--- a/src/HOL/Arith.ML Tue May 20 11:34:26 1997 +0200
+++ b/src/HOL/Arith.ML Tue May 20 11:37:57 1997 +0200
@@ -4,7 +4,7 @@
Copyright 1993 University of Cambridge
Proofs about elementary arithmetic: addition, multiplication, etc.
-Tests definitions and simplifier.
+Some from the Hoare example from Norbert Galm
*)
open Arith;
@@ -124,287 +124,6 @@
qed "add_pred";
Addsimps [add_pred];
-(*** Multiplication ***)
-
-(*right annihilation in product*)
-qed_goal "mult_0_right" Arith.thy "m * 0 = 0"
- (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
-
-(*right Sucessor law for multiplication*)
-qed_goal "mult_Suc_right" Arith.thy "m * Suc(n) = m + (m * n)"
- (fn _ => [nat_ind_tac "m" 1,
- ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
-
-Addsimps [mult_0_right,mult_Suc_right];
-
-goal Arith.thy "1 * n = n";
-by (Asm_simp_tac 1);
-qed "mult_1";
-
-goal Arith.thy "n * 1 = n";
-by (Asm_simp_tac 1);
-qed "mult_1_right";
-
-(*Commutative law for multiplication*)
-qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)"
- (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
-
-(*addition distributes over multiplication*)
-qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)"
- (fn _ => [nat_ind_tac "m" 1,
- ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
-
-qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)"
- (fn _ => [nat_ind_tac "m" 1,
- ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
-
-(*Associative law for multiplication*)
-qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)"
- (fn _ => [nat_ind_tac "m" 1,
- ALLGOALS (asm_simp_tac (!simpset addsimps [add_mult_distrib]))]);
-
-qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)"
- (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
- rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
-
-val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
-
-(*** Difference ***)
-
-qed_goal "pred_Suc_diff" Arith.thy "pred(Suc m - n) = m - n"
- (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
-Addsimps [pred_Suc_diff];
-
-qed_goal "diff_self_eq_0" Arith.thy "m - m = 0"
- (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
-Addsimps [diff_self_eq_0];
-
-(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
-val [prem] = goal Arith.thy "[| ~ m<n |] ==> n+(m-n) = (m::nat)";
-by (rtac (prem RS rev_mp) 1);
-by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (ALLGOALS (Asm_simp_tac));
-qed "add_diff_inverse";
-
-
-(*** Remainder ***)
-
-goal Arith.thy "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 Arith.thy "!!m::nat. m - n <= m";
-by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
-by (ALLGOALS Asm_simp_tac);
-qed "diff_le_self";
-
-goal Arith.thy "!!n::nat. (n+m) - n = m";
-by (nat_ind_tac "n" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "diff_add_inverse";
-
-goal Arith.thy "!!n::nat.(m+n) - n = m";
-by (res_inst_tac [("m1","m")] (add_commute RS ssubst) 1);
-by (REPEAT (ares_tac [diff_add_inverse] 1));
-qed "diff_add_inverse2";
-
-goal Arith.thy "!!k::nat. (k+m) - (k+n) = m - n";
-by (nat_ind_tac "k" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "diff_cancel";
-Addsimps [diff_cancel];
-
-goal Arith.thy "!!m::nat. (m+k) - (n+k) = m - n";
-val add_commute_k = read_instantiate [("n","k")] add_commute;
-by (asm_simp_tac (!simpset addsimps ([add_commute_k])) 1);
-qed "diff_cancel2";
-Addsimps [diff_cancel2];
-
-goal Arith.thy "!!n::nat. n - (n+m) = 0";
-by (nat_ind_tac "n" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "diff_add_0";
-Addsimps [diff_add_0];
-
-(** Difference distributes over multiplication **)
-
-goal Arith.thy "!!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);
