src/HOL/Arith.ML
 author clasohm Tue Jan 30 15:24:36 1996 +0100 (1996-01-30) changeset 1465 5d7a7e439cec parent 1398 b8de98c2c29c child 1475 7f5a4cd08209 permissions -rw-r--r--
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 clasohm@1465 ` 1` ```(* Title: HOL/Arith.ML ``` clasohm@923 ` 2` ``` ID: \$Id\$ ``` clasohm@1465 ` 3` ``` Author: Lawrence C Paulson, Cambridge University Computer Laboratory ``` clasohm@923 ` 4` ``` Copyright 1993 University of Cambridge ``` clasohm@923 ` 5` clasohm@923 ` 6` ```Proofs about elementary arithmetic: addition, multiplication, etc. ``` clasohm@923 ` 7` ```Tests definitions and simplifier. ``` clasohm@923 ` 8` ```*) ``` clasohm@923 ` 9` clasohm@923 ` 10` ```open Arith; ``` clasohm@923 ` 11` clasohm@923 ` 12` ```(*** Basic rewrite rules for the arithmetic operators ***) ``` clasohm@923 ` 13` clasohm@923 ` 14` ```val [pred_0, pred_Suc] = nat_recs pred_def; ``` clasohm@923 ` 15` ```val [add_0,add_Suc] = nat_recs add_def; ``` clasohm@923 ` 16` ```val [mult_0,mult_Suc] = nat_recs mult_def; ``` nipkow@1301 ` 17` ```Addsimps [pred_0,pred_Suc,add_0,add_Suc,mult_0,mult_Suc]; ``` nipkow@1301 ` 18` nipkow@1301 ` 19` ```(** pred **) ``` nipkow@1301 ` 20` nipkow@1301 ` 21` ```val prems = goal Arith.thy "n ~= 0 ==> Suc(pred n) = n"; ``` nipkow@1301 ` 22` ```by(res_inst_tac [("n","n")] natE 1); ``` nipkow@1301 ` 23` ```by(cut_facts_tac prems 1); ``` nipkow@1301 ` 24` ```by(ALLGOALS Asm_full_simp_tac); ``` nipkow@1301 ` 25` ```qed "Suc_pred"; ``` nipkow@1301 ` 26` ```Addsimps [Suc_pred]; ``` clasohm@923 ` 27` clasohm@923 ` 28` ```(** Difference **) ``` clasohm@923 ` 29` clasohm@923 ` 30` ```val diff_0 = diff_def RS def_nat_rec_0; ``` clasohm@923 ` 31` clasohm@923 ` 32` ```qed_goalw "diff_0_eq_0" Arith.thy [diff_def, pred_def] ``` clasohm@923 ` 33` ``` "0 - n = 0" ``` clasohm@1264 ` 34` ``` (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 35` clasohm@923 ` 36` ```(*Must simplify BEFORE the induction!! (Else we get a critical pair) ``` clasohm@923 ` 37` ``` Suc(m) - Suc(n) rewrites to pred(Suc(m) - n) *) ``` clasohm@923 ` 38` ```qed_goalw "diff_Suc_Suc" Arith.thy [diff_def, pred_def] ``` clasohm@923 ` 39` ``` "Suc(m) - Suc(n) = m - n" ``` clasohm@923 ` 40` ``` (fn _ => ``` clasohm@1264 ` 41` ``` [Simp_tac 1, nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 42` nipkow@1301 ` 43` ```Addsimps [diff_0, diff_0_eq_0, diff_Suc_Suc]; ``` clasohm@923 ` 44` clasohm@923 ` 45` clasohm@923 ` 46` ```(**** Inductive properties of the operators ****) ``` clasohm@923 ` 47` clasohm@923 ` 48` ```(*** Addition ***) ``` clasohm@923 ` 49` clasohm@923 ` 50` ```qed_goal "add_0_right" Arith.thy "m + 0 = m" ``` clasohm@1264 ` 51` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 52` clasohm@923 ` 53` ```qed_goal "add_Suc_right" Arith.thy "m + Suc(n) = Suc(m+n)" ``` clasohm@1264 ` 54` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 55` clasohm@1264 ` 56` ```Addsimps [add_0_right,add_Suc_right]; ``` clasohm@923 ` 57` clasohm@923 ` 58` ```(*Associative law for