src/HOL/Arith.ML
 author nipkow Fri Oct 16 17:32:06 1998 +0200 (1998-10-16) changeset 5654 8b872d546b9e parent 5604 cd17004d09e1 child 5758 27a2b36efd95 permissions -rw-r--r--
Installed trans_tac in solver of simpset().
 clasohm@1465 ` 1` ```(* Title: HOL/Arith.ML ``` clasohm@923 ` 2` ``` ID: \$Id\$ ``` clasohm@1465 ` 3` ``` Author: Lawrence C Paulson, Cambridge University Computer Laboratory ``` paulson@4736 ` 4` ``` Copyright 1998 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` paulson@4732 ` 15` ```qed_goal "diff_0_eq_0" thy ``` clasohm@923 ` 16` ``` "0 - n = 0" ``` paulson@3339 ` 17` ``` (fn _ => [induct_tac "n" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 18` paulson@5429 ` 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) *) ``` paulson@4732 ` 21` ```qed_goal "diff_Suc_Suc" 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 *) ``` paulson@5143 ` 31` ```Goal "0 < n ==> Suc(n-1) = n"; ``` berghofe@5183 ` 32` ```by (asm_simp_tac (simpset() addsplits [nat.split]) 1); ``` nipkow@4360 ` 33` ```qed "Suc_pred"; ``` nipkow@4360 ` 34` ```Addsimps [Suc_pred]; ``` nipkow@4360 ` 35` nipkow@4360 ` 36` ```Delsimps [diff_Suc]; ``` nipkow@4360 ` 37` clasohm@923 ` 38` clasohm@923 ` 39` ```(**** Inductive properties of the operators ****) ``` clasohm@923 ` 40` clasohm@923 ` 41` ```(*** Addition ***) ``` clasohm@923 ` 42` paulson@4732 ` 43` ```qed_goal "add_0_right" thy "m + 0 = m" ``` paulson@3339 ` 44` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 45` paulson@4732 ` 46` ```qed_goal "add_Suc_right" thy "m + Suc(n) = Suc(m+n)" ``` paulson@3339 ` 47` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 48` clasohm@1264 ` 49` ```Addsimps [add_0_right,add_Suc_right]; ``` clasohm@923 ` 50` clasohm@923 ` 51` ```(*Associative law for addition*) ``` paulson@4732 ` 52` ```qed_goal "add_assoc" thy "(m + n) + k = m + ((n + k)::nat)" ``` paulson@3339 ` 53` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 54` clasohm@923 ` 55` ```(*Commutative law for addition*) ``` paulson@4732 ` 56` ```qed_goal "add_commute" thy "m + n = n + (m::nat)" ``` paulson@3339 ` 57` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` clasohm@923 ` 58` paulson@4732 ` 59` ```qed_goal "add_left_commute" thy "x+(y+z)=y+((x+z)::nat)" ``` clasohm@923 ` 60` ``` (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1, ``` clasohm@923 ` 61` ``` rtac (add_commute RS arg_cong) 1]); ``` clasohm@923 ` 62` clasohm@923 ` 63` ```(*Addition is an AC-operator*) ``` clasohm@923 ` 64` ```val add_ac = [add_assoc, add_commute, add_left_commute]; ``` clasohm@923 ` 65` paulson@5429 ` 66` ```Goal "(k + m = k + n) = (m=(n::nat))"; ``` paulson@3339 ` 67` ```by (induct_tac "k" 1); ``` clasohm@1264 ` 68` ```by (Simp_tac 1); ``` clasohm@1264 ` 69` ```by (Asm_simp_tac 1); ``` clasohm@923 ` 70` ```qed "add_left_cancel"; ``` clasohm@923 ` 71` paulson@5429 ` 72` ```Goal "(m + k = n + k) = (m=(n::nat))"; ``` 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_right_cancel"; ``` clasohm@923 ` 77` paulson@5429 ` 78` ```Goal "(k + m <= k + n) = (m<=(n::nat))"; ``` 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_left_cancel_le"; ``` clasohm@923 ` 83` paulson@5429 ` 84` ```Goal "(k + m < k + n) = (m<(n::nat))"; ``` 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_less"; ``` clasohm@923 ` 89` nipkow@1327 ` 90` ```Addsimps [add_left_cancel, add_right_cancel, ``` nipkow@1327 ` 91` ``` add_left_cancel_le, add_left_cancel_less]; ``` nipkow@1327 ` 92` paulson@3339 ` 93` ```(** Reasoning about m+0=0, etc. **) ``` paulson@3339 ` 94` wenzelm@5069 ` 95` ```Goal "(m+n = 0) = (m=0 & n=0)"; ``` nipkow@5598 ` 96` ```by (exhaust_tac "m" 1); ``` nipkow@5598 ` 97` ```by (Auto_tac); ``` nipkow@1327 ` 98` ```qed "add_is_0"; ``` nipkow@4360 ` 99` ```AddIffs [add_is_0]; ``` nipkow@1327 ` 100` nipkow@5598 ` 101` ```Goal "(0 = m+n) = (m=0 & n=0)"; ``` nipkow@5598 ` 102` ```by (exhaust_tac "m" 1); ``` nipkow@5598 ` 103` ```by (Auto_tac); ``` nipkow@5598 ` 104` ```qed "zero_is_add"; ``` nipkow@5598 ` 105` ```AddIffs [zero_is_add]; ``` nipkow@5598 ` 106` nipkow@5598 ` 107` ```Goal "(m+n=1) = (m=1 & n=0 | m=0 & n=1)"; ``` nipkow@5598 ` 108` ```by(exhaust_tac "m" 1); ``` nipkow@5598 ` 109` ```by(Auto_tac); ``` nipkow@5598 ` 110` ```qed "add_is_1"; ``` nipkow@5598 ` 111` nipkow@5598 ` 112` ```Goal "(1=m+n) = (m=1 & n=0 | m=0 & n=1)"; ``` nipkow@5598 ` 113` ```by(exhaust_tac "m" 1); ``` nipkow@5598 ` 114` ```by(Auto_tac); ``` nipkow@5598 ` 115` ```qed "one_is_add"; ``` nipkow@5598 ` 116` wenzelm@5069 ` 117` ```Goal "(0 m+(n-(Suc k)) = (m+n)-(Suc k)" *) ``` paulson@5143 ` 130` ```Goal "0 m + (n-1) = (m+n)-1"; ``` nipkow@4360 ` 131` ```by (exhaust_tac "m" 1); ``` nipkow@4360 ` 132` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc] ``` berghofe@5183 ` 133` ``` addsplits [nat.split]))); ``` nipkow@1327 ` 134` ```qed "add_pred"; ``` nipkow@1327 ` 135` ```Addsimps [add_pred]; ``` nipkow@1327 ` 136` paulson@5429 ` 137` ```Goal "m + n = m ==> n = 0"; ``` paulson@5078 ` 138` ```by (dtac (add_0_right RS ssubst) 1); ``` paulson@5078 ` 139` ```by (asm_full_simp_tac (simpset() addsimps [add_assoc] ``` paulson@5078 ` 140` ``` delsimps [add_0_right]) 1); ``` paulson@5078 ` 141` ```qed "add_eq_self_zero"; ``` paulson@5078 ` 142` paulson@1626 ` 143` clasohm@923 ` 144` ```(**** Additional theorems about "less than" ****) ``` clasohm@923 ` 145` paulson@5078 ` 146` ```(*Deleted less_natE; instead use less_eq_Suc_add RS exE*) ``` paulson@5143 ` 147` ```Goal "m (? k. n=Suc(m+k))"; ``` paulson@3339 ` 148` ```by (induct_tac "n" 1); ``` paulson@5604 ` 149` ```by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less]))); ``` wenzelm@4089 ` 150` ```by (blast_tac (claset() addSEs [less_SucE] ``` paulson@5497 ` 151` ``` addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1); ``` nipkow@1485 ` 152` ```qed_spec_mp "less_eq_Suc_add"; ``` clasohm@923 ` 153` wenzelm@5069 ` 154` ```Goal "n <= ((m + n)::nat)"; ``` paulson@3339 ` 155` ```by (induct_tac "m" 1); ``` clasohm@1264 ` 156` ```by (ALLGOALS Simp_tac); ``` clasohm@923 ` 157` ```qed "le_add2"; ``` clasohm@923 ` 158` wenzelm@5069 ` 159` ```Goal "n <= ((n + m)::nat)"; ``` wenzelm@4089 ` 160` ```by (simp_tac (simpset() addsimps add_ac) 1); ``` clasohm@923 ` 161` ```by (rtac le_add2 1); ``` clasohm@923 ` 162` ```qed "le_add1"; ``` clasohm@923 ` 163` clasohm@923 ` 164` ```bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans))); ``` clasohm@923 ` 165` ```bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans))); ``` clasohm@923 ` 166` paulson@5429 ` 167` ```Goal "(m i <= j+m"*) ``` clasohm@923 ` 173` ```bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans)); ``` clasohm@923 ` 174` clasohm@923 ` 175` ```(*"i <= j ==> i <= m+j"*) ``` clasohm@923 ` 176` ```bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans)); ``` clasohm@923 ` 177` clasohm@923 ` 178` ```(*"i < j ==> i < j+m"*) ``` clasohm@923 ` 