author  paulson 
Fri, 16 Feb 1996 18:00:47 +0100  
changeset 1512  ce37c64244c0 
parent 1461  6bcb44e4d6e5 
child 1609  5324067d993f 
permissions  rwrr 
1461  1 
(* Title: ZF/arith.ML 
0  2 
ID: $Id$ 
1461  3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
0  4 
Copyright 1992 University of Cambridge 
5 

6 
For arith.thy. Arithmetic operators and their definitions 

7 

8 
Proofs about elementary arithmetic: addition, multiplication, etc. 

9 

10 
Could prove def_rec_0, def_rec_succ... 

11 
*) 

12 

13 
open Arith; 

14 

15 
(*"Difference" is subtraction of natural numbers. 

16 
There are no negative numbers; we have 

17 
m # n = 0 iff m<=n and m # n = succ(k) iff m>n. 

18 
Also, rec(m, 0, %z w.z) is pred(m). 

19 
*) 

20 

21 
(** rec  better than nat_rec; the succ case has no type requirement! **) 

22 

23 
val rec_trans = rec_def RS def_transrec RS trans; 

24 

25 
goal Arith.thy "rec(0,a,b) = a"; 

26 
by (rtac rec_trans 1); 

27 
by (rtac nat_case_0 1); 

760  28 
qed "rec_0"; 
0  29 

30 
goal Arith.thy "rec(succ(m),a,b) = b(m, rec(m,a,b))"; 

31 
by (rtac rec_trans 1); 

14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

32 
by (simp_tac (ZF_ss addsimps [nat_case_succ, nat_succI]) 1); 
760  33 
qed "rec_succ"; 
0  34 

35 
val major::prems = goal Arith.thy 

36 
"[ n: nat; \ 

37 
\ a: C(0); \ 

38 
\ !!m z. [ m: nat; z: C(m) ] ==> b(m,z): C(succ(m)) \ 

39 
\ ] ==> rec(n,a,b) : C(n)"; 

40 
by (rtac (major RS nat_induct) 1); 

41 
by (ALLGOALS 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

42 
(asm_simp_tac (ZF_ss addsimps (prems@[rec_0,rec_succ])))); 
760  43 
qed "rec_type"; 
0  44 

435  45 
val nat_le_refl = nat_into_Ord RS le_refl; 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

46 

3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

47 
val nat_typechecks = [rec_type, nat_0I, nat_1I, nat_succI, Ord_nat]; 
0  48 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

49 
val nat_simps = [rec_0, rec_succ, not_lt0, nat_0_le, le0_iff, succ_le_iff, 
1461  50 
nat_le_refl]; 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

51 

3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

52 
val nat_ss = ZF_ss addsimps (nat_simps @ nat_typechecks); 
0  53 

54 

55 
(** Addition **) 

56 

760  57 
qed_goalw "add_type" Arith.thy [add_def] 
0  58 
"[ m:nat; n:nat ] ==> m #+ n : nat" 
59 
(fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]); 

60 

760  61 
qed_goalw "add_0" Arith.thy [add_def] 
0  62 
"0 #+ n = n" 
63 
(fn _ => [ (rtac rec_0 1) ]); 

64 

760  65 
qed_goalw "add_succ" Arith.thy [add_def] 
0  66 
"succ(m) #+ n = succ(m #+ n)" 
67 
(fn _=> [ (rtac rec_succ 1) ]); 

68 

69 
(** Multiplication **) 

70 

760  71 
qed_goalw "mult_type" Arith.thy [mult_def] 
0  72 
"[ m:nat; n:nat ] ==> m #* n : nat" 
73 
(fn prems=> 

74 
[ (typechk_tac (prems@[add_type]@nat_typechecks@ZF_typechecks)) ]); 

75 

760  76 
qed_goalw "mult_0" Arith.thy [mult_def] 
0  77 
"0 #* n = 0" 
78 
(fn _ => [ (rtac rec_0 1) ]); 

79 

760  80 
qed_goalw "mult_succ" Arith.thy [mult_def] 
0  81 
"succ(m) #* n = n #+ (m #* n)" 
82 
(fn _ => [ (rtac rec_succ 1) ]); 

83 

84 
(** Difference **) 

85 

760  86 
qed_goalw "diff_type" Arith.thy [diff_def] 
0  87 
"[ m:nat; n:nat ] ==> m # n : nat" 
88 
(fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]); 

89 

760  90 
qed_goalw "diff_0" Arith.thy [diff_def] 
0  91 
"m # 0 = m" 
92 
(fn _ => [ (rtac rec_0 1) ]); 