addition*) ``` clasohm@923 ` 59` ```qed_goal "add_assoc" Arith.thy "(m + n) + k = m + ((n + k)::nat)" ``` clasohm@1264 ` 60` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 61` clasohm@923 ` 62` ```(*Commutative law for addition*) ``` clasohm@923 ` 63` ```qed_goal "add_commute" Arith.thy "m + n = n + (m::nat)" ``` clasohm@1264 ` 64` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 65` clasohm@923 ` 66` ```qed_goal "add_left_commute" Arith.thy "x+(y+z)=y+((x+z)::nat)" ``` clasohm@923 ` 67` ``` (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1, ``` clasohm@923 ` 68` ``` rtac (add_commute RS arg_cong) 1]); ``` clasohm@923 ` 69` clasohm@923 ` 70` ```(*Addition is an AC-operator*) ``` clasohm@923 ` 71` ```val add_ac = [add_assoc, add_commute, add_left_commute]; ``` clasohm@923 ` 72` clasohm@923 ` 73` ```goal Arith.thy "!!k::nat. (k + m = k + n) = (m=n)"; ``` clasohm@923 ` 74` ```by (nat_ind_tac "k" 1); ``` clasohm@1264 ` 75` ```by (Simp_tac 1); ``` clasohm@1264 ` 76` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 77` ```qed "add_left_cancel"; ``` clasohm@923 ` 78` clasohm@923 ` 79` ```goal Arith.thy "!!k::nat. (m + k = n + k) = (m=n)"; ``` clasohm@923 ` 80` ```by (nat_ind_tac "k" 1); ``` clasohm@1264 ` 81` ```by (Simp_tac 1); ``` clasohm@1264 ` 82` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 83` ```qed "add_right_cancel"; ``` clasohm@923 ` 84` clasohm@923 ` 85` ```goal Arith.thy "!!k::nat. (k + m <= k + n) = (m<=n)"; ``` clasohm@923 ` 86` ```by (nat_ind_tac "k" 1); ``` clasohm@1264 ` 87` ```by (Simp_tac 1); ``` clasohm@1264 ` 88` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 89` ```qed "add_left_cancel_le"; ``` clasohm@923 ` 90` clasohm@923 ` 91` ```goal Arith.thy "!!k::nat. (k + m < k + n) = (m m + pred n = pred(m+n)"; ``` nipkow@1327 ` 107` ```by (nat_ind_tac "m" 1); ``` nipkow@1327 ` 108` ```by (ALLGOALS Asm_simp_tac); ``` nipkow@1327 ` 109` ```qed "add_pred"; ``` nipkow@1327 ` 110` ```Addsimps [add_pred]; ``` nipkow@1327 ` 111` clasohm@923 ` 112` ```(*** Multiplication ***) ``` clasohm@923 ` 113` clasohm@923 ` 114` ```(*right annihilation in product*) ``` clasohm@923 ` 115` ```qed_goal "mult_0_right" Arith.thy "m * 0 = 0" ``` clasohm@1264 ` 116` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 117` clasohm@923 ` 118` ```(*right Sucessor law for multiplication*) ``` clasohm@923 ` 119` ```qed_goal "mult_Suc_right" Arith.thy "m * Suc(n) = m + (m * n)" ``` clasohm@923 ` 120` ``` (fn _ => [nat_ind_tac "m" 1, ``` clasohm@1264 ` 121` ``` ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]); ``` clasohm@923 ` 122` clasohm@1264 ` 123` ```Addsimps [mult_0_right,mult_Suc_right]; ``` clasohm@923 ` 124` clasohm@923 ` 125` ```(*Commutative law for multiplication*) ``` clasohm@923 ` 126` ```qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)" ``` clasohm@1264 ` 127` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 128` clasohm@923 ` 129` ```(*addition distributes over multiplication*) ``` clasohm@923 ` 130` ```qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)" ``` clasohm@923 ` 131` ``` (fn _ => [nat_ind_tac "m" 1, ``` clasohm@1264 ` 132` ``` ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]); ``` clasohm@923 ` 133` clasohm@923 ` 134` ```qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)" ``` clasohm@923 ` 135` ``` (fn _ => [nat_ind_tac "m" 1, ``` clasohm@1264 ` 136` ``` ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]); ``` clasohm@923 ` 137` clasohm@1264 ` 138` ```Addsimps [add_mult_distrib,add_mult_distrib2]; ``` clasohm@923 ` 139` clasohm@923 ` 140` ```(*Associative law for multiplication*) ``` clasohm@923 ` 141` ```qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)" ``` clasohm@1264 ` 142` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 143` clasohm@923 ` 144` ```qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)" ``` clasohm@923 ` 145` ``` (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1, ``` clasohm@923 ` 146` ``` rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]); ``` clasohm@923 ` 147` clasohm@923 ` 148` ```val mult_ac = [mult_assoc,mult_commute,mult_left_commute]; ``` clasohm@923 ` 149` clasohm@923 ` 150` ```(*** Difference ***) ``` clasohm@923 ` 151` clasohm@923 ` 152` ```qed_goal "diff_self_eq_0" Arith.thy "m - m = 0" ``` clasohm@1264 ` 153` ``` (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 154` clasohm@923 ` 155` ```(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *) ``` clasohm@923 ` 156` ```val [prem] = goal Arith.thy "[| ~ m n+(m-n) = (m::nat)"; ``` clasohm@923 ` 157` ```by (rtac (prem RS rev_mp) 1); ``` clasohm@923 ` 158` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@1264 ` 159` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 160` ```qed "add_diff_inverse"; ``` clasohm@923 ` 161` clasohm@923 ` 162` clasohm@923 ` 163` ```(*** Remainder ***) ``` clasohm@923 ` 164` clasohm@923 ` 165` ```goal Arith.thy "m - n < Suc(m)"; ``` clasohm@923 ` 166` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@923 ` 167` ```by (etac less_SucE 3); ``` clasohm@1264 ` 168` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 169` ```qed "diff_less_Suc"; ``` clasohm@923 ` 170` clasohm@923 ` 171` ```goal Arith.thy "!!m::nat. m - n <= m"; ``` clasohm@923 ` 172` ```by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1); ``` clasohm@1264 ` 173` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 174` ```qed "diff_le_self"; ``` clasohm@923 ` 175` clasohm@923 ` 176` ```goal Arith.thy "!!n::nat. (n+m) - n = m"; ``` clasohm@923 ` 177` ```by (nat_ind_tac "n" 1); ``` clasohm@1264 ` 178` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 179` ```qed "diff_add_inverse"; ``` clasohm@923 ` 180` clasohm@923 ` 181` ```goal Arith.thy "!!n::nat. n - (n+m) = 0"; ``` clasohm@923 ` 182` ```by (nat_ind_tac "n" 1); ``` clasohm@1264 ` 183` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 184` ```qed "diff_add_0"; ``` clasohm@923 ` 185` clasohm@923 ` 186` ```(*In ordinary notation: if 0 m - n < m"; ``` clasohm@923 ` 188` ```by (subgoal_tac "0 ~ m m - n < m" 