src/HOL/Arith.ML
 author nipkow Sat Dec 06 17:05:41 1997 +0100 (1997-12-06) changeset 4378 e52f864c5b88 parent 4360 40e5c97e988d child 4389 1865cb8df116 permissions -rw-r--r--
Got rid of some preds and replaced some n~=0 by 0<n.
 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. ``` paulson@3234 ` 7` ```Some from the Hoare example from Norbert Galm ``` clasohm@923 ` 8` ```*) ``` clasohm@923 ` 9` clasohm@923 ` 10` ```(*** Basic rewrite rules for the arithmetic operators ***) ``` clasohm@923 ` 11` nipkow@3896 ` 12` clasohm@923 ` 13` ```(** Difference **) ``` clasohm@923 ` 14` nipkow@4360 ` 15` ```qed_goal "diff_0_eq_0" Arith.thy ``` clasohm@923 ` 16` ``` "0 - n = 0" ``` paulson@3339 ` 17` ``` (fn _ => [induct_tac "n" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 18` clasohm@923 ` 19` ```(*Must simplify BEFORE the induction!! (Else we get a critical pair) ``` clasohm@923 ` 20` ``` Suc(m) - Suc(n) rewrites to pred(Suc(m) - n) *) ``` nipkow@4360 ` 21` ```qed_goal "diff_Suc_Suc" Arith.thy ``` clasohm@923 ` 22` ``` "Suc(m) - Suc(n) = m - n" ``` clasohm@923 ` 23` ``` (fn _ => ``` paulson@3339 ` 24` ``` [Simp_tac 1, induct_tac "n" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 25` pusch@2682 ` 26` ```Addsimps [diff_0_eq_0, diff_Suc_Suc]; ``` clasohm@923 ` 27` nipkow@4360 ` 28` ```(* Could be (and is, below) generalized in various ways; ``` nipkow@4360 ` 29` ``` However, none of the generalizations are currently in the simpset, ``` nipkow@4360 ` 30` ``` and I dread to think what happens if I put them in *) ``` nipkow@4360 ` 31` ```goal Arith.thy "!!n. 0 < n ==> Suc(n-1) = n"; ``` nipkow@4360 ` 32` ```by(asm_simp_tac (simpset() addsplits [expand_nat_case]) 1); ``` nipkow@4360 ` 33` ```qed "Suc_pred"; ``` nipkow@4360 ` 34` ```Addsimps [Suc_pred]; ``` nipkow@4360 ` 35` nipkow@4360 ` 36` ```(* Generalize? *) ``` nipkow@4360 ` 37` ```goal Arith.thy "!!n. 0 n-1 < n"; ``` nipkow@4360 ` 38` ```by(asm_simp_tac (simpset() addsplits [expand_nat_case]) 1); ``` nipkow@4360 ` 39` ```qed "pred_less"; ``` nipkow@4360 ` 40` ```Addsimps [pred_less]; ``` nipkow@4360 ` 41` nipkow@4360 ` 42` ```Delsimps [diff_Suc]; ``` nipkow@4360 ` 43` clasohm@923 ` 44` clasohm@923 ` 45` ```(**** Inductive properties of the operators ****) ``` clasohm@923 ` 46` clasohm@923 ` 47` ```(*** Addition ***) ``` clasohm@923 ` 48` clasohm@923 ` 49` ```qed_goal "add_0_right" Arith.thy "m + 0 = m" ``` paulson@3339 ` 50` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 51` clasohm@923 ` 52` ```qed_goal "add_Suc_right" Arith.thy "m + Suc(n) = Suc(m+n)" ``` paulson@3339 ` 53` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 54` clasohm@1264 ` 55` ```Addsimps [add_0_right,add_Suc_right]; ``` clasohm@923 ` 56` clasohm@923 ` 57` ```(*Associative law for addition*) ``` clasohm@923 ` 58` ```qed_goal "add_assoc" Arith.thy "(m + n) + k = m + ((n + k)::nat)" ``` paulson@3339 ` 59` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 60` clasohm@923 ` 61` ```(*Commutative law for addition*) ``` clasohm@923 ` 62` ```qed_goal "add_commute" Arith.thy "m + n = n + (m::nat)" ``` paulson@3339 ` 63` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 64` clasohm@923 ` 65` ```qed_goal "add_left_commute" Arith.thy "x+(y+z)=y+((x+z)::nat)" ``` clasohm@923 ` 66` ``` (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1, ``` clasohm@923 ` 67` ``` rtac (add_commute RS arg_cong) 1]); ``` clasohm@923 ` 68` clasohm@923 ` 69` ```(*Addition is an AC-operator*) ``` clasohm@923 ` 70` ```val add_ac = [add_assoc, add_commute, add_left_commute]; ``` clasohm@923 ` 71` clasohm@923 ` 72` ```goal Arith.thy "!!k::nat. (k + m = k + n) = (m=n)"; ``` paulson@3339 ` 73` ```by (induct_tac "k" 1); ``` clasohm@1264 ` 74` ```by (Simp_tac 1); ``` clasohm@1264 ` 75` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 76` ```qed "add_left_cancel"; ``` clasohm@923 ` 77` clasohm@923 ` 78` ```goal Arith.thy "!!k::nat. (m + k = n + k) = (m=n)"; ``` paulson@3339 ` 79` ```by (induct_tac "k" 1); ``` clasohm@1264 ` 80` ```by (Simp_tac 1); ``` clasohm@1264 ` 81` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 82` ```qed "add_right_cancel"; ``` clasohm@923 ` 83` clasohm@923 ` 84` ```goal Arith.thy "!!k::nat. (k + m <= k + n) = (m<=n)"; ``` paulson@3339 ` 85` ```by (induct_tac "k" 1); ``` clasohm@1264 ` 86` ```by (Simp_tac 1); ``` clasohm@1264 ` 87` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 88` ```qed "add_left_cancel_le"; ``` clasohm@923 ` 89` clasohm@923 ` 90` ```goal Arith.thy "!!k::nat. (k + m < k + n) = (m m+(n-(Suc k)) = (m+n)-(Suc k)" *) ``` nipkow@4378 ` 120` ```goal Arith.thy "!!n. 0 m + (n-1) = (m+n)-1"; ``` nipkow@4360 ` 121` ```by (exhaust_tac "m" 1); ``` nipkow@4360 ` 122` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc] ``` nipkow@4360 ` 123` ``` addsplits [expand_nat_case]))); ``` nipkow@1327 ` 124` ```qed "add_pred"; ``` nipkow@1327 ` 125` ```Addsimps [add_pred]; ``` nipkow@1327 ` 126` paulson@1626 ` 127` clasohm@923 ` 128` ```(**** Additional theorems about "less than" ****) ``` clasohm@923 ` 129` paulson@3339 ` 130` ```goal Arith.thy "i (EX k. j = Suc(i+k))"; ``` paulson@3339 ` 131` ```by (induct_tac "j" 1); ``` paulson@1909 ` 132` ```by (Simp_tac 1); ``` wenzelm@4089 ` 133` ```by (blast_tac (claset() addSEs [less_SucE] ``` paulson@3339 ` 134` ``` addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1); ``` paulson@1909 ` 135` ```val lemma = result(); ``` paulson@1909 ` 136` paulson@3339 ` 137` ```(* [| i Q |] ==> Q *) ``` paulson@3339 ` 138` ```bind_thm ("less_natE", lemma RS mp RS exE); ``` paulson@3339 ` 139` clasohm@923 ` 140` ```goal Arith.thy "!!m. m (? k. n=Suc(m+k))"; ``` paulson@3339 ` 141` ```by (induct_tac "n" 1); ``` wenzelm@4089 ` 142` ```by (ALLGOALS (simp_tac (simpset() addsimps [less_Suc_eq]))); ``` wenzelm@4089 ` 143` ```by (blast_tac (claset() addSEs [less_SucE] ``` paulson@3339 ` 144` ``` addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1); ``` nipkow@1485 ` 145` ```qed_spec_mp "less_eq_Suc_add"; ``` clasohm@923 ` 146` clasohm@923 ` 147` ```goal Arith.thy "n <= ((m + n)::nat)"; ``` paulson@3339 ` 148` ```by (induct_tac "m" 1); ``` clasohm@1264 ` 149` ```by (ALLGOALS Simp_tac); ``` clasohm@923 ` 150` ```by (etac le_trans 1); ``` clasohm@923 ` 151` ```by (rtac (lessI RS less_imp_le) 1); ``` clasohm@923 ` 152` ```qed "le_add2"; ``` clasohm@923 ` 153` clasohm@923 ` 154` ```goal Arith.thy "n <= ((n + m)::nat)"; ``` wenzelm@4089 ` 155` ```by (simp_tac (simpset() addsimps add_ac) 1); ``` clasohm@923 ` 156` ```by (rtac le_add2 1); ``` clasohm@923 ` 157` ```qed "le_add1"; ``` clasohm@923 ` 158` clasohm@923 ` 159` ```bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans))); ``` clasohm@923 ` 160` ```bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans))); ``` clasohm@923 ` 161` clasohm@923 ` 162` ```(*"i <= j ==> i <= j+m"*) ``` clasohm@923 ` 163` ```bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans)); ``` clasohm@923 ` 164` clasohm@923 ` 165` ```(*"i <= j ==> i <= m+j"*) ``` clasohm@923 ` 166` ```bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans)); ``` clasohm@923 ` 167` clasohm@923 ` 168` ```(*"i < j ==> i < j+m"*) ``` clasohm@923 ` 169` ```bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans)); ``` clasohm@923 ` 170` clasohm@923 ` 171` ```(*"i < j ==> i < m+j"*) ``` clasohm@923 ` 172` ```bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans)); ``` clasohm@923 ` 173` nipkow@1152 ` 174` ```goal Arith.thy "!!i. i+j < (k::nat) ==> i m <= n+k"; ``` paulson@1552 ` 192` ```by (etac le_trans 1); ``` paulson@1552 ` 193` ```by (rtac le_add1 1); ``` clasohm@923 ` 194` ```qed "le_imp_add_le"; ``` clasohm@923 ` 195` clasohm@923 ` 196` ```goal Arith.thy "!!k::nat. m < n ==> m < n+k"; ``` paulson@1552 ` 197` ```by (etac less_le_trans 1); ``` paulson@1552 ` 198` ```by (rtac le_add1 1); ``` clasohm@923 ` 199` ```qed "less_imp_add_less"; ``` clasohm@923 ` 200` clasohm@923 ` 201` ```goal Arith.thy "m+k<=n --> m<=(n::nat)"; ``` paulson@3339 ` 202` ```by (induct_tac "k" 1); ``` clasohm@1264 ` 203` ```by (ALLGOALS Asm_simp_tac); ``` wenzelm@4089 ` 204` ```by (blast_tac (claset() addDs [Suc_leD]) 1); ``` nipkow@1485 ` 205` ```qed_spec_mp "add_leD1"; ``` clasohm@923 ` 206` paulson@2498 ` 207` ```goal Arith.thy "!!n::nat. m+k<=n ==> k<=n"; ``` wenzelm@4089 ` 208` ```by (full_simp_tac (simpset() addsimps [add_commute]) 1); ``` paulson@2498 ` 209` ```by (etac add_leD1 1); ``` paulson@2498 ` 210` ```qed_spec_mp "add_leD2"; ``` paulson@2498 ` 211` paulson@2498 ` 212` ```goal Arith.thy "!!n::nat. m+k<=n ==> m<=n & k<=n"; ``` wenzelm@4089 ` 213` ```by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1); ``` paulson@2498 ` 214` ```bind_thm ("add_leE", result() RS conjE); ``` paulson@2498 ` 215` clasohm@923 ` 216` ```goal Arith.thy "!!k l::nat. [| k m i + k < j + k"; ``` paulson@3339 ` 230` ```by (induct_tac "k" 1); ``` clasohm@1264 ` 231` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 232` ```qed "add_less_mono1"; ``` clasohm@923 ` 233` clasohm@923 ` 234` ```(*strict, in both arguments*) ``` clasohm@923 ` 235` ```goal Arith.thy "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l"; ``` clasohm@923 ` 236` ```by (rtac (add_less_mono1 RS less_trans) 1); ``` lcp@1198 ` 237` ```by (REPEAT (assume_tac 1)); ``` paulson@3339 ` 238` ```by (induct_tac "j" 1); ``` clasohm@1264 ` 239` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 240` ```qed "add_less_mono"; ``` clasohm@923 ` 241` clasohm@923 ` 242` ```(*A [clumsy] way of lifting < monotonicity to <= monotonicity *) ``` clasohm@923 ` 243` ```val [lt_mono,le] = goal Arith.thy ``` clasohm@1465 ` 244` ``` "[| !!i j::nat. i f(i) < f(j); \ ``` clasohm@1465 ` 245` ```\ i <= j \ ``` clasohm@923 ` 246` ```\ |] ==> f(i) <= (f(j)::nat)"; ``` clasohm@923 ` 247` ```by (cut_facts_tac [le] 1); ``` wenzelm@4089 ` 248` ```by (asm_full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1); ``` wenzelm@4089 ` 249` ```by (blast_tac (claset() addSIs [lt_mono]) 1); ``` clasohm@923 ` 250` ```qed "less_mono_imp_le_mono"; ``` clasohm@923 ` 251` clasohm@923 ` 252` ```(*non-strict, in 1st argument*) ``` clasohm@923 ` 253` ```goal Arith.thy "!!i j k::nat. i<=j ==> i + k <= j + k"; ``` wenzelm@3842 ` 254` ```by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1); ``` paulson@1552 ` 255` ```by (etac add_less_mono1 1); ``` clasohm@923 ` 256` ```by (assume_tac 1); ``` clasohm@923 ` 257` ```qed "add_le_mono1"; ``` clasohm@923 ` 258` clasohm@923 ` 259` ```(*non-strict, in both arguments*) ``` clasohm@923 ` 260` ```goal Arith.thy "!!k l::nat. [|i<=j; k<=l |] ==> i + k <= j + l"; ``` clasohm@923 ` 261` ```by (etac (add_le_mono1 RS le_trans) 1); ``` wenzelm@4089 ` 262` ```by (simp_tac (simpset() addsimps [add_commute]) 1); ``` clasohm@923 ` 263` ```(*j moves to the end because it is free while k, l are bound*) ``` paulson@1552 ` 264` ```by (etac add_le_mono1 1); ``` clasohm@923 ` 265` ```qed "add_le_mono"; ``` paulson@1713 ` 266` paulson@3234 ` 267` paulson@3234 ` 268` ```(*** Multiplication ***) ``` paulson@3234 ` 269` paulson@3234 ` 270` ```(*right annihilation in product*) ``` paulson@3234 ` 271` ```qed_goal "mult_0_right" Arith.thy "m * 0 = 0" ``` paulson@3339 ` 272` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` paulson@3234 ` 273` paulson@3293 ` 274` ```(*right successor law for multiplication*) ``` paulson@3234 ` 275` ```qed_goal "mult_Suc_right" Arith.thy "m * Suc(n) = m + (m * n)" ``` paulson@3339 ` 276` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 277` ``` ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); ``` paulson@3234 ` 278` paulson@3293 ` 279` ```Addsimps [mult_0_right, mult_Suc_right]; ``` paulson@3234 ` 280` paulson@3234 ` 281` ```goal Arith.thy "1 * n = n"; ``` paulson@3234 ` 282` ```by (Asm_simp_tac 1); ``` paulson@3234 ` 283` ```qed "mult_1"; ``` paulson@3234 ` 284` paulson@3234 ` 285` ```goal Arith.thy "n * 1 = n"; ``` paulson@3234 ` 286` ```by (Asm_simp_tac 1); ``` paulson@3234 ` 287` ```qed "mult_1_right"; ``` paulson@3234 ` 288` paulson@3234 ` 289` ```(*Commutative law for multiplication*) ``` paulson@3234 ` 290` ```qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)" ``` paulson@3339 ` 291` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` paulson@3234 ` 292` paulson@3234 ` 293` ```(*addition distributes over multiplication*) ``` paulson@3234 ` 294` ```qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)" ``` paulson@3339 ` 295` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 296` ``` ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); ``` paulson@3234 ` 297` paulson@3234 ` 298` ```qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)" ``` paulson@3339 ` 299` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 300` ``` ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); ``` paulson@3234 ` 301` paulson@3234 ` 302` ```(*Associative law for multiplication*) ``` paulson@3234 ` 303` ```qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)" ``` paulson@3339 ` 304` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 305` ``` ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]); ``` paulson@3234 ` 306` paulson@3234 ` 307` ```qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)" ``` paulson@3234 ` 308` ``` (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1, ``` paulson@3234 ` 309` ``` rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]); ``` paulson@3234 ` 310` paulson@3234 ` 311` ```val mult_ac = [mult_assoc,mult_commute,mult_left_commute]; ``` paulson@3234 ` 312` paulson@3293 ` 313` ```goal Arith.thy "(m*n = 0) = (m=0 | n=0)"; ``` paulson@3339 ` 314` ```by (induct_tac "m" 1); ``` paulson@3339 ` 315` ```by (induct_tac "n" 2); ``` paulson@3293 ` 316` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3293 ` 317` ```qed "mult_is_0"; ``` paulson@3293 ` 318` ```Addsimps [mult_is_0]; ``` paulson@3293 ` 319` paulson@4158 ` 320` ```goal Arith.thy "!!m::nat. m <= m*m"; ``` paulson@4158 ` 321` ```by (induct_tac "m" 1); ``` paulson@4158 ` 322` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym]))); ``` paulson@4158 ` 323` ```by (etac (le_add2 RSN (2,le_trans)) 1); ``` paulson@4158 ` 324` ```qed "le_square"; ``` paulson@4158 ` 325` paulson@3234 ` 326` paulson@3234 ` 327` ```(*** Difference ***) ``` paulson@3234 ` 328` paulson@3234 ` 329` paulson@3234 ` 330` ```qed_goal "diff_self_eq_0" Arith.thy "m - m = 0" ``` paulson@3339 ` 331` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` paulson@3234 ` 332` ```Addsimps [diff_self_eq_0]; ``` paulson@3234 ` 333` paulson@3234 ` 334` ```(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *) ``` paulson@3381 ` 335` ```goal Arith.thy "~ m n+(m-n) = (m::nat)"; ``` paulson@3234 ` 336` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3352 ` 337` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3381 ` 338` ```qed_spec_mp "add_diff_inverse"; ``` paulson@3381 ` 339` paulson@3381 ` 340` ```goal Arith.thy "!!m. n<=m ==> n+(m-n) = (m::nat)"; ``` wenzelm@4089 ` 341` ```by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1); ``` paulson@3381 ` 342` ```qed "le_add_diff_inverse"; ``` paulson@3234 ` 343` paulson@3381 ` 344` ```goal Arith.thy "!!m. n<=m ==> (m-n)+n = (m::nat)"; ``` wenzelm@4089 ` 345` ```by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1); ``` paulson@3381 ` 346` ```qed "le_add_diff_inverse2"; ``` paulson@3381 ` 347` paulson@3381 ` 348` ```Addsimps [le_add_diff_inverse, le_add_diff_inverse2]; ``` paulson@3234 ` 349` paulson@3234 ` 350` paulson@3234 ` 351` ```(*** More results about difference ***) ``` paulson@3234 ` 352` paulson@3352 ` 353` ```val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)"; ``` paulson@3352 ` 354` ```by (rtac (prem RS rev_mp) 1); ``` paulson@3352 ` 355` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3352 ` 356` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3352 ` 357` ```qed "Suc_diff_n"; ``` paulson@3352 ` 358` paulson@3234 ` 359` ```goal Arith.thy "m - n < Suc(m)"; ``` paulson@3234 ` 360` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3234 ` 361` ```by (etac less_SucE 3); ``` wenzelm@4089 ` 362` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq]))); ``` paulson@3234 ` 363` ```qed "diff_less_Suc"; ``` paulson@3234 ` 364` paulson@3234 ` 365` ```goal Arith.thy "!!m::nat. m - n <= m"; ``` paulson@3234 ` 366` ```by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1); ``` paulson@3234 ` 367` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 368` ```qed "diff_le_self"; ``` paulson@3903 ` 369` ```Addsimps [diff_le_self]; ``` paulson@3234 ` 370` paulson@3352 ` 371` ```goal Arith.thy "!!i::nat. i-j-k = i - (j+k)"; ``` paulson@3352 ` 372` ```by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); ``` paulson@3352 ` 373` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3352 ` 374` ```qed "diff_diff_left"; ``` paulson@3352 ` 375` nipkow@4360 ` 376` ```(* This is a trivial consequence of diff_diff_left; ``` nipkow@4360 ` 377` ``` could be got rid of if diff_diff_left were in the simpset... ``` nipkow@4360 ` 378` ```*) ``` nipkow@4360 ` 379` ```goal Arith.thy "(Suc m - n)-1 = m - n"; ``` nipkow@4360 ` 380` ```by(simp_tac (simpset() addsimps [diff_diff_left]) 1); ``` nipkow@4360 ` 381` ```qed "pred_Suc_diff"; ``` nipkow@4360 ` 382` ```Addsimps [pred_Suc_diff]; ``` nipkow@4360 ` 383` wenzelm@3396 ` 384` ```(*This and the next few suggested by Florian Kammueller*) ``` paulson@3352 ` 385` ```goal Arith.thy "!!i::nat. i-j-k = i-k-j"; ``` wenzelm@4089 ` 386` ```by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1); ``` paulson@3352 ` 387` ```qed "diff_commute"; ``` paulson@3352 ` 388` paulson@3352 ` 389` ```goal Arith.thy "!!i j k:: nat. k<=j --> j<=i --> i - (j - k) = i - j + k"; ``` paulson@3352 ` 390` ```by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); ``` paulson@3352 ` 391` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3352 ` 392` ```by (asm_simp_tac ``` wenzelm@4089 ` 393` ``` (simpset() addsimps [Suc_diff_n, le_imp_less_Suc, le_Suc_eq]) 1); ``` paulson@3352 ` 394` ```qed_spec_mp "diff_diff_right"; ``` paulson@3352 ` 395` paulson@3352 ` 396` ```goal Arith.thy "!!i j k:: nat. k<=j --> (i + j) - k = i + (j - k)"; ``` paulson@3352 ` 397` ```by (res_inst_tac [("m","j"),("n","k")] diff_induct 1); ``` paulson@3352 ` 398` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3352 ` 399` ```qed_spec_mp "diff_add_assoc"; ``` paulson@3352 ` 400` paulson@3234 ` 401` ```goal Arith.thy "!!n::nat. (n+m) - n = m"; ``` paulson@3339 ` 402` ```by (induct_tac "n" 1); ``` paulson@3234 ` 403` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 404` ```qed "diff_add_inverse"; ``` paulson@3234 ` 405` ```Addsimps [diff_add_inverse]; ``` paulson@3234 ` 406` paulson@3234 ` 407` ```goal Arith.thy "!!n::nat.