-qed "diff_mult_distrib" ;
-
-goal Arith.thy "!!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*)
-
-
-(** Less-then properties **)
-
-(*In ordinary notation: if 0<n and n<=m then m-n < m *)
-goal Arith.thy "!!m. [| 0<n; ~ m<n |] ==> m - n < m";
-by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
-by (Blast_tac 1);
-by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (ALLGOALS(asm_simp_tac(!simpset addsimps [diff_less_Suc])));
-qed "diff_less";
-
-val wf_less_trans = wf_pred_nat RS wf_trancl RSN (2, def_wfrec RS trans);
-
-goalw Nat.thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
-by (rtac refl 1);
-qed "less_eq";
-
-goal Arith.thy "(%m. m mod n) = wfrec (trancl pred_nat) \
- \ (%f j. if j<n then j else f (j-n))";
-by (simp_tac (HOL_ss addsimps [mod_def]) 1);
-val mod_def1 = result() RS eq_reflection;
-
-goal Arith.thy "!!m. m<n ==> m mod n = m";
-by (rtac (mod_def1 RS wf_less_trans) 1);
-by (Asm_simp_tac 1);
-qed "mod_less";
-
-goal Arith.thy "!!m. [| 0<n; ~m<n |] ==> m mod n = (m-n) mod n";
-by (rtac (mod_def1 RS wf_less_trans) 1);
-by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
-qed "mod_geq";
-
-
-(*** Quotient ***)
-
-goal Arith.thy "(%m. m div n) = wfrec (trancl pred_nat) \
- \ (%f j. if j<n then 0 else Suc (f (j-n)))";
-by (simp_tac (HOL_ss addsimps [div_def]) 1);
-val div_def1 = result() RS eq_reflection;
-
-goal Arith.thy "!!m. m<n ==> m div n = 0";
-by (rtac (div_def1 RS wf_less_trans) 1);
-by (Asm_simp_tac 1);
-qed "div_less";
-
-goal Arith.thy "!!M. [| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)";
-by (rtac (div_def1 RS wf_less_trans) 1);
-by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
-qed "div_geq";
-
-(*Main Result about quotient and remainder.*)
-goal Arith.thy "!!m. 0<n ==> (m div n)*n + m mod n = m";
-by (res_inst_tac [("n","m")] less_induct 1);
-by (rename_tac "k" 1); (*Variable name used in line below*)
-by (case_tac "k<n" 1);
-by (ALLGOALS (asm_simp_tac(!simpset addsimps ([add_assoc] @
- [mod_less, mod_geq, div_less, div_geq,
- add_diff_inverse, diff_less]))));
-qed "mod_div_equality";
-
-
-(*** More results about difference ***)
-
-val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0";
-by (rtac (prem RS rev_mp) 1);
-by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
-by (ALLGOALS (Asm_simp_tac));
-qed "less_imp_diff_is_0";
-
-val prems = goal Arith.thy "m-n = 0 --> n-m = 0 --> m=n";
-by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
-qed_spec_mp "diffs0_imp_equal";
-
-val [prem] = goal Arith.thy "m<n ==> 0<n-m";
-by (rtac (prem RS rev_mp) 1);
-by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (ALLGOALS (Asm_simp_tac));
-qed "less_imp_diff_positive";
-
-val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
-by (rtac (prem RS rev_mp) 1);
-by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
-by (ALLGOALS (Asm_simp_tac));
-qed "Suc_diff_n";
-
-goal Arith.thy "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
-by (simp_tac (!simpset addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]
- setloop (split_tac [expand_if])) 1);
-qed "if_Suc_diff_n";
-
-goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
-by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
-by (ALLGOALS (strip_tac THEN' Simp_tac THEN' TRY o Blast_tac));
-qed "zero_induct_lemma";
-
-val prems = goal Arith.thy "[| 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";
-
-(*13 July 1992: loaded in 105.7s*)
-
-
-(*** Further facts about mod (mainly for mutilated checkerboard ***)
-
-goal Arith.thy
- "!!m n. 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_refl2 RS not_sym]) 2);