1); ``` clasohm@923 ` 189` ```by (fast_tac HOL_cs 1); ``` clasohm@923 ` 190` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@1264 ` 191` ```by (ALLGOALS(asm_simp_tac(!simpset addsimps [diff_less_Suc]))); ``` nipkow@1398 ` 192` ```qed "diff_less"; ``` clasohm@923 ` 193` clasohm@923 ` 194` ```val wf_less_trans = wf_pred_nat RS wf_trancl RSN (2, def_wfrec RS trans); ``` clasohm@923 ` 195` clasohm@972 ` 196` ```goalw Nat.thy [less_def] "(m,n) : pred_nat^+ = (m m mod n = m"; ``` clasohm@923 ` 201` ```by (rtac (mod_def RS wf_less_trans) 1); ``` clasohm@1264 ` 202` ```by(Asm_simp_tac 1); ``` clasohm@923 ` 203` ```qed "mod_less"; ``` clasohm@923 ` 204` clasohm@923 ` 205` ```goal Arith.thy "!!m. [| 0 m mod n = (m-n) mod n"; ``` clasohm@923 ` 206` ```by (rtac (mod_def RS wf_less_trans) 1); ``` nipkow@1398 ` 207` ```by(asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1); ``` clasohm@923 ` 208` ```qed "mod_geq"; ``` clasohm@923 ` 209` clasohm@923 ` 210` clasohm@923 ` 211` ```(*** Quotient ***) ``` clasohm@923 ` 212` clasohm@923 ` 213` ```goal Arith.thy "!!m. m m div n = 0"; ``` clasohm@923 ` 214` ```by (rtac (div_def RS wf_less_trans) 1); ``` clasohm@1264 ` 215` ```by(Asm_simp_tac 1); ``` clasohm@923 ` 216` ```qed "div_less"; ``` clasohm@923 ` 217` clasohm@923 ` 218` ```goal Arith.thy "!!M. [| 0 m div n = Suc((m-n) div n)"; ``` clasohm@923 ` 219` ```by (rtac (div_def RS wf_less_trans) 1); ``` nipkow@1398 ` 220` ```by(asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1); ``` clasohm@923 ` 221` ```qed "div_geq"; ``` clasohm@923 ` 222` clasohm@923 ` 223` ```(*Main Result about quotient and remainder.*) ``` clasohm@923 ` 224` ```goal Arith.thy "!!m. 0 (m div n)*n + m mod n = m"; ``` clasohm@923 ` 225` ```by (res_inst_tac [("n","m")] less_induct 1); ``` clasohm@923 ` 226` ```by (rename_tac "k" 1); (*Variable name used in line below*) ``` clasohm@923 ` 227` ```by (case_tac "k m-n = 0"; ``` clasohm@923 ` 237` ```by (rtac (prem RS rev_mp) 1); ``` clasohm@923 ` 238` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@1264 ` 239` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 240` ```qed "less_imp_diff_is_0"; ``` clasohm@923 ` 241` clasohm@923 ` 242` ```val prems = goal Arith.thy "m-n = 0 --> n-m = 0 --> m=n"; ``` clasohm@923 ` 243` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@1264 ` 244` ```by (REPEAT(Simp_tac 1 THEN TRY(atac 1))); ``` clasohm@923 ` 245` ```qed "diffs0_imp_equal_lemma"; ``` clasohm@923 ` 246` clasohm@923 ` 247` ```(* [| m-n = 0; n-m = 0 |] ==> m=n *) ``` clasohm@923 ` 248` ```bind_thm ("diffs0_imp_equal", (diffs0_imp_equal_lemma RS mp RS mp)); ``` clasohm@923 ` 249` clasohm@923 ` 250` ```val [prem] = goal Arith.thy "m 0 Suc(m)-n = Suc(m-n)"; ``` clasohm@923 ` 257` ```by (rtac (prem RS rev_mp) 1); ``` clasohm@923 ` 258` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` clasohm@1264 ` 259` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 260` ```qed "Suc_diff_n"; ``` clasohm@923 ` 261` nipkow@1398 ` 262` ```goal Arith.thy "Suc(m)-n = (if m (!n. P(Suc(n))--> P(n)) --> P(k-i)"; ``` clasohm@923 ` 268` ```by (res_inst_tac [("m","k"),("n","i")] diff_induct 1); ``` clasohm@1264 ` 269` ```by (ALLGOALS (strip_tac THEN' Simp_tac THEN' TRY o fast_tac HOL_cs)); ``` clasohm@923 ` 270` ```qed "zero_induct_lemma"; ``` clasohm@923 ` 271` clasohm@923 ` 272` ```val prems = goal Arith.thy "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)"; ``` clasohm@923 ` 273` ```by (rtac (diff_self_eq_0 RS subst) 1); ``` clasohm@923 ` 274` ```by (rtac (zero_induct_lemma RS mp RS mp) 1); ``` clasohm@923 ` 275` ```by (REPEAT (ares_tac ([impI,allI]@prems) 1)); ``` clasohm@923 ` 276` ```qed "zero_induct"; ``` clasohm@923 ` 277` clasohm@923 ` 278` ```(*13 July 1992: loaded in 105.7s*) ``` clasohm@923 ` 279` clasohm@923 ` 280` ```(**** Additional theorems about "less than" ****) ``` clasohm@923 ` 281` clasohm@923 ` 282` ```goal Arith.thy "!!m. m (? k. n=Suc(m+k))"; ``` clasohm@923 ` 283` ```by (nat_ind_tac "n" 1); ``` clasohm@1264 ` 284` ```by (ALLGOALS(Simp_tac)); ``` clasohm@923 ` 285` ```by (REPEAT_FIRST (ares_tac [conjI, impI])); ``` clasohm@923 ` 286` ```by (res_inst_tac [("x","0")] exI 2); ``` clasohm@1264 ` 287` ```by (Simp_tac 2); ``` clasohm@923 ` 288` ```by (safe_tac HOL_cs); ``` clasohm@923 ` 289` ```by (res_inst_tac [("x","Suc(k)")] exI 1); ``` clasohm@1264 ` 290` ```by (Simp_tac 1); ``` clasohm@923 ` 291` ```val less_eq_Suc_add_lemma = result(); ``` clasohm@923 ` 292` clasohm@923 ` 293` ```(*"m ? k. n = Suc(m+k)"*) ``` clasohm@923 ` 294` ```bind_thm ("less_eq_Suc_add", less_eq_Suc_add_lemma RS mp); ``` clasohm@923 ` 295` clasohm@923 ` 296` clasohm@923 ` 297` ```goal Arith.thy "n <= ((m + n)::nat)"; ``` clasohm@923 ` 298` ```by (nat_ind_tac "m" 1); ``` clasohm@1264 ` 299` ```by (ALLGOALS Simp_tac); ``` clasohm@923 ` 300` ```by (etac le_trans 1); ``` clasohm@923 ` 301` ```by (rtac (lessI RS less_imp_le) 1); ``` clasohm@923 ` 302` ```qed "le_add2"; ``` clasohm@923 ` 303` clasohm@923 ` 304` ```goal Arith.thy "n <= ((n + m)::nat)"; ``` clasohm@1264 ` 305` ```by (simp_tac (!simpset addsimps add_ac) 1); ``` clasohm@923 ` 306` ```by (rtac le_add2 1); ``` clasohm@923 ` 307` ```qed "le_add1"; ``` clasohm@923 ` 308` clasohm@923 ` 309` ```bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans))); ``` clasohm@923 ` 310` ```bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans))); ``` clasohm@923 ` 311` clasohm@923 ` 312` ```(*"i <= j ==> i <= j+m"*) ``` clasohm@923 ` 313` ```bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans)); ``` clasohm@923 ` 314` clasohm@923 ` 315` ```(*"i <= j ==> i <= m+j"*) ``` clasohm@923 ` 316` ```bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans)); ``` clasohm@923 ` 317` clasohm@923 ` 318` ```(*"i < j ==> i < j+m"*) ``` clasohm@923 ` 319` ```bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans)); ``` clasohm@923 ` 320` clasohm@923 ` 321` ```(*"i < j ==> i < m+j"*) ``` clasohm@923 ` 322` ```bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans)); ``` clasohm@923 ` 323` nipkow@1152 ` 324` ```goal