93 

760  94 
qed_goalw "diff_0_eq_0" Arith.thy [diff_def] 
0  95 
"n:nat ==> 0 # n = 0" 
96 
(fn [prem]=> 

97 
[ (rtac (prem RS nat_induct) 1), 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

98 
(ALLGOALS (asm_simp_tac nat_ss)) ]); 
0  99 

100 
(*Must simplify BEFORE the induction!! (Else we get a critical pair) 

101 
succ(m) # succ(n) rewrites to pred(succ(m) # n) *) 

760  102 
qed_goalw "diff_succ_succ" Arith.thy [diff_def] 
0  103 
"[ m:nat; n:nat ] ==> succ(m) # succ(n) = m # n" 
104 
(fn prems=> 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

105 
[ (asm_simp_tac (nat_ss addsimps prems) 1), 
0  106 
(nat_ind_tac "n" prems 1), 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

107 
(ALLGOALS (asm_simp_tac (nat_ss addsimps prems))) ]); 
0  108 

109 
val prems = goal Arith.thy 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

110 
"[ m:nat; n:nat ] ==> m # n le m"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

111 
by (rtac (prems MRS diff_induct) 1); 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

112 
by (etac leE 3); 
0  113 
by (ALLGOALS 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

114 
(asm_simp_tac 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

115 
(nat_ss addsimps (prems @ [le_iff, diff_0, diff_0_eq_0, 
1461  116 
diff_succ_succ, nat_into_Ord])))); 
760  117 
qed "diff_le_self"; 
0  118 

119 
(*** Simplification over add, mult, diff ***) 

120 

121 
val arith_typechecks = [add_type, mult_type, diff_type]; 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

122 
val arith_simps = [add_0, add_succ, 
1461  123 
mult_0, mult_succ, 
124 
diff_0, diff_0_eq_0, diff_succ_succ]; 

0  125 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

126 
val arith_ss = nat_ss addsimps (arith_simps@arith_typechecks); 
0  127 

128 
(*** Addition ***) 

129 

130 
(*Associative law for addition*) 

760  131 
qed_goal "add_assoc" Arith.thy 
0  132 
"m:nat ==> (m #+ n) #+ k = m #+ (n #+ k)" 
133 
(fn prems=> 

134 
[ (nat_ind_tac "m" prems 1), 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

135 
(ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]); 
0  136 

137 
(*The following two lemmas are used for add_commute and sometimes 

138 
elsewhere, since they are safe for rewriting.*) 

760  139 
qed_goal "add_0_right" Arith.thy 
0  140 
"m:nat ==> m #+ 0 = m" 
141 
(fn prems=> 

142 
[ (nat_ind_tac "m" prems 1), 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

143 
(ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]); 
0  144 

760  145 
qed_goal "add_succ_right" Arith.thy 
0  146 
"m:nat ==> m #+ succ(n) = succ(m #+ n)" 
147 
(fn prems=> 

148 
[ (nat_ind_tac "m" prems 1), 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

149 
(ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]); 
0  150 

151 
(*Commutative law for addition*) 

760  152 
qed_goal "add_commute" Arith.thy 
435  153 
"!!m n. [ m:nat; n:nat ] ==> m #+ n = n #+ m" 
154 
(fn _ => 

155 
[ (nat_ind_tac "n" [] 1), 

0  156 
(ALLGOALS 
435  157 
(asm_simp_tac (arith_ss addsimps [add_0_right, add_succ_right]))) ]); 
158 

437  159 
(*for a/c rewriting*) 
760  160 
qed_goal "add_left_commute" Arith.thy 
437  161 
"!!m n k. [ m:nat; n:nat ] ==> m#+(n#+k)=n#+(m#+k)" 
162 
(fn _ => [asm_simp_tac (ZF_ss addsimps [add_assoc RS sym, add_commute]) 1]); 

435  163 

164 
(*Addition is an ACoperator*) 

165 
val add_ac = [add_assoc, add_commute, add_left_commute]; 

0  166 

167 
(*Cancellation law on the left*) 

437  168 
val [eqn,knat] = goal Arith.thy 
169 
"[ k #+ m = k #+ n; k:nat ] ==> m=n"; 

0  170 
by (rtac (eqn RS rev_mp) 1); 
171 
by (nat_ind_tac "k" [knat] 1); 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

172 
by (ALLGOALS (simp_tac arith_ss)); 
0  173 
by (fast_tac ZF_cs 1); 
760  174 
qed "add_left_cancel"; 
0  175 

