src/ZF/Arith.ML
 author clasohm Thu Sep 16 12:20:38 1993 +0200 (1993-09-16) changeset 0 a5a9c433f639 child 6 8ce8c4d13d4d permissions -rw-r--r--
Initial revision
 clasohm@0 ` 1` ```(* Title: ZF/arith.ML ``` clasohm@0 ` 2` ``` ID: \$Id\$ ``` clasohm@0 ` 3` ``` Author: Lawrence C Paulson, Cambridge University Computer Laboratory ``` clasohm@0 ` 4` ``` Copyright 1992 University of Cambridge ``` clasohm@0 ` 5` clasohm@0 ` 6` ```For arith.thy. Arithmetic operators and their definitions ``` clasohm@0 ` 7` clasohm@0 ` 8` ```Proofs about elementary arithmetic: addition, multiplication, etc. ``` clasohm@0 ` 9` clasohm@0 ` 10` ```Could prove def_rec_0, def_rec_succ... ``` clasohm@0 ` 11` ```*) ``` clasohm@0 ` 12` clasohm@0 ` 13` ```open Arith; ``` clasohm@0 ` 14` clasohm@0 ` 15` ```(*"Difference" is subtraction of natural numbers. ``` clasohm@0 ` 16` ``` There are no negative numbers; we have ``` clasohm@0 ` 17` ``` m #- n = 0 iff m<=n and m #- n = succ(k) iff m>n. ``` clasohm@0 ` 18` ``` Also, rec(m, 0, %z w.z) is pred(m). ``` clasohm@0 ` 19` ```*) ``` clasohm@0 ` 20` clasohm@0 ` 21` ```(** rec -- better than nat_rec; the succ case has no type requirement! **) ``` clasohm@0 ` 22` clasohm@0 ` 23` ```val rec_trans = rec_def RS def_transrec RS trans; ``` clasohm@0 ` 24` clasohm@0 ` 25` ```goal Arith.thy "rec(0,a,b) = a"; ``` clasohm@0 ` 26` ```by (rtac rec_trans 1); ``` clasohm@0 ` 27` ```by (rtac nat_case_0 1); ``` clasohm@0 ` 28` ```val rec_0 = result(); ``` clasohm@0 ` 29` clasohm@0 ` 30` ```goal Arith.thy "rec(succ(m),a,b) = b(m, rec(m,a,b))"; ``` clasohm@0 ` 31` ```val rec_ss = ZF_ss ``` clasohm@0 ` 32` ``` addcongs (mk_typed_congs Arith.thy [("b", "[i,i]=>i")]) ``` clasohm@0 ` 33` ``` addrews [nat_case_succ, nat_succI]; ``` clasohm@0 ` 34` ```by (rtac rec_trans 1); ``` clasohm@0 ` 35` ```by (SIMP_TAC rec_ss 1); ``` clasohm@0 ` 36` ```val rec_succ = result(); ``` clasohm@0 ` 37` clasohm@0 ` 38` ```val major::prems = goal Arith.thy ``` clasohm@0 ` 39` ``` "[| n: nat; \ ``` clasohm@0 ` 40` ```\ a: C(0); \ ``` clasohm@0 ` 41` ```\ !!m z. [| m: nat; z: C(m) |] ==> b(m,z): C(succ(m)) \ ``` clasohm@0 ` 42` ```\ |] ==> rec(n,a,b) : C(n)"; ``` clasohm@0 ` 43` ```by (rtac (major RS nat_induct) 1); ``` clasohm@0 ` 44` ```by (ALLGOALS ``` clasohm@0 ` 45` ``` (ASM_SIMP_TAC (ZF_ss addrews (prems@[rec_0,rec_succ])))); ``` clasohm@0 ` 46` ```val rec_type = result(); ``` clasohm@0 ` 47` clasohm@0 ` 48` ```val prems = goalw Arith.thy [rec_def] ``` clasohm@0 ` 49` ``` "[| n=n'; a=a'; !!m z. b(m,z)=b'(m,z) \ ``` clasohm@0 ` 50` ```\ |] ==> rec(n,a,b)=rec(n',a',b')"; ``` clasohm@0 ` 51` ```by (SIMP_TAC (ZF_ss addcongs [transrec_cong,nat_case_cong] ``` clasohm@0 ` 52` ``` addrews (prems