Arith.thy "!!i. i+j < (k::nat) ==> i m <= n+k"; ``` clasohm@1465 ` 332` ```by (etac le_trans 1); ``` clasohm@1465 ` 333` ```by (rtac le_add1 1); ``` clasohm@923 ` 334` ```qed "le_imp_add_le"; ``` clasohm@923 ` 335` clasohm@923 ` 336` ```goal Arith.thy "!!k::nat. m < n ==> m < n+k"; ``` clasohm@1465 ` 337` ```by (etac less_le_trans 1); ``` clasohm@1465 ` 338` ```by (rtac le_add1 1); ``` clasohm@923 ` 339` ```qed "less_imp_add_less"; ``` clasohm@923 ` 340` clasohm@923 ` 341` ```goal Arith.thy "m+k<=n --> m<=(n::nat)"; ``` clasohm@923 ` 342` ```by (nat_ind_tac "k" 1); ``` clasohm@1264 ` 343` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 344` ```by (fast_tac (HOL_cs addDs [Suc_leD]) 1); ``` clasohm@923 ` 345` ```val add_leD1_lemma = result(); ``` clasohm@1264 ` 346` ```bind_thm ("add_leD1", add_leD1_lemma RS mp); ``` clasohm@923 ` 347` clasohm@923 ` 348` ```goal Arith.thy "!!k l::nat. [| k m i + k < j + k"; ``` clasohm@923 ` 364` ```by (nat_ind_tac "k" 1); ``` clasohm@1264 ` 365` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 366` ```qed "add_less_mono1"; ``` clasohm@923 ` 367` clasohm@923 ` 368` ```(*strict, in both arguments*) ``` clasohm@923 ` 369` ```goal Arith.thy "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l"; ``` clasohm@923 ` 370` ```by (rtac (add_less_mono1 RS less_trans) 1); ``` lcp@1198 ` 371` ```by (REPEAT (assume_tac 1)); ``` clasohm@923 ` 372` ```by (nat_ind_tac "j" 1); ``` clasohm@1264 ` 373` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 374` ```qed "add_less_mono"; ``` clasohm@923 ` 375` clasohm@923 ` 376` ```(*A [clumsy] way of lifting < monotonicity to <= monotonicity *) ``` clasohm@923 ` 377` ```val [lt_mono,le] = goal Arith.thy ``` clasohm@1465 ` 378` ``` "[| !!i j::nat. i f(i) < f(j); \ ``` clasohm@1465 ` 379` ```\ i <= j \ ``` clasohm@923 ` 380` ```\ |] ==> f(i) <= (f(j)::nat)"; ``` clasohm@923 ` 381` ```by (cut_facts_tac [le] 1); ``` clasohm@1264 ` 382` ```by (asm_full_simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); ``` clasohm@923 ` 383` ```by (fast_tac (HOL_cs addSIs [lt_mono]) 1); ``` clasohm@923 ` 384` ```qed "less_mono_imp_le_mono"; ``` clasohm@923 ` 385` clasohm@923 ` 386` ```(*non-strict, in 1st argument*) ``` clasohm@923 ` 387` ```goal Arith.thy "!!i j k::nat. i<=j ==> i + k <= j + k"; ``` clasohm@923 ` 388` ```by (res_inst_tac [("f", "%j.j+k")] less_mono_imp_le_mono 1); ``` clasohm@1465 ` 389` ```by (etac add_less_mono1 1); ``` clasohm@923 ` 390` ```by (assume_tac 1); ``` clasohm@923 ` 391` ```qed "add_le_mono1"; ``` clasohm@923 ` 392` clasohm@923 ` 393` ```(*non-strict, in both arguments*) ``` clasohm@923 ` 394` ```goal Arith.thy "!!k l::nat. [|i<=j; k<=l |] ==> i + k <= j + l"; ``` clasohm@923 ` 395` ```by (etac (add_le_mono1 RS le_trans) 1); ``` clasohm@1264 ` 396` ```by (simp_tac (!simpset addsimps [add_commute]) 1); ``` clasohm@923 ` 397` ```(*j moves to the end because it is free while k, l are bound*) ``` clasohm@1465 ` 398` ```by (etac add_le_mono1 1); ``` clasohm@923 ` 399` ```qed "add_le_mono"; ```