-(* case n <= Suc(na) *)
-by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, mod_geq]) 1);
-by (etac (le_imp_less_or_eq RS disjE) 1);
-by (asm_simp_tac (!simpset addsimps [Suc_diff_n]) 1);
-by (asm_full_simp_tac (!simpset addsimps [not_less_eq RS sym,
- diff_less, mod_geq]) 1);
-by (asm_simp_tac (!simpset addsimps [mod_less]) 1);
-qed "mod_Suc";
-
-goal Arith.thy "!!m n. 0<n ==> m mod n < n";
-by (res_inst_tac [("n","m")] less_induct 1);
-by (excluded_middle_tac "na<n" 1);
-(*case na<n*)
-by (asm_simp_tac (!simpset addsimps [mod_less]) 2);
-(*case n le na*)
-by (asm_full_simp_tac (!simpset addsimps [mod_geq, diff_less]) 1);
-qed "mod_less_divisor";
-
-
-(** Evens and Odds **)
-
-(*With less_zeroE, causes case analysis on b<2*)
-AddSEs [less_SucE];
-
-goal thy "!!k b. 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 (!simpset setloop split_tac [expand_if]) 1);
-by (Blast_tac 1);
-qed "mod2_cases";
-
-goal thy "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 (Step_tac 1);
-by (ALLGOALS (asm_simp_tac (!simpset addsimps [mod_Suc])));
-qed "mod2_Suc_Suc";
-Addsimps [mod2_Suc_Suc];
-
-goal thy "(m+m) mod 2 = 0";
-by (nat_ind_tac "m" 1);
-by (simp_tac (!simpset addsimps [mod_less]) 1);
-by (asm_simp_tac (!simpset addsimps [mod2_Suc_Suc, add_Suc_right]) 1);
-qed "mod2_add_self";
-Addsimps [mod2_add_self];
-
-Delrules [less_SucE];
-
(**** Additional theorems about "less than" ****)
@@ -455,6 +174,16 @@
by (blast_tac (!claset addDs [Suc_lessD]) 1);
qed "add_lessD1";
+goal Arith.thy "!!i::nat. ~ (i+j < i)";
+br notI 1;
+be (add_lessD1 RS less_irrefl) 1;
+qed "not_add_less1";
+
+goal Arith.thy "!!i::nat. ~ (j+i < i)";
+by (simp_tac (!simpset addsimps [add_commute, not_add_less1]) 1);
+qed "not_add_less2";
+AddIffs [not_add_less1, not_add_less2];
+
goal Arith.thy "!!k::nat. m <= n ==> m <= n+k";
by (etac le_trans 1);
by (rtac le_add1 1);
@@ -531,6 +260,327 @@
by (etac add_le_mono1 1);
qed "add_le_mono";
+
+(*** Multiplication ***)
+
+(*right annihilation in product*)
+qed_goal "mult_0_right" Arith.thy "m * 0 = 0"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
+
+(*right Sucessor law for multiplication*)
+qed_goal "mult_Suc_right" Arith.thy "m * Suc(n) = m + (m * n)"
+ (fn _ => [nat_ind_tac "m" 1,
+ ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
+
+Addsimps [mult_0_right,mult_Suc_right];
+
+goal Arith.thy "1 * n = n";
+by (Asm_simp_tac 1);
+qed "mult_1";
+
+goal Arith.thy "n * 1 = n";
+by (Asm_simp_tac 1);
+qed "mult_1_right";
+
+(*Commutative law for multiplication*)
+qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
+
+(*addition distributes over multiplication*)
+qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)"
+ (fn _ => [nat_ind_tac "m" 1,
+ ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
+
+qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)"
+ (fn _ => [nat_ind_tac "m" 1,
+ ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
+
+(*Associative law for multiplication*)
+qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)"
+ (fn _ => [nat_ind_tac "m" 1,
+ ALLGOALS (asm_simp_tac (!simpset addsimps [add_mult_distrib]))]);
+
+qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)"
+ (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
+ rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
+
+val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
+
+
+(*** Difference ***)
+
+qed_goal "pred_Suc_diff" Arith.thy "pred(Suc m - n) = m - n"
+ (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
+Addsimps [pred_Suc_diff];
+
+qed_goal "diff_self_eq_0" Arith.thy "m - m = 0"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