RL [sym])) 1); ``` clasohm@0 ` 53` ```val rec_cong = result(); ``` clasohm@0 ` 54` clasohm@0 ` 55` ```val nat_typechecks = [rec_type,nat_0I,nat_1I,nat_succI,Ord_nat]; ``` clasohm@0 ` 56` clasohm@0 ` 57` ```val nat_ss = ZF_ss addcongs [nat_case_cong,rec_cong] ``` clasohm@0 ` 58` ``` addrews ([rec_0,rec_succ] @ nat_typechecks); ``` clasohm@0 ` 59` clasohm@0 ` 60` clasohm@0 ` 61` ```(** Addition **) ``` clasohm@0 ` 62` clasohm@0 ` 63` ```val add_type = prove_goalw Arith.thy [add_def] ``` clasohm@0 ` 64` ``` "[| m:nat; n:nat |] ==> m #+ n : nat" ``` clasohm@0 ` 65` ``` (fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]); ``` clasohm@0 ` 66` clasohm@0 ` 67` ```val add_0 = prove_goalw Arith.thy [add_def] ``` clasohm@0 ` 68` ``` "0 #+ n = n" ``` clasohm@0 ` 69` ``` (fn _ => [ (rtac rec_0 1) ]); ``` clasohm@0 ` 70` clasohm@0 ` 71` ```val add_succ = prove_goalw Arith.thy [add_def] ``` clasohm@0 ` 72` ``` "succ(m) #+ n = succ(m #+ n)" ``` clasohm@0 ` 73` ``` (fn _=> [ (rtac rec_succ 1) ]); ``` clasohm@0 ` 74` clasohm@0 ` 75` ```(** Multiplication **) ``` clasohm@0 ` 76` clasohm@0 ` 77` ```val mult_type = prove_goalw Arith.thy [mult_def] ``` clasohm@0 ` 78` ``` "[| m:nat; n:nat |] ==> m #* n : nat" ``` clasohm@0 ` 79` ``` (fn prems=> ``` clasohm@0 ` 80` ``` [ (typechk_tac (prems@[add_type]@nat_typechecks@ZF_typechecks)) ]); ``` clasohm@0 ` 81` clasohm@0 ` 82` ```val mult_0 = prove_goalw Arith.thy [mult_def] ``` clasohm@0 ` 83` ``` "0 #* n = 0" ``` clasohm@0 ` 84` ``` (fn _ => [ (rtac rec_0 1) ]); ``` clasohm@0 ` 85` clasohm@0 ` 86` ```val mult_succ = prove_goalw Arith.thy [mult_def] ``` clasohm@0 ` 87` ``` "succ(m) #* n = n #+ (m #* n)" ``` clasohm@0 ` 88` ``` (fn _ => [ (rtac rec_succ 1) ]); ``` clasohm@0 ` 89` clasohm@0 ` 90` ```(** Difference **) ``` clasohm@0 ` 91` clasohm@0 ` 92` ```val diff_type = prove_goalw Arith.thy [diff_def] ``` clasohm@0 ` 93` ``` "[| m:nat; n:nat |] ==> m #- n : nat" ``` clasohm@0 ` 94` ``` (fn prems=> [ (typechk_tac (prems@nat_typechecks@ZF_typechecks)) ]); ``` clasohm@0 ` 95` clasohm@0 ` 96` ```val diff_0 = prove_goalw Arith.thy [diff_def] ``` clasohm@0 ` 97` ``` "m #- 0 = m" ``` clasohm@0 ` 98` ``` (fn _ => [ (rtac rec_0 1) ]); ``` clasohm@0 ` 99` clasohm@0 ` 100` ```val diff_0_eq_0 = prove_goalw Arith.thy [diff_def] ``` clasohm@0 ` 101` ``` "n:nat ==> 0 #- n = 0" ``` clasohm@0 ` 102` ``` (fn [prem]=> ``` clasohm@0 ` 103` ``` [ (rtac (prem RS nat_induct) 1), ``` clasohm@0 ` 104` ``` (ALLGOALS (ASM_SIMP_TAC nat_ss)) ]); ``` clasohm@0 ` 105` clasohm@0 ` 106` ```(*Must simplify BEFORE the induction!! (Else we get a critical pair) ``` clasohm@0 ` 107` ``` succ(m) #- succ(n) rewrites to pred(succ(m) #- n) *) ``` clasohm@0 ` 108` ```val