176 
(*** Multiplication ***) 

177 

178 
(*right annihilation in product*) 

760  179 
qed_goal "mult_0_right" Arith.thy 
435  180 
"!!m. m:nat ==> m #* 0 = 0" 
181 
(fn _=> 

182 
[ (nat_ind_tac "m" [] 1), 

183 
(ALLGOALS (asm_simp_tac arith_ss)) ]); 

0  184 

185 
(*right successor law for multiplication*) 

760  186 
qed_goal "mult_succ_right" Arith.thy 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

187 
"!!m n. [ m:nat; n:nat ] ==> m #* succ(n) = m #+ (m #* n)" 
435  188 
(fn _ => 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

189 
[ (nat_ind_tac "m" [] 1), 
435  190 
(ALLGOALS (asm_simp_tac (arith_ss addsimps add_ac))) ]); 
0  191 

192 
(*Commutative law for multiplication*) 

760  193 
qed_goal "mult_commute" Arith.thy 
0  194 
"[ m:nat; n:nat ] ==> m #* n = n #* m" 
195 
(fn prems=> 

196 
[ (nat_ind_tac "m" prems 1), 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

197 
(ALLGOALS (asm_simp_tac 
1461  198 
(arith_ss addsimps (prems@[mult_0_right, mult_succ_right])))) ]); 
0  199 

200 
(*addition distributes over multiplication*) 

760  201 
qed_goal "add_mult_distrib" Arith.thy 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

202 
"!!m n. [ m:nat; k:nat ] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)" 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

203 
(fn _=> 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

204 
[ (etac nat_induct 1), 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

205 
(ALLGOALS (asm_simp_tac (arith_ss addsimps [add_assoc RS sym]))) ]); 
0  206 

207 
(*Distributive law on the left; requires an extra typing premise*) 

760  208 
qed_goal "add_mult_distrib_left" Arith.thy 
435  209 
"!!m. [ m:nat; n:nat; k:nat ] ==> k #* (m #+ n) = (k #* m) #+ (k #* n)" 
0  210 
(fn prems=> 
435  211 
[ (nat_ind_tac "m" [] 1), 
212 
(asm_simp_tac (arith_ss addsimps [mult_0_right]) 1), 

213 
(asm_simp_tac (arith_ss addsimps ([mult_succ_right] @ add_ac)) 1) ]); 

0  214 

215 
(*Associative law for multiplication*) 

760  216 
qed_goal "mult_assoc" Arith.thy 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

217 
"!!m n k. [ m:nat; n:nat; k:nat ] ==> (m #* n) #* k = m #* (n #* k)" 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

218 
(fn _=> 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

219 
[ (etac nat_induct 1), 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

220 
(ALLGOALS (asm_simp_tac (arith_ss addsimps [add_mult_distrib]))) ]); 
0  221 

437  222 
(*for a/c rewriting*) 
760  223 
qed_goal "mult_left_commute" Arith.thy 
437  224 
"!!m n k. [ m:nat; n:nat; k:nat ] ==> m #* (n #* k) = n #* (m #* k)" 
225 
(fn _ => [rtac (mult_commute RS trans) 1, 

226 
rtac (mult_assoc RS trans) 3, 

1461  227 
rtac (mult_commute RS subst_context) 6, 
228 
REPEAT (ares_tac [mult_type] 1)]); 

437  229 

230 
val mult_ac = [mult_assoc,mult_commute,mult_left_commute]; 

231 

0  232 

233 
(*** Difference ***) 

234 

760  235 
qed_goal "diff_self_eq_0" Arith.thy 
0  236 
"m:nat ==> m # m = 0" 
237 
(fn prems=> 

238 
[ (nat_ind_tac "m" prems 1), 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

239 
(ALLGOALS (asm_simp_tac (arith_ss addsimps prems))) ]); 
0  240 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

241 
(*Addition is the inverse of subtraction*) 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

242 
goal Arith.thy "!!m n. [ n le m; m:nat ] ==> n #+ (m#n) = m"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

243 
by (forward_tac [lt_nat_in_nat] 1); 
127  244 
by (etac nat_succI 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

245 
by (etac rev_mp 1); 
0  246 
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

247 
by (ALLGOALS (asm_simp_tac arith_ss)); 
760  248 
qed "add_diff_inverse"; 
0  249 

250 
(*Subtraction is the inverse of addition. *) 

251 
val [mnat,nnat] = goal Arith.thy 

437  252 
"[ m:nat; n:nat ] ==> (n#+m) # n = m"; 
0  253 
by (rtac (nnat RS nat_induct) 1); 
6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