+Addsimps [diff_self_eq_0];
+
+(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
+val [prem] = goal Arith.thy "[| ~ m<n |] ==> n+(m-n) = (m::nat)";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS (Asm_simp_tac));
+qed "add_diff_inverse";
+
+Delsimps [diff_Suc];
+
+
+(*** More results about difference ***)
+
+goal Arith.thy "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 Arith.thy "!!m::nat. m - n <= m";
+by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
+by (ALLGOALS Asm_simp_tac);
+qed "diff_le_self";
+
+goal Arith.thy "!!n::nat. (n+m) - n = m";
+by (nat_ind_tac "n" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "diff_add_inverse";
+Addsimps [diff_add_inverse];
+
+goal Arith.thy "!!n::nat.(m+n) - n = m";
+by (res_inst_tac [("m1","m")] (add_commute RS ssubst) 1);
+by (REPEAT (ares_tac [diff_add_inverse] 1));
+qed "diff_add_inverse2";
+Addsimps [diff_add_inverse2];
+
+val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
+by (ALLGOALS (Asm_simp_tac));
+qed "less_imp_diff_is_0";
+
+val prems = goal Arith.thy "m-n = 0 --> n-m = 0 --> m=n";
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
+qed_spec_mp "diffs0_imp_equal";
+
+val [prem] = goal Arith.thy "m<n ==> 0<n-m";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS (Asm_simp_tac));
+qed "less_imp_diff_positive";
+
+val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS (Asm_simp_tac));
+qed "Suc_diff_n";
+
+goal Arith.thy "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
+by (simp_tac (!simpset addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]
+ setloop (split_tac [expand_if])) 1);
+qed "if_Suc_diff_n";
+
+goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
+by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
+by (ALLGOALS (strip_tac THEN' Simp_tac THEN' TRY o Blast_tac));
+qed "zero_induct_lemma";
+
+val prems = goal Arith.thy "[| 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 Arith.thy "!!k::nat. (k+m) - (k+n) = m - n";
+by (nat_ind_tac "k" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "diff_cancel";
+Addsimps [diff_cancel];
+
+goal Arith.thy "!!m::nat. (m+k) - (n+k) = m - n";
+val add_commute_k = read_instantiate [("n","k")] add_commute;
+by (asm_simp_tac (!simpset addsimps ([add_commute_k])) 1);
+qed "diff_cancel2";
+Addsimps [diff_cancel2];
+
+(*From Clemens Ballarin*)
+goal Arith.thy "!!n::nat. [| k<=n; n<=m |] ==> (m-k) - (n-k) = m-n";
+by (subgoal_tac "k<=n --> n<=m --> (m-k) - (n-k) = m-n" 1);
+by (Asm_full_simp_tac 1);
+by (nat_ind_tac "k" 1);
+by (Simp_tac 1);
+(* Induction step *)
+by (subgoal_tac "Suc na <= m --> n <= m --> Suc na <= n --> \
+\ Suc (m - Suc na) - Suc (n - Suc na) = m-n" 1);
+by (Asm_full_simp_tac 1);
+by (blast_tac (!claset addIs [le_trans]) 1);
+by (auto_tac (!claset addIs [Suc_leD], !simpset delsimps [diff_Suc_Suc]));
+by (asm_full_simp_tac (!simpset delsimps [Suc_less_eq]
+ addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1);
+qed "diff_right_cancel";
+
+goal Arith.thy "!!n::nat. n - (n+m) = 0";
+by (nat_ind_tac "n" 1);
+by (ALLGOALS Asm_simp_tac);
+qed "diff_add_0";
+Addsimps [diff_add_0];
+
+(** Difference distributes over multiplication **)
+
+goal Arith.thy "!!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);
+qed "diff_mult_distrib" ;
+
+goal Arith.thy "!!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*)
+
+
+(** Less-then properties **)
+
+(*In ordinary notation: if 0<n and n<=m then m-n < m *)
+goal Arith.thy "!!m. [| 0<n; ~ m<n |] ==> m - n < m";
+by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
+by (Blast_tac 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS(asm_simp_tac(!simpset addsimps [diff_less_Suc])));