diff_succ_succ = prove_goalw Arith.thy [diff_def] ``` clasohm@0 ` 109` ``` "[| m:nat; n:nat |] ==> succ(m) #- succ(n) = m #- n" ``` clasohm@0 ` 110` ``` (fn prems=> ``` clasohm@0 ` 111` ``` [ (ASM_SIMP_TAC (nat_ss addrews prems) 1), ``` clasohm@0 ` 112` ``` (nat_ind_tac "n" prems 1), ``` clasohm@0 ` 113` ``` (ALLGOALS (ASM_SIMP_TAC (nat_ss addrews prems))) ]); ``` clasohm@0 ` 114` clasohm@0 ` 115` ```val prems = goal Arith.thy ``` clasohm@0 ` 116` ``` "[| m:nat; n:nat |] ==> m #- n : succ(m)"; ``` clasohm@0 ` 117` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@0 ` 118` ```by (resolve_tac prems 1); ``` clasohm@0 ` 119` ```by (resolve_tac prems 1); ``` clasohm@0 ` 120` ```by (etac succE 3); ``` clasohm@0 ` 121` ```by (ALLGOALS ``` clasohm@0 ` 122` ``` (ASM_SIMP_TAC ``` clasohm@0 ` 123` ``` (nat_ss addrews (prems@[diff_0,diff_0_eq_0,diff_succ_succ])))); ``` clasohm@0 ` 124` ```val diff_leq = result(); ``` clasohm@0 ` 125` clasohm@0 ` 126` ```(*** Simplification over add, mult, diff ***) ``` clasohm@0 ` 127` clasohm@0 ` 128` ```val arith_typechecks = [add_type, mult_type, diff_type]; ``` clasohm@0 ` 129` ```val arith_rews = [add_0, add_succ, ``` clasohm@0 ` 130` ``` mult_0, mult_succ, ``` clasohm@0 ` 131` ``` diff_0, diff_0_eq_0, diff_succ_succ]; ``` clasohm@0 ` 132` clasohm@0 ` 133` ```val arith_congs = mk_congs Arith.thy ["op #+", "op #-", "op #*"]; ``` clasohm@0 ` 134` clasohm@0 ` 135` ```val arith_ss = nat_ss addcongs arith_congs ``` clasohm@0 ` 136` ``` addrews (arith_rews@arith_typechecks); ``` clasohm@0 ` 137` clasohm@0 ` 138` ```(*** Addition ***) ``` clasohm@0 ` 139` clasohm@0 ` 140` ```(*Associative law for addition*) ``` clasohm@0 ` 141` ```val add_assoc = prove_goal Arith.thy ``` clasohm@0 ` 142` ``` "m:nat ==> (m #+ n) #+ k = m #+ (n #+ k)" ``` clasohm@0 ` 143` ``` (fn prems=> ``` clasohm@0 ` 144` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 145` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]); ``` clasohm@0 ` 146` clasohm@0 ` 147` ```(*The following two lemmas are used for add_commute and sometimes ``` clasohm@0 ` 148` ``` elsewhere, since they are safe for rewriting.*) ``` clasohm@0 ` 149` ```val add_0_right = prove_goal Arith.thy ``` clasohm@0 ` 150` ``` "m:nat ==> m #+ 0 = m" ``` clasohm@0 ` 151` ``` (fn prems=> ``` clasohm@0 ` 152` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 153` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]); ``` clasohm@0 ` 154` clasohm@0 ` 155` ```val add_succ_right = prove_goal Arith.thy ``` clasohm@0 ` 156` ``` "m:nat ==> m #+ succ(n) = succ(m #+ n)" ``` clasohm@0 ` 157` ``` (fn prems=> ``` clasohm@0 ` 158` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 159` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]); ``` clasohm@0 ` 160` clasohm@0 ` 161` ```(*Commutative law for addition*) ``` clasohm@0 ` 162` ```val add_commute = prove_goal Arith.thy ``` clasohm@0 ` 163` ``` "[| m:nat; n:nat |] ==> m #+ n = n #+ m" ``` clasohm@0 ` 164` ``` (fn prems=> ``` clasohm@0 ` 165` ``` [ (nat_ind_tac "n" prems 1), ``` clasohm@0 ` 166` ``` (ALLGOALS ``` clasohm@0 ` 167` ``` (ASM_SIMP_TAC ``` clasohm@0 ` 168` ``` (arith_ss addrews (prems@[add_0_right, add_succ_right])))) ]); ``` clasohm@0 ` 169` clasohm@0 ` 170` ```(*Cancellation law on the left*) ``` clasohm@0 ` 171` ```val [knat,eqn] = goal Arith.thy ``` clasohm@0 ` 172` ``` "[| k:nat; k #+ m = k #+ n |] ==> m=n"; ``` clasohm@0 ` 173` ```by (rtac (eqn RS rev_mp) 1); ``` clasohm@0 ` 174` ```by (nat_ind_tac "k" [knat] 1); ``` clasohm@0 ` 175` ```by (ALLGOALS (SIMP_TAC arith_ss)); ``` clasohm@0 ` 176` ```by (fast_tac ZF_cs 1); ``` clasohm@0 ` 177` ```val add_left_cancel = result(); ``` clasohm@0 ` 178` clasohm@0 ` 179` ```(*** Multiplication ***) ``` clasohm@0 ` 180` clasohm@0 ` 181` ```(*right annihilation in product*) ``` clasohm@0 ` 182` ```val mult_0_right = prove_goal Arith.thy ``` clasohm@0 ` 183` ``` "m:nat ==> m #* 0 = 0" ``` clasohm@0 ` 184` ``` (fn prems=> ``` clasohm@0 ` 185` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 186` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]); ``` clasohm@0 ` 187` clasohm@0 ` 188` ```(*right successor law for multiplication*) ``` clasohm@0 ` 189` ```val mult_succ_right = prove_goal Arith.thy ``` clasohm@0 ` 190` ``` "[| m:nat; n:nat |] ==> m #* succ(n) = m #+ (m #* n)" ``` clasohm@0 ` 191` ``` (fn prems=> ``` clasohm@0 ` 192` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 193` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews ([add_assoc RS sym]@prems)))), ``` clasohm@0 ` 194` ``` (*The final goal requires the commutative law for addition*) ``` clasohm@0 ` 195` ``` (REPEAT (ares_tac (prems@[refl,add_commute]@ZF_congs@arith_congs) 1)) ]); ``` clasohm@0 ` 196` clasohm@0 ` 197` ```(*Commutative law for multiplication*) ``` clasohm@0 ` 198` ```val mult_commute = prove_goal Arith.thy ``` clasohm@0 ` 199` ``` "[| m:nat; n:nat |] ==> m #* n = n #* m" ``` clasohm@0 ` 200` ``` (fn prems=> ``` clasohm@0 ` 201` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 202` ``` (ALLGOALS (ASM_SIMP_TAC ``` clasohm@0 ` 203` ``` (arith_ss addrews (prems@[mult_0_right, mult_succ_right])))) ]); ``` clasohm@0 ` 204` clasohm@0 ` 205` ```(*addition distributes over multiplication*) ``` clasohm@0 ` 206` ```val add_mult_distrib = prove_goal Arith.thy ``` clasohm@0 ` 207` ``` "[| m:nat; k:nat |] ==> (m #+ n) #* k = (m #* k) #+ (n #* k)" ``` clasohm@0 ` 208` ``` (fn