(m+n) - n = m"; ``` wenzelm@4089 ` 408` ```by (simp_tac (simpset() addsimps [diff_add_assoc]) 1); ``` paulson@3234 ` 409` ```qed "diff_add_inverse2"; ``` paulson@3234 ` 410` ```Addsimps [diff_add_inverse2]; ``` paulson@3234 ` 411` paulson@3366 ` 412` ```goal Arith.thy "!!i j k::nat. i<=j ==> (j-i=k) = (j=k+i)"; ``` paulson@3724 ` 413` ```by Safe_tac; ``` paulson@3381 ` 414` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3366 ` 415` ```qed "le_imp_diff_is_add"; ``` paulson@3366 ` 416` paulson@3234 ` 417` ```val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0"; ``` paulson@3234 ` 418` ```by (rtac (prem RS rev_mp) 1); ``` paulson@3234 ` 419` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` wenzelm@4089 ` 420` ```by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1); ``` paulson@3352 ` 421` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 422` ```qed "less_imp_diff_is_0"; ``` paulson@3234 ` 423` paulson@3234 ` 424` ```val prems = goal Arith.thy "m-n = 0 --> n-m = 0 --> m=n"; ``` paulson@3234 ` 425` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3234 ` 426` ```by (REPEAT(Simp_tac 1 THEN TRY(atac 1))); ``` paulson@3234 ` 427` ```qed_spec_mp "diffs0_imp_equal"; ``` paulson@3234 ` 428` paulson@3234 ` 429` ```val [prem] = goal Arith.thy "m 0 (!n. P(Suc(n))--> P(n)) --> P(k-i)"; ``` paulson@3234 ` 441` ```by (res_inst_tac [("m","k"),("n","i")] diff_induct 1); ``` paulson@3718 ` 442` ```by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac)); ``` paulson@3234 ` 443` ```qed "zero_induct_lemma"; ``` paulson@3234 ` 444` paulson@3234 ` 445` ```val prems = goal Arith.thy "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)"; ``` paulson@3234 ` 446` ```by (rtac (diff_self_eq_0 RS subst) 1); ``` paulson@3234 ` 447` ```by (rtac (zero_induct_lemma RS mp RS mp) 1); ``` paulson@3234 ` 448` ```by (REPEAT (ares_tac ([impI,allI]@prems) 1)); ``` paulson@3234 ` 449` ```qed "zero_induct"; ``` paulson@3234 ` 450` paulson@3234 ` 451` ```goal Arith.thy "!!k::nat. (k+m) - (k+n) = m - n"; ``` paulson@3339 ` 452` ```by (induct_tac "k" 1); ``` paulson@3234 ` 453` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 454` ```qed "diff_cancel"; ``` paulson@3234 ` 455` ```Addsimps [diff_cancel]; ``` paulson@3234 ` 456` paulson@3234 ` 457` ```goal Arith.thy "!!m::nat. (m+k) - (n+k) = m - n"; ``` paulson@3234 ` 458` ```val add_commute_k = read_instantiate [("n","k")] add_commute; ``` wenzelm@4089 ` 459` ```by (asm_simp_tac (simpset() addsimps ([add_commute_k])) 1); ``` paulson@3234 ` 460` ```qed "diff_cancel2"; ``` paulson@3234 ` 461` ```Addsimps [diff_cancel2]; ``` paulson@3234 ` 462` paulson@3234 ` 463` ```(*From Clemens Ballarin*) ``` paulson@3234 ` 464` ```goal Arith.thy "!!n::nat. [| k<=n; n<=m |] ==> (m-k) - (n-k) = m-n"; ``` paulson@3234 ` 465` ```by (subgoal_tac "k<=n --> n<=m --> (m-k) - (n-k) = m-n" 1); ``` paulson@3234 ` 466` ```by (Asm_full_simp_tac 1); ``` paulson@3339 ` 467` ```by (induct_tac "k" 1); ``` paulson@3234 ` 468` ```by (Simp_tac 1); ``` paulson@3234 ` 469` ```(* Induction step *) ``` paulson@3234 ` 470` ```by (subgoal_tac "Suc na <= m --> n <= m --> Suc na <= n --> \ ``` paulson@3234 ` 471` ```\ Suc (m - Suc na) - Suc (n - Suc na) = m-n" 1); ``` paulson@3234 ` 472` ```by (Asm_full_simp_tac 1); ``` wenzelm@4089 ` 473` ```by (blast_tac (claset() addIs [le_trans]) 1); ``` wenzelm@4089 ` 474` ```by (auto_tac (claset() addIs [Suc_leD], simpset() delsimps [diff_Suc_Suc])); ``` wenzelm@4089 ` 475` ```by (asm_full_simp_tac (simpset() delsimps [Suc_less_eq] ``` paulson@3234 ` 476` ``` addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1); ``` paulson@3234 ` 477` ```qed "diff_right_cancel"; ``` paulson@3234 ` 478` paulson@3234 ` 479` ```goal Arith.thy "!!n::nat. n - (n+m) = 0"; ``` paulson@3339 ` 480` ```by (induct_tac "n" 1); ``` paulson@3234 ` 481` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 482` ```qed "diff_add_0"; ``` paulson@3234 ` 483` ```Addsimps [diff_add_0]; ``` paulson@3234 ` 484` paulson@3234 ` 485` ```(** Difference distributes over multiplication **) ``` paulson@3234 ` 486` paulson@3234 ` 487` ```goal Arith.thy "!!m::nat. (m - n) * k = (m * k) - (n * k)"; ``` paulson@3234 ` 488` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3234 ` 489` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 490` ```qed "diff_mult_distrib" ; ``` paulson@3234 ` 491` paulson@3234 ` 492` ```goal Arith.thy "!!m::nat. k * (m - n) = (k * m) - (k * n)"; ``` paulson@3234 ` 493` ```val mult_commute_k = read_instantiate [("m","k")] mult_commute; ``` wenzelm@4089 ` 494` ```by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1); ``` paulson@3234 ` 495` ```qed "diff_mult_distrib2" ; ``` paulson@3234 ` 496` ```(*NOT added as rewrites, since sometimes they are used from right-to-left*) ``` paulson@3234 ` 497` paulson@3234 ` 498` paulson@1713 ` 499` ```(*** Monotonicity of Multiplication ***) ``` paulson@1713 ` 500` paulson@1713 ` 501` ```goal Arith.thy "!!i::nat. i<=j ==> i*k<=j*k"; ``` paulson@3339 ` 502` ```by (induct_tac "k" 1); ``` wenzelm@4089 ` 503` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono]))); ``` paulson@1713 ` 504` ```qed "mult_le_mono1"; ``` paulson@1713 ` 505` paulson@1713 ` 506` ```(*<=monotonicity, BOTH arguments*) ``` paulson@1713 ` 507` ```goal Arith.thy "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l"; ``` paulson@2007 ` 508` ```by (etac (mult_le_mono1 RS le_trans) 1); ``` paulson@1713 ` 509` ```by (rtac le_trans 1); ``` paulson@2007 ` 510` ```by (stac mult_commute 2); ``` paulson@2007 ` 511` ```by (etac mult_le_mono1 2); ``` wenzelm@4089 ` 512` ```by (simp_tac (simpset() addsimps [mult_commute]) 1); ``` paulson@1713 ` 513` ```qed "mult_le_mono"; ``` paulson@1713 ` 514` paulson@1713 ` 515` ```(*strict, in 1st argument; proof is by induction on k>0*) ``` paulson@1713 ` 516` ```goal Arith.thy "!!i::nat. [| i k*i < k*j"; ``` paulson@3339 ` 517` ```by (eres_inst_tac [("i","0")] less_natE 1); ``` paulson@1713 ` 518` ```by (Asm_simp_tac 1); ``` paulson@3339 ` 519` ```by (induct_tac "x" 1); ``` wenzelm@4089 ` 520` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono]))); ``` paulson@1713 ` 521` ```qed "mult_less_mono2"; ``` paulson@1713 ` 522` paulson@3234 ` 523` ```goal Arith.thy "!!i::nat. [| i i*k < j*k"; ``` paulson@3457 ` 524` ```by (dtac mult_less_mono2 1); ``` wenzelm@4089 ` 525` ```by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute]))); ``` paulson@3234 ` 526` ```qed "mult_less_mono1"; ``` paulson@3234 ` 527` paulson@1713 ` 528` ```goal Arith.thy "(0 < m*n) = (0 (m*k < n*k) = (m (k*m < k*n) = (m (m*k = n*k) = (m=n)"; ``` paulson@3234 ` 567` ```by (cut_facts_tac [less_linear] 1); ``` paulson@3724 ` 568` ```by Safe_tac; ``` paulson@3457 ` 569` ```by (assume_tac 2); ``` paulson@3234 ` 570` ```by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac)); ``` paulson@3234 ` 571` ```by (ALLGOALS Asm_full_simp_tac); ``` paulson@3234 ` 572` ```qed "mult_cancel2"; ``` paulson@3234 ` 573` paulson@3234 ` 574` ```goal Arith.thy "!!k. 0 (k*m = k*n) = (m=n)"; ``` paulson@3457 ` 575` ```by (dtac mult_cancel2 1); ``` wenzelm@4089 ` 576` ```by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1); ``` paulson@3234 ` 577` ```qed "mult_cancel1"; ``` paulson@3234 ` 578` ```Addsimps [mult_cancel1, mult_cancel2]; ``` paulson@3234 ` 579` wenzelm@4297 ` 580` ```goal Arith.thy "(Suc k * m = Suc k * n) = (m = n)"; ``` wenzelm@4297 ` 581` ```br mult_cancel1 1; ``` wenzelm@4297 ` 582` ```by (Simp_tac 1); ``` wenzelm@4297 ` 583` ```qed "Suc_mult_cancel1"; ``` wenzelm@4297 ` 584` paulson@3234 ` 585` paulson@1795 ` 586` ```(** Lemma for gcd **) ``` paulson@1795 ` 587` paulson@1795 ` 588` ```goal Arith.thy "!!m n. m = m*n ==> n=1 | m=0"; ``` paulson@1795 ` 589` ```by (dtac sym 1); ``` paulson@1795 ` 590` ```by (rtac disjCI 1); ``` paulson@1795 ` 591` ```by (rtac nat_less_cases 1 THEN assume_tac 2); ``` wenzelm@4089 ` 592` ```by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1); ``` nipkow@4356 ` 593` ```by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1); ``` paulson@1795 ` 594` ```qed "mult_eq_self_implies_10"; ``` paulson@1795 ` 595` paulson@1795 ` 596` paulson@3234 ` 597` ```(*** Subtraction