+qed "diff_less";
+
+val wf_less_trans = [eq_reflection, wf_pred_nat RS wf_trancl] MRS
+ def_wfrec RS trans;
+
+goalw Nat.thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
+by (rtac refl 1);
+qed "less_eq";
+
+(*** Remainder ***)
+
+goal Arith.thy "(%m. m mod n) = wfrec (trancl pred_nat) \
+ \ (%f j. if j<n then j else f (j-n))";
+by (simp_tac (!simpset addsimps [mod_def]) 1);
+qed "mod_eq";
+
+goal Arith.thy "!!m. m<n ==> m mod n = m";
+by (rtac (mod_eq RS wf_less_trans) 1);
+by (Asm_simp_tac 1);
+qed "mod_less";
+
+goal Arith.thy "!!m. [| 0<n; ~m<n |] ==> m mod n = (m-n) mod n";
+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";
+
+goal thy "!!n. 0<n ==> n mod n = 0";
+by (rtac (mod_eq RS wf_less_trans) 1);
+by (asm_simp_tac (!simpset addsimps [mod_less, diff_self_eq_0,
+ cut_def, less_eq]) 1);
+qed "mod_nn_is_0";
+
+goal thy "!!n. 0<n ==> (m+n) mod n = m mod n";
+by (subgoal_tac "(n + m) mod n = (n+m-n) mod n" 1);
+by (stac (mod_geq RS sym) 2);
+by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [add_commute])));
+qed "mod_eq_add";
+
+goal thy "!!n. 0<n ==> m*n mod n = 0";
+by (nat_ind_tac "m" 1);
+by (asm_simp_tac (!simpset addsimps [mod_less]) 1);
+by (dres_inst_tac [("m","m*n")] mod_eq_add 1);
+by (asm_full_simp_tac (!simpset addsimps [add_commute]) 1);
+qed "mod_prod_nn_is_0";
+
+
+(*** Quotient ***)
+
+goal Arith.thy "(%m. m div n) = wfrec (trancl pred_nat) \
+ \ (%f j. if j<n then 0 else Suc (f (j-n)))";
+by (simp_tac (!simpset addsimps [div_def]) 1);
+qed "div_eq";
+
+goal Arith.thy "!!m. m<n ==> m div n = 0";
+by (rtac (div_eq RS wf_less_trans) 1);
+by (Asm_simp_tac 1);
+qed "div_less";
+
+goal Arith.thy "!!M. [| 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";
+
+(*Main Result about quotient and remainder.*)
+goal Arith.thy "!!m. 0<n ==> (m div n)*n + m mod n = m";
+by (res_inst_tac [("n","m")] less_induct 1);
+by (rename_tac "k" 1); (*Variable name used in line below*)
+by (case_tac "k<n" 1);
+by (ALLGOALS (asm_simp_tac(!simpset addsimps ([add_assoc] @
+ [mod_less, mod_geq, div_less, div_geq,
+ add_diff_inverse, diff_less]))));
+qed "mod_div_equality";
+
+
+(*** Further facts about mod (mainly for mutilated checkerboard ***)
+
+goal Arith.thy
+ "!!m n. 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_refl2 RS not_sym]) 2);
+(* case n <= Suc(na) *)
+by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, mod_geq]) 1);
+by (etac (le_imp_less_or_eq RS disjE) 1);
+by (asm_simp_tac (!simpset addsimps [Suc_diff_n]) 1);
+by (asm_full_simp_tac (!simpset addsimps [not_less_eq RS sym,
+ diff_less, mod_geq]) 1);
+by (asm_simp_tac (!simpset addsimps [mod_less]) 1);
+qed "mod_Suc";
+
+goal Arith.thy "!!m n. 0<n ==> m mod n < n";
+by (res_inst_tac [("n","m")] less_induct 1);
+by (excluded_middle_tac "na<n" 1);
+(*case na<n*)
+by (asm_simp_tac (!simpset addsimps [mod_less]) 2);
+(*case n le na*)
+by (asm_full_simp_tac (!simpset addsimps [mod_geq, diff_less]) 1);
+qed "mod_less_divisor";
+
+
+(** Evens and Odds **)
+
+(*With less_zeroE, causes case analysis on b<2*)
+AddSEs [less_SucE];
+
+goal thy "!!k b. 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 (!simpset setloop split_tac [expand_if]) 1);
+by (Blast_tac 1);
+qed "mod2_cases";
+
+goal thy "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 (Step_tac 1);
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [mod_Suc])));
+qed "mod2_Suc_Suc";
+Addsimps [mod2_Suc_Suc];
+
+goal thy "(m+m) mod 2 = 0";
+by (nat_ind_tac "m" 1);
+by (simp_tac (!simpset addsimps [mod_less]) 1);
+by (asm_simp_tac (!simpset addsimps [mod2_Suc_Suc, add_Suc_right]) 1);