prems=> ``` clasohm@0 ` 209` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 210` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews ([add_assoc RS sym]@prems)))) ]); ``` clasohm@0 ` 211` clasohm@0 ` 212` ```(*Distributive law on the left; requires an extra typing premise*) ``` clasohm@0 ` 213` ```val add_mult_distrib_left = prove_goal Arith.thy ``` clasohm@0 ` 214` ``` "[| m:nat; n:nat; k:nat |] ==> k #* (m #+ n) = (k #* m) #+ (k #* n)" ``` clasohm@0 ` 215` ``` (fn prems=> ``` clasohm@0 ` 216` ``` let val mult_commute' = read_instantiate [("m","k")] mult_commute ``` clasohm@0 ` 217` ``` val ss = arith_ss addrews ([mult_commute',add_mult_distrib]@prems) ``` clasohm@0 ` 218` ``` in [ (SIMP_TAC ss 1) ] ``` clasohm@0 ` 219` ``` end); ``` clasohm@0 ` 220` clasohm@0 ` 221` ```(*Associative law for multiplication*) ``` clasohm@0 ` 222` ```val mult_assoc = prove_goal Arith.thy ``` clasohm@0 ` 223` ``` "[| m:nat; n:nat; k:nat |] ==> (m #* n) #* k = m #* (n #* k)" ``` clasohm@0 ` 224` ``` (fn prems=> ``` clasohm@0 ` 225` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 226` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews (prems@[add_mult_distrib])))) ]); ``` clasohm@0 ` 227` clasohm@0 ` 228` clasohm@0 ` 229` ```(*** Difference ***) ``` clasohm@0 ` 230` clasohm@0 ` 231` ```val diff_self_eq_0 = prove_goal Arith.thy ``` clasohm@0 ` 232` ``` "m:nat ==> m #- m = 0" ``` clasohm@0 ` 233` ``` (fn prems=> ``` clasohm@0 ` 234` ``` [ (nat_ind_tac "m" prems 1), ``` clasohm@0 ` 235` ``` (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews prems))) ]); ``` clasohm@0 ` 236` clasohm@0 ` 237` ```(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *) ``` clasohm@0 ` 238` ```val notless::prems = goal Arith.thy ``` clasohm@0 ` 239` ``` "[| ~m:n; m:nat; n:nat |] ==> n #+ (m#-n) = m"; ``` clasohm@0 ` 240` ```by (rtac (notless RS rev_mp) 1); ``` clasohm@0 ` 241` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@0 ` 242` ```by (resolve_tac prems 1); ``` clasohm@0 ` 243` ```by (resolve_tac prems 1); ``` clasohm@0 ` 244` ```by (ALLGOALS (ASM_SIMP_TAC ``` clasohm@0 ` 245` ``` (arith_ss addrews (prems@[succ_mem_succ_iff, Ord_0_mem_succ, ``` clasohm@0 ` 246` ``` naturals_are_ordinals])))); ``` clasohm@0 ` 247` ```val add_diff_inverse = result(); ``` clasohm@0 ` 248` clasohm@0 ` 249` clasohm@0 ` 250` ```(*Subtraction is the inverse of addition. *) ``` clasohm@0 ` 251` ```val [mnat,nnat] = goal Arith.thy ``` clasohm@0 ` 252` ``` "[| m:nat; n:nat |] ==> (n#+m) #-n = m"; ``` clasohm@0 ` 253` ```by (rtac (nnat RS nat_induct) 1); ``` clasohm@0 ` 254` ```by (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews [mnat]))); ``` clasohm@0 ` 255` ```val diff_add_inverse = result(); ``` clasohm@0 ` 256` clasohm@0 ` 257` ```val [mnat,nnat] = goal Arith.thy ``` clasohm@0 ` 258` ``` "[| m:nat; n:nat |] ==> n #- (n#+m) = 0"; ``` clasohm@0 ` 259` ```by (rtac (nnat RS nat_induct) 1); ``` clasohm@0 ` 260` ```by (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews [mnat]))); ``` clasohm@0 ` 261` ```val diff_add_0 = result(); ``` clasohm@0 ` 262` clasohm@0 ` 263` clasohm@0 ` 264` ```(*** Remainder ***) ``` clasohm@0 ` 265` clasohm@0 ` 266` ```(*In ordinary notation: if 0 m #- n : m"; ``` clasohm@0 ` 269` ```by (cut_facts_tac prems 1); ``` clasohm@0 ` 270` ```by (etac rev_mp 1); ``` clasohm@0 ` 271` ```by (etac rev_mp 1); ``` clasohm@0 ` 272` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@0 ` 273` ```by (resolve_tac prems 1); ``` clasohm@0 ` 274` ```by (resolve_tac prems 1); ``` clasohm@0 ` 275` ```by (ALLGOALS (ASM_SIMP_TAC ``` clasohm@0 ` 276` ``` (nat_ss addrews (prems@[diff_leq,diff_succ_succ])))); ``` clasohm@0 ` 277` ```val div_termination = result(); ``` clasohm@0 ` 278` clasohm@0 ` 279` ```val div_rls = ``` clasohm@0 ` 280` ``` [Ord_transrec_type, apply_type, div_termination, if_type] @ ``` clasohm@0 ` 281` ``` nat_typechecks; ``` clasohm@0 ` 282` clasohm@0 ` 283` ```(*Type checking depends upon termination!*) ``` clasohm@0 ` 284` ```val prems = goalw Arith.thy [mod_def] ``` clasohm@0 ` 285` ``` "[| 0:n; m:nat; n:nat |] ==> m mod n : nat"; ``` clasohm@0 ` 286` ```by (REPEAT (ares_tac (prems @ div_rls) 1 ORELSE etac Ord_trans 1)); ``` clasohm@0 ` 287` ```val mod_type = result(); ``` clasohm@0 ` 288` clasohm@0 ` 289` ```val div_ss = ZF_ss addrews [naturals_are_ordinals,div_termination]; ``` clasohm@0 ` 290` clasohm@0 ` 291` ```val prems = goal Arith.thy "[| 0:n; m:n; m:nat; n:nat |] ==> m mod n = m"; ``` clasohm@0 ` 292` ```by (rtac (mod_def RS def_transrec RS trans) 1); ``` clasohm@0 ` 293` ```by (SIMP_TAC (div_ss addrews prems) 1); ``` clasohm@0 ` 294` ```val mod_less = result(); ``` clasohm@0 ` 295` clasohm@0 ` 296` ```val prems = goal Arith.thy ``` clasohm@0 ` 297` ``` "[| 0:n; ~m:n; m:nat; n:nat |] ==> m mod n = (m#-n) mod n"; ``` clasohm@0 ` 298` ```by (rtac (mod_def RS def_transrec RS trans) 1); ``` clasohm@0 ` 299` ```by (SIMP_TAC (div_ss addrews prems) 1); ``` clasohm@0 ` 300` ```val mod_geq = result(); ``` clasohm@0 ` 301` clasohm@0 ` 302` ```(*** Quotient ***) ``` clasohm@0 ` 303` clasohm@0 ` 304` ```(*Type checking depends upon termination!