laws -- from Clemens Ballarin ***) ``` paulson@3234 ` 598` paulson@3234 ` 599` ```goal Arith.thy "!! a b c::nat. [| a < b; c <= a |] ==> a-c < b-c"; ``` paulson@3234 ` 600` ```by (subgoal_tac "c+(a-c) < c+(b-c)" 1); ``` paulson@3381 ` 601` ```by (Full_simp_tac 1); ``` paulson@3234 ` 602` ```by (subgoal_tac "c <= b" 1); ``` wenzelm@4089 ` 603` ```by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2); ``` paulson@3381 ` 604` ```by (Asm_simp_tac 1); ``` paulson@3234 ` 605` ```qed "diff_less_mono"; ``` paulson@3234 ` 606` paulson@3234 ` 607` ```goal Arith.thy "!! a b c::nat. a+b < c ==> a < c-b"; ``` paulson@3457 ` 608` ```by (dtac diff_less_mono 1); ``` paulson@3457 ` 609` ```by (rtac le_add2 1); ``` paulson@3234 ` 610` ```by (Asm_full_simp_tac 1); ``` paulson@3234 ` 611` ```qed "add_less_imp_less_diff"; ``` paulson@3234 ` 612` paulson@3234 ` 613` ```goal Arith.thy "!! n. n <= m ==> Suc m - n = Suc (m - n)"; ``` paulson@3457 ` 614` ```by (rtac Suc_diff_n 1); ``` wenzelm@4089 ` 615` ```by (asm_full_simp_tac (simpset() addsimps [le_eq_less_Suc]) 1); ``` paulson@3234 ` 616` ```qed "Suc_diff_le"; ``` paulson@3234 ` 617` paulson@3234 ` 618` ```goal Arith.thy "!! n. Suc i <= n ==> Suc (n - Suc i) = n - i"; ``` paulson@3234 ` 619` ```by (asm_full_simp_tac ``` wenzelm@4089 ` 620` ``` (simpset() addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1); ``` paulson@3234 ` 621` ```qed "Suc_diff_Suc"; ``` paulson@3234 ` 622` paulson@3234 ` 623` ```goal Arith.thy "!! i::nat. i <= n ==> n - (n - i) = i"; ``` paulson@3903 ` 624` ```by (etac rev_mp 1); ``` paulson@3903 ` 625` ```by (res_inst_tac [("m","n"),("n","i")] diff_induct 1); ``` wenzelm@4089 ` 626` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [Suc_diff_le]))); ``` paulson@3234 ` 627` ```qed "diff_diff_cancel"; ``` paulson@3381 ` 628` ```Addsimps [diff_diff_cancel]; ``` paulson@3234 ` 629` paulson@3234 ` 630` ```goal Arith.thy "!!k::nat. k <= n ==> m <= n + m - k"; ``` paulson@3457 ` 631` ```by (etac rev_mp 1); ``` paulson@3234 ` 632` ```by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1); ``` paulson@3234 ` 633` ```by (Simp_tac 1); ``` wenzelm@4089 ` 634` ```by (simp_tac (simpset() addsimps [less_add_Suc2, less_imp_le]) 1); ``` paulson@3234 ` 635` ```by (Simp_tac 1); ``` paulson@3234 ` 636` ```qed "le_add_diff"; ``` paulson@3234 ` 637` paulson@3234 ` 638` nipkow@3484 ` 639` ```(** (Anti)Monotonicity of subtraction -- by Stefan Merz **) ``` nipkow@3484 ` 640` nipkow@3484 ` 641` ```(* Monotonicity of subtraction in first argument *) ``` nipkow@3484 ` 642` ```goal Arith.thy "!!n::nat. m<=n --> (m-l) <= (n-l)"; ``` nipkow@3484 ` 643` ```by (induct_tac "n" 1); ``` nipkow@3484 ` 644` ```by (Simp_tac 1); ``` wenzelm@4089 ` 645` ```by (simp_tac (simpset() addsimps [le_Suc_eq]) 1); ``` nipkow@3484 ` 646` ```by (rtac impI 1); ``` nipkow@3484 ` 647` ```by (etac impE 1); ``` nipkow@3484 ` 648` ```by (atac 1); ``` nipkow@3484 ` 649` ```by (etac le_trans 1); ``` nipkow@3484 ` 650` ```by (res_inst_tac [("m1","n")] (pred_Suc_diff RS subst) 1); ``` nipkow@4360 ` 651` ```by (simp_tac (simpset() addsimps [diff_Suc] addsplits [expand_nat_case]) 1); ``` nipkow@3484 ` 652` ```qed_spec_mp "diff_le_mono"; ``` nipkow@3484 ` 653` nipkow@3484 ` 654` ```goal Arith.thy "!!n::nat. m<=n ==> (l-n) <= (l-m)"; ``` nipkow@3484 ` 655` ```by (induct_tac "l" 1); ``` nipkow@3484 ` 656` ```by (Simp_tac 1); ``` nipkow@3484 ` 657` ```by (case_tac "n <= l" 1); ``` nipkow@3484 ` 658` ```by (subgoal_tac "m <= l" 1); ``` wenzelm@4089 ` 659` ```by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1); ``` wenzelm@4089 ` 660` ```by (fast_tac (claset() addEs [le_trans]) 1); ``` nipkow@3484 ` 661` ```by (dtac not_leE 1); ``` wenzelm@4089 ` 662` ```by (asm_simp_tac (simpset() addsimps [if_Suc_diff_n]) 1); ``` nipkow@3484 ` 663` ```qed_spec_mp "diff_le_mono2"; ```