+qed "mod2_add_self";
+Addsimps [mod2_add_self];
+
+Delrules [less_SucE];
+
+
(*** Monotonicity of Multiplication ***)
goal Arith.thy "!!i::nat. i<=j ==> i*k<=j*k";
@@ -555,6 +605,11 @@
by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_less_mono])));
qed "mult_less_mono2";
+goal Arith.thy "!!i::nat. [| i<j; 0<k |] ==> i*k < j*k";
+bd mult_less_mono2 1;
+by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [mult_commute])));
+qed "mult_less_mono1";
+
goal Arith.thy "(0 < m*n) = (0<m & 0<n)";
by (nat_ind_tac "m" 1);
by (nat_ind_tac "n" 2);
@@ -569,6 +624,42 @@
by (fast_tac (!claset addss !simpset) 1);
qed "mult_eq_1_iff";
+goal Arith.thy "!!k. 0<k ==> (m*k < n*k) = (m<n)";
+by (safe_tac (!claset addSIs [mult_less_mono1]));
+by (cut_facts_tac [less_linear] 1);
+by (blast_tac (!claset addDs [mult_less_mono1] addEs [less_asym]) 1);
+qed "mult_less_cancel2";
+
+goal Arith.thy "!!k. 0<k ==> (k*m < k*n) = (m<n)";
+bd mult_less_cancel2 1;
+by (asm_full_simp_tac (!simpset addsimps [mult_commute]) 1);
+qed "mult_less_cancel1";
+Addsimps [mult_less_cancel1, mult_less_cancel2];
+
+goal Arith.thy "!!k. 0<k ==> (m*k = n*k) = (m=n)";
+by (cut_facts_tac [less_linear] 1);
+by(Step_tac 1);
+ba 2;
+by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
+by (ALLGOALS Asm_full_simp_tac);
+qed "mult_cancel2";
+
+goal Arith.thy "!!k. 0<k ==> (k*m = k*n) = (m=n)";
+bd mult_cancel2 1;
+by (asm_full_simp_tac (!simpset addsimps [mult_commute]) 1);
+qed "mult_cancel1";
+Addsimps [mult_cancel1, mult_cancel2];
+
+
+(*** More division laws ***)
+
+goal thy "!!n. 0<n ==> m*n div n = m";
+by (cut_inst_tac [("m", "m*n")] mod_div_equality 1);
+ba 1;
+by (asm_full_simp_tac (!simpset addsimps [mod_prod_nn_is_0]) 1);
+qed "div_prod_nn_is_m";
+Addsimps [div_prod_nn_is_m];
+
(*Cancellation law for division*)
goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n";
by (res_inst_tac [("n","m")] less_induct 1);
@@ -582,6 +673,7 @@
by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le,
le_refl RS mult_le_mono]) 1);
qed "div_cancel";
+Addsimps [div_cancel];
goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)";
by (res_inst_tac [("n","m")] less_induct 1);
@@ -609,3 +701,45 @@
qed "mult_eq_self_implies_10";
+(*** Subtraction laws -- from Clemens Ballarin ***)
+
+goal Arith.thy "!! a b c::nat. [| a < b; c <= a |] ==> a-c < b-c";
+by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
+by (Asm_full_simp_tac 1);
+by (subgoal_tac "c <= b" 1);
+by (blast_tac (!claset addIs [less_imp_le, le_trans]) 2);
+by (asm_simp_tac (!simpset addsimps [leD RS add_diff_inverse]) 1);
+qed "diff_less_mono";
+
+goal Arith.thy "!! a b c::nat. a+b < c ==> a < c-b";
+bd diff_less_mono 1;
+br le_add2 1;
+by (Asm_full_simp_tac 1);
+qed "add_less_imp_less_diff";
+
+goal Arith.thy "!! n. n <= m ==> Suc m - n = Suc (m - n)";
+br Suc_diff_n 1;
+by (asm_full_simp_tac (!simpset addsimps [le_eq_less_Suc]) 1);
+qed "Suc_diff_le";
+
+goal Arith.thy "!! n. Suc i <= n ==> Suc (n - Suc i) = n - i";
+by (asm_full_simp_tac
+ (!simpset addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1);
+qed "Suc_diff_Suc";
+
+goal Arith.thy "!! i::nat. i <= n ==> n - (n - i) = i";
+by (subgoal_tac "(n-i) + (n - (n-i)) = (n-i) + i" 1);
+by (Asm_full_simp_tac 1);
+by (asm_simp_tac (!simpset addsimps [leD RS add_diff_inverse, diff_le_self,
+ add_commute]) 1);
+qed "diff_diff_cancel";
+
+goal Arith.thy "!!k::nat. k <= n ==> m <= n + m - k";
+be rev_mp 1;
+by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
+by (Simp_tac 1);
+by (simp_tac (!simpset addsimps [less_add_Suc2, less_imp_le]) 1);
+by (Simp_tac 1);
+qed "le_add_diff";
+
+