*) ``` clasohm@0 ` 305` ```val prems = goalw Arith.thy [div_def] ``` clasohm@0 ` 306` ``` "[| 0:n; m:nat; n:nat |] ==> m div n : nat"; ``` clasohm@0 ` 307` ```by (REPEAT (ares_tac (prems @ div_rls) 1 ORELSE etac Ord_trans 1)); ``` clasohm@0 ` 308` ```val div_type = result(); ``` clasohm@0 ` 309` clasohm@0 ` 310` ```val prems = goal Arith.thy ``` clasohm@0 ` 311` ``` "[| 0:n; m:n; m:nat; n:nat |] ==> m div n = 0"; ``` clasohm@0 ` 312` ```by (rtac (div_def RS def_transrec RS trans) 1); ``` clasohm@0 ` 313` ```by (SIMP_TAC (div_ss addrews prems) 1); ``` clasohm@0 ` 314` ```val div_less = result(); ``` clasohm@0 ` 315` clasohm@0 ` 316` ```val prems = goal Arith.thy ``` clasohm@0 ` 317` ``` "[| 0:n; ~m:n; m:nat; n:nat |] ==> m div n = succ((m#-n) div n)"; ``` clasohm@0 ` 318` ```by (rtac (div_def RS def_transrec RS trans) 1); ``` clasohm@0 ` 319` ```by (SIMP_TAC (div_ss addrews prems) 1); ``` clasohm@0 ` 320` ```val div_geq = result(); ``` clasohm@0 ` 321` clasohm@0 ` 322` ```(*Main Result.*) ``` clasohm@0 ` 323` ```val prems = goal Arith.thy ``` clasohm@0 ` 324` ``` "[| 0:n; m:nat; n:nat |] ==> (m div n)#*n #+ m mod n = m"; ``` clasohm@0 ` 325` ```by (res_inst_tac [("i","m")] complete_induct 1); ``` clasohm@0 ` 326` ```by (resolve_tac prems 1); ``` clasohm@0 ` 327` ```by (res_inst_tac [("Q","x:n")] (excluded_middle RS disjE) 1); ``` clasohm@0 ` 328` ```by (ALLGOALS ``` clasohm@0 ` 329` ``` (ASM_SIMP_TAC ``` clasohm@0 ` 330` ``` (arith_ss addrews ([mod_type,div_type] @ prems @ ``` clasohm@0 ` 331` ``` [mod_less,mod_geq, div_less, div_geq, ``` clasohm@0 ` 332` ``` add_assoc, add_diff_inverse, div_termination])))); ``` clasohm@0 ` 333` ```val mod_div_equality = result(); ``` clasohm@0 ` 334` clasohm@0 ` 335` clasohm@0 ` 336` ```(**** Additional theorems about "less than" ****) ``` clasohm@0 ` 337` clasohm@0 ` 338` ```val [mnat,nnat] = goal Arith.thy ``` clasohm@0 ` 339` ``` "[| m:nat; n:nat |] ==> ~ (m #+ n) : n"; ``` clasohm@0 ` 340` ```by (rtac (mnat RS nat_induct) 1); ``` clasohm@0 ` 341` ```by (ALLGOALS (ASM_SIMP_TAC (arith_ss addrews [mem_not_refl]))); ``` clasohm@0 ` 342` ```by (rtac notI 1); ``` clasohm@0 ` 343` ```by (etac notE 1); ``` clasohm@0 ` 344` ```by (etac (succI1 RS Ord_trans) 1); ``` clasohm@0 ` 345` ```by (rtac (nnat RS naturals_are_ordinals) 1); ``` clasohm@0 ` 346` ```val add_not_less_self = result(); ``` clasohm@0 ` 347` clasohm@0 ` 348` ```val [mnat,nnat] = goal Arith.thy ``` clasohm@0 ` 349` ``` "[| m:nat; n:nat |] ==> m : succ(m #+ n)"; ``` clasohm@0 ` 350` ```by (rtac (mnat RS nat_induct) 1); ``` clasohm@0 ` 351` ```(*May not simplify even with ZF_ss because it would expand m:succ(...) *) ``` clasohm@0 ` 352` ```by (rtac (add_0 RS ssubst) 1); ``` clasohm@0 ` 353` ```by (rtac (add_succ RS ssubst) 2); ``` clasohm@0 ` 354` ```by (REPEAT (ares_tac [nnat, Ord_0_mem_succ, succ_mem_succI, ``` clasohm@0 ` 355` ``` naturals_are_ordinals, nat_succI, add_type] 1)); ``` clasohm@0 ` 356` ```val add_less_succ_self = result(); ```