254 
by (ALLGOALS (asm_simp_tac (arith_ss addsimps [mnat]))); 
760  255 
qed "diff_add_inverse"; 
0  256 

437  257 
goal Arith.thy 
258 
"!!m n. [ m:nat; n:nat ] ==> (m#+n) # n = m"; 

259 
by (res_inst_tac [("m1","m")] (add_commute RS ssubst) 1); 

260 
by (REPEAT (ares_tac [diff_add_inverse] 1)); 

760  261 
qed "diff_add_inverse2"; 
437  262 

0  263 
val [mnat,nnat] = goal Arith.thy 
264 
"[ m:nat; n:nat ] ==> n # (n#+m) = 0"; 

265 
by (rtac (nnat RS nat_induct) 1); 

6
8ce8c4d13d4d
Installation of new simplifier for ZF. Deleted all congruence rules not
lcp
parents:
0
diff
changeset

266 
by (ALLGOALS (asm_simp_tac (arith_ss addsimps [mnat]))); 
760  267 
qed "diff_add_0"; 
0  268 

269 

270 
(*** Remainder ***) 

271 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

272 
goal Arith.thy "!!m n. [ 0<n; n le m; m:nat ] ==> m # n < m"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

273 
by (forward_tac [lt_nat_in_nat] 1 THEN etac nat_succI 1); 
0  274 
by (etac rev_mp 1); 
275 
by (etac rev_mp 1); 

276 
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

277 
by (ALLGOALS (asm_simp_tac (nat_ss addsimps [diff_le_self,diff_succ_succ]))); 
760  278 
qed "div_termination"; 
0  279 

1461  280 
val div_rls = (*for mod and div*) 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

281 
nat_typechecks @ 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

282 
[Ord_transrec_type, apply_type, div_termination RS ltD, if_type, 
435  283 
nat_into_Ord, not_lt_iff_le RS iffD1]; 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

284 

435  285 
val div_ss = ZF_ss addsimps [nat_into_Ord, div_termination RS ltD, 
1461  286 
not_lt_iff_le RS iffD2]; 
0  287 

288 
(*Type checking depends upon termination!*) 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

289 
goalw Arith.thy [mod_def] "!!m n. [ 0<n; m:nat; n:nat ] ==> m mod n : nat"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

290 
by (REPEAT (ares_tac div_rls 1 ORELSE etac lt_trans 1)); 
760  291 
qed "mod_type"; 
0  292 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

293 
goal Arith.thy "!!m n. [ 0<n; m<n ] ==> m mod n = m"; 
0  294 
by (rtac (mod_def RS def_transrec RS trans) 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

295 
by (asm_simp_tac div_ss 1); 
760  296 
qed "mod_less"; 
0  297 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

298 
goal Arith.thy "!!m n. [ 0<n; n le m; m:nat ] ==> m mod n = (m#n) mod n"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

299 
by (forward_tac [lt_nat_in_nat] 1 THEN etac nat_succI 1); 
0  300 
by (rtac (mod_def RS def_transrec RS trans) 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

301 
by (asm_simp_tac div_ss 1); 
760  302 
qed "mod_geq"; 
0  303 

304 
(*** Quotient ***) 

305 

306 
(*Type checking depends upon termination!*) 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

307 
goalw Arith.thy [div_def] 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

308 
"!!m n. [ 0<n; m:nat; n:nat ] ==> m div n : nat"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

309 
by (REPEAT (ares_tac div_rls 1 ORELSE etac lt_trans 1)); 
760  310 
qed "div_type"; 
0  311 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

312 
goal Arith.thy "!!m n. [ 0<n; m<n ] ==> m div n = 0"; 
0  313 
by (rtac (div_def RS def_transrec RS trans) 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

314 
by (asm_simp_tac div_ss 1); 
760  315 
qed "div_less"; 
0  316 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

317 
goal Arith.thy 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

318 
"!!m n. [ 0<n; n le m; m:nat ] ==> m div n = succ((m#n) div n)"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

319 
by (forward_tac [lt_nat_in_nat] 1 THEN etac nat_succI 1); 
0  320 
by (rtac (div_def RS def_transrec RS trans) 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

321 
by (asm_simp_tac div_ss 1); 
760  322 
qed "div_geq"; 
0  323 

324 
(*Main Result.*) 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

325 
goal Arith.thy 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

326 
"!!m n. [ 0<n; m:nat; n:nat ] ==> (m div n)#*n #+ m mod n = m"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

