind_thm("not_less_Least", wf_less RS wellorder_not_less_Least);
Goal "(S::nat set) ~= {} ==> EX x:S. ALL y:S. x <= y";
by (cut_facts_tac [wf_pred_nat RS wf_trancl RS (wf_eq_minimal RS iffD1)] 1);
by (dres_inst_tac [("x","S")] spec 1);
by (Asm_full_simp_tac 1);
by (etac impE 1);
by (Force_tac 1);
by (force_tac (claset(), simpset() addsimps [less_eq,not_le_iff_less]) 1);
qed "nonempty_has_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";
(** 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";
Goalw [min_def] "min (Suc m) (Suc n) = Suc(min m n)";
by (Simp_tac 1);
qed "min_Suc_Suc";
Addsimps [min_0L,min_0R,min_Suc_Suc];
Goalw [max_def] "max 0 n = (n::nat)";
by (Simp_tac 1);
qed "max_0L";
Goalw [max_def] "max n 0 = (n::nat)";
by (Simp_tac 1);
qed "max_0R";
Goalw [max_def] "max (Suc m) (Suc n) = Suc(max m n)";
by (Simp_tac 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-1) = 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::nat. (m+n=1) = (m=1 & n=0 | m=0 & n=1)";
by (case_tac "m" 1);
by (Auto_tac);
qed "add_is_1";
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 * n = n";
by (Asm_simp_tac 1);
qed "mult_1";
Goal "n * 1 = 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 "(1 <= 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 = 1) = (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 "!!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";