327 
by (etac complete_induct 1); 
437  328 
by (excluded_middle_tac "x<n" 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

329 
(*case x<n*) 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

330 
by (asm_simp_tac (arith_ss addsimps [mod_less, div_less]) 2); 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

331 
(*case n le x*) 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

332 
by (asm_full_simp_tac 
435  333 
(arith_ss addsimps [not_lt_iff_le, nat_into_Ord, 
1461  334 
mod_geq, div_geq, add_assoc, 
335 
div_termination RS ltD, add_diff_inverse]) 1); 

760  336 
qed "mod_div_equality"; 
0  337 

338 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

339 
(**** Additional theorems about "le" ****) 
0  340 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

341 
goal Arith.thy "!!m n. [ m:nat; n:nat ] ==> m le m #+ n"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

342 
by (etac nat_induct 1); 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

343 
by (ALLGOALS (asm_simp_tac arith_ss)); 
760  344 
qed "add_le_self"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

345 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

346 
goal Arith.thy "!!m n. [ m:nat; n:nat ] ==> m le n #+ m"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

347 
by (rtac (add_commute RS ssubst) 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

348 
by (REPEAT (ares_tac [add_le_self] 1)); 
760  349 
qed "add_le_self2"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

350 

1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

351 
(** Monotonicity of addition **) 
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

352 

1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

353 
(*strict, in 1st argument*) 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

354 
goal Arith.thy "!!i j k. [ i<j; j:nat; k:nat ] ==> i#+k < j#+k"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

355 
by (forward_tac [lt_nat_in_nat] 1); 
127  356 
by (assume_tac 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

357 
by (etac succ_lt_induct 1); 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

358 
by (ALLGOALS (asm_simp_tac (arith_ss addsimps [leI]))); 
760  359 
qed "add_lt_mono1"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

360 

1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

361 
(*strict, in both arguments*) 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

362 
goal Arith.thy "!!i j k l. [ i<j; k<l; j:nat; l:nat ] ==> i#+k < j#+l"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

363 
by (rtac (add_lt_mono1 RS lt_trans) 1); 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

364 
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat] 1)); 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

365 
by (EVERY [rtac (add_commute RS ssubst) 1, 
1461  366 
rtac (add_commute RS ssubst) 3, 
367 
rtac add_lt_mono1 5]); 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

368 
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat] 1)); 
760  369 
qed "add_lt_mono"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

370 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

371 
(*A [clumsy] way of lifting < monotonicity to le monotonicity *) 
435  372 
val lt_mono::ford::prems = goal Ordinal.thy 
1461  373 
"[ !!i j. [ i<j; j:k ] ==> f(i) < f(j); \ 
374 
\ !!i. i:k ==> Ord(f(i)); \ 

375 
\ i le j; j:k \ 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

376 
\ ] ==> f(i) le f(j)"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

377 
by (cut_facts_tac prems 1); 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

378 
by (fast_tac (lt_cs addSIs [lt_mono,ford] addSEs [leE]) 1); 
760  379 
qed "Ord_lt_mono_imp_le_mono"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

380 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

381 
(*le monotonicity, 1st argument*) 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

382 
goal Arith.thy 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

383 
"!!i j k. [ i le j; j:nat; k:nat ] ==> i#+k le j#+k"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

384 
by (res_inst_tac [("f", "%j.j#+k")] Ord_lt_mono_imp_le_mono 1); 
435  385 
by (REPEAT (ares_tac [add_lt_mono1, add_type RS nat_into_Ord] 1)); 
760  386 
qed "add_le_mono1"; 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

387 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

388 
(* le monotonicity, BOTH arguments*) 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

389 
goal Arith.thy 
25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

390 
"!!i j k. [ i le j; k le l; j:nat; l:nat ] ==> i#+k le j#+l"; 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

391 
by (rtac (add_le_mono1 RS le_trans) 1); 
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

392 
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat, nat_succI] 1)); 
14
1c0926788772
ex/{bin.ML,comb.ML,prop.ML}: replaced NewSext by Syntax.simple_sext
lcp
parents:
6
diff
changeset

393 
by (EVERY [rtac (add_commute RS ssubst) 1, 
1461  394 
rtac (add_commute RS ssubst) 3, 
395 
rtac add_le_mono1 5]); 

25
3ac1c0c0016e
ordinal: DEFINITION of < and le to replace : and <= on ordinals! Many
lcp
parents:
14
diff
changeset

396 
by (REPEAT (eresolve_tac [asm_rl, lt_nat_in_nat, nat_succI] 1)); 
760  397 
qed "add_le_mono"; 