179` ```bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans)); ``` clasohm@923 ` 180` clasohm@923 ` 181` ```(*"i < j ==> i < m+j"*) ``` clasohm@923 ` 182` ```bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans)); ``` clasohm@923 ` 183` nipkow@5654 ` 184` ```Goal "i+j < (k::nat) --> i m<=(n::nat)"; ``` paulson@3339 ` 200` ```by (induct_tac "k" 1); ``` paulson@5497 ` 201` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps))); ``` nipkow@1485 ` 202` ```qed_spec_mp "add_leD1"; ``` clasohm@923 ` 203` paulson@5429 ` 204` ```Goal "m+k<=n ==> k<=(n::nat)"; ``` wenzelm@4089 ` 205` ```by (full_simp_tac (simpset() addsimps [add_commute]) 1); ``` paulson@2498 ` 206` ```by (etac add_leD1 1); ``` paulson@2498 ` 207` ```qed_spec_mp "add_leD2"; ``` paulson@2498 ` 208` paulson@5429 ` 209` ```Goal "m+k<=n ==> m<=n & k<=(n::nat)"; ``` wenzelm@4089 ` 210` ```by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1); ``` paulson@2498 ` 211` ```bind_thm ("add_leE", result() RS conjE); ``` paulson@2498 ` 212` paulson@5429 ` 213` ```(*needs !!k for add_ac to work*) ``` paulson@5429 ` 214` ```Goal "!!k:: nat. [| k m i + k < j + (k::nat)"; ``` paulson@3339 ` 226` ```by (induct_tac "k" 1); ``` clasohm@1264 ` 227` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 228` ```qed "add_less_mono1"; ``` clasohm@923 ` 229` clasohm@923 ` 230` ```(*strict, in both arguments*) ``` paulson@5429 ` 231` ```Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)"; ``` clasohm@923 ` 232` ```by (rtac (add_less_mono1 RS less_trans) 1); ``` lcp@1198 ` 233` ```by (REPEAT (assume_tac 1)); ``` paulson@3339 ` 234` ```by (induct_tac "j" 1); ``` clasohm@1264 ` 235` ```by (ALLGOALS Asm_simp_tac); ``` clasohm@923 ` 236` ```qed "add_less_mono"; ``` clasohm@923 ` 237` clasohm@923 ` 238` ```(*A [clumsy] way of lifting < monotonicity to <= monotonicity *) ``` paulson@5316 ` 239` ```val [lt_mono,le] = Goal ``` clasohm@1465 ` 240` ``` "[| !!i j::nat. i f(i) < f(j); \ ``` clasohm@1465 ` 241` ```\ i <= j \ ``` clasohm@923 ` 242` ```\ |] ==> f(i) <= (f(j)::nat)"; ``` clasohm@923 ` 243` ```by (cut_facts_tac [le] 1); ``` paulson@5604 ` 244` ```by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1); ``` wenzelm@4089 ` 245` ```by (blast_tac (claset() addSIs [lt_mono]) 1); ``` clasohm@923 ` 246` ```qed "less_mono_imp_le_mono"; ``` clasohm@923 ` 247` clasohm@923 ` 248` ```(*non-strict, in 1st argument*) ``` paulson@5429 ` 249` ```Goal "i<=j ==> i + k <= j + (k::nat)"; ``` wenzelm@3842 ` 250` ```by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1); ``` paulson@1552 ` 251` ```by (etac add_less_mono1 1); ``` clasohm@923 ` 252` ```by (assume_tac 1); ``` clasohm@923 ` 253` ```qed "add_le_mono1"; ``` clasohm@923 ` 254` clasohm@923 ` 255` ```(*non-strict, in both arguments*) ``` paulson@5429 ` 256` ```Goal "[|i<=j; k<=l |] ==> i + k <= j + (l::nat)"; ``` clasohm@923 ` 257` ```by (etac (add_le_mono1 RS le_trans) 1); ``` wenzelm@4089 ` 258` ```by (simp_tac (simpset() addsimps [add_commute]) 1); ``` clasohm@923 ` 259` ```qed "add_le_mono"; ``` paulson@1713 ` 260` paulson@3234 ` 261` paulson@3234 ` 262` ```(*** Multiplication ***) ``` paulson@3234 ` 263` paulson@3234 ` 264` ```(*right annihilation in product*) ``` paulson@4732 ` 265` ```qed_goal "mult_0_right" thy "m * 0 = 0" ``` paulson@3339 ` 266` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` paulson@3234 ` 267` paulson@3293 ` 268` ```(*right successor law for multiplication*) ``` paulson@4732 ` 269` ```qed_goal "mult_Suc_right" thy "m * Suc(n) = m + (m * n)" ``` paulson@3339 ` 270` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 271` ``` ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); ``` paulson@3234 ` 272` paulson@3293 ` 273` ```Addsimps [mult_0_right, mult_Suc_right]; ``` paulson@3234 ` 274` wenzelm@5069 ` 275` ```Goal "1 * n = n"; ``` paulson@3234 ` 276` ```by (Asm_simp_tac 1); ``` paulson@3234 ` 277` ```qed "mult_1"; ``` paulson@3234 ` 278` wenzelm@5069 ` 279` ```Goal "n * 1 = n"; ``` paulson@3234 ` 280` ```by (Asm_simp_tac 1); ``` paulson@3234 ` 281` ```qed "mult_1_right"; ``` paulson@3234 ` 282` paulson@3234 ` 283` ```(*Commutative law for multiplication*) ``` paulson@4732 ` 284` ```qed_goal "mult_commute" thy "m * n = n * (m::nat)" ``` paulson@3339 ` 285` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` paulson@3234 ` 286` paulson@3234 ` 287` ```(*addition distributes over multiplication*) ``` paulson@4732 ` 288` ```qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)" ``` paulson@3339 ` 289` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 290` ``` ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); ``` paulson@3234 ` 291` paulson@4732 ` 292` ```qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)" ``` paulson@3339 ` 293` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 294` ``` ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); ``` paulson@3234 ` 295` paulson@3234 ` 296` ```(*Associative law for multiplication*) ``` paulson@4732 ` 297` ```qed_goal "mult_assoc" thy "(m * n) * k = m * ((n * k)::nat)" ``` paulson@3339 ` 298` ``` (fn _ => [induct_tac "m" 1, ``` wenzelm@4089 ` 299` ``` ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]); ``` paulson@3234 ` 300` paulson@4732 ` 301` ```qed_goal "mult_left_commute" thy "x*(y*z) = y*((x*z)::nat)" ``` paulson@3234 ` 302` ``` (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1, ``` paulson@3234 ` 303` ``` rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]); ``` paulson@3234 ` 304` paulson@3234 ` 305` ```val mult_ac = [mult_assoc,mult_commute,mult_left_commute]; ``` paulson@3234 ` 306` wenzelm@5069 ` 307` ```Goal "(m*n = 0) = (m=0 | n=0)"; ``` paulson@3339 ` 308` ```by (induct_tac "m" 1); ``` paulson@3339 ` 309` ```by (induct_tac "n" 2); ``` paulson@3293 ` 310` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3293 ` 311` ```qed "mult_is_0"; ``` paulson@3293 ` 312` ```Addsimps [mult_is_0]; ``` paulson@3293 ` 313` paulson@5429 ` 314` ```Goal "m <= m*(m::nat)"; ``` paulson@4158 ` 315` ```by (induct_tac "m" 1); ``` paulson@4158 ` 316` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym]))); ``` paulson@4158 ` 317` ```by (etac (le_add2 RSN (2,le_trans)) 1); ``` paulson@4158 ` 318` ```qed "le_square"; ``` paulson@4158 ` 319` paulson@3234 ` 320` paulson@3234 ` 321` ```(*** Difference ***) ``` paulson@3234 ` 322` paulson@3234 ` 323` paulson@4732 ` 324` ```qed_goal "diff_self_eq_0" thy "m - m = 0" ``` paulson@3339 ` 325` ``` (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); ``` paulson@3234 ` 326` ```Addsimps [diff_self_eq_0]; ``` paulson@3234 ` 327` paulson@3234 ` 328` ```(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *) ``` wenzelm@5069 ` 329` ```Goal "~ m n+(m-n) = (m::nat)"; ``` paulson@3234 ` 330` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3352 ` 331` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3381 ` 332` ```qed_spec_mp "add_diff_inverse"; ``` paulson@3381 ` 333` paulson@5143 ` 334` ```Goal "n<=m ==> n+(m-n) = (m::nat)"; ``` wenzelm@4089 ` 335` ```by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1); ``` paulson@3381 ` 336` ```qed "le_add_diff_inverse"; ``` paulson@3234 ` 337` paulson@5143 ` 338` ```Goal "n<=m ==> (m-n)+n = (m::nat)"; ``` wenzelm@4089 ` 339` ```by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1); ``` paulson@3381 ` 340` ```qed "le_add_diff_inverse2"; ``` paulson@3381 ` 341` paulson@3381 ` 342` ```Addsimps [le_add_diff_inverse, le_add_diff_inverse2]; ``` paulson@3234 ` 343` paulson@3234 ` 344` paulson@3234 ` 345` ```(*** More results about difference ***) ``` paulson@3234 ` 346` paulson@5414 ` 347` ```Goal "n <= m ==> Suc(m)-n = Suc(m-n)"; ``` paulson@5316 ` 348` ```by (etac rev_mp 1); ``` paulson@3352 ` 349` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3352 ` 350` ```by (ALLGOALS Asm_simp_tac); ``` paulson@5414 ` 351` ```qed "Suc_diff_le"; ``` paulson@3352 ` 352` paulson@5429 ` 353` ```Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)"; ``` paulson@5429 ` 354` ```by (res_inst_tac [("m","n"),("n","l")] diff_induct 1); ``` paulson@5429 ` 355` ```by (ALLGOALS Asm_simp_tac); ``` paulson@5429 ` 356` ```qed_spec_mp "Suc_diff_add_le"; ``` paulson@5429 ` 357` wenzelm@5069 ` 358` ```Goal "m - n < Suc(m)"; ``` paulson@3234 ` 359` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3234 ` 360` ```by (etac less_SucE 3); ``` wenzelm@4089 ` 361` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq]))); ``` paulson@3234 ` 362` ```qed "diff_less_Suc"; ``` paulson@3234 ` 363` paulson@5429 ` 364` ```Goal "m - n <= (m::nat)"; ``` paulson@3234 ` 365` ```by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1); ``` paulson@3234 ` 366` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 367` ```qed "diff_le_self"; ``` paulson@3903 ` 368` ```Addsimps [diff_le_self]; ``` paulson@3234 ` 369` paulson@4732 ` 370` ```(* j j-n < k *) ``` paulson@4732 ` 371` ```bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans); ``` paulson@4732 ` 372` wenzelm@5069 ` 373` ```Goal "!!i::nat. i-j-k = i - (j+k)"; ``` paulson@3352 ` 374` ```by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); ``` paulson@3352 ` 375` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3352 ` 376` ```qed "diff_diff_left"; ``` paulson@3352 ` 377` wenzelm@5069 ` 378` ```Goal "(Suc m - n) - Suc k = m - n - k"; ``` wenzelm@4423 ` 379` ```by (simp_tac (simpset() addsimps [diff_diff_left]) 1); ``` paulson@4736 ` 380` ```qed "Suc_diff_diff"; ``` paulson@4736 ` 381` ```Addsimps [Suc_diff_diff]; ``` nipkow@4360 ` 382` paulson@5143 ` 383` ```Goal "0 n - Suc i < n"; ``` berghofe@5183 ` 384` ```by (exhaust_tac "n" 1); ``` paulson@4732 ` 385` ```by Safe_tac; ``` paulson@5497 ` 386` ```by (asm_simp_tac (simpset() addsimps le_simps) 1); ``` paulson@4732 ` 387` ```qed "diff_Suc_less"; ``` paulson@4732 ` 388` ```Addsimps [diff_Suc_less]; ``` paulson@4732 ` 389` paulson@5329 ` 390` ```Goal "i n - Suc i < n - i"; ``` paulson@5329 ` 391` ```by (exhaust_tac "n" 1); ``` paulson@5497 ` 392` ```by (auto_tac (claset(), ``` paulson@5537 ` 393` ``` simpset() addsimps [Suc_diff_le]@le_simps)); ``` paulson@5329 ` 394` ```qed "diff_Suc_less_diff"; ``` paulson@5329 ` 395` wenzelm@3396 ` 396` ```(*This and the next few suggested by Florian Kammueller*) ``` wenzelm@5069 ` 397` ```Goal "!!i::nat. i-j-k = i-k-j"; ``` wenzelm@4089 ` 398` ```by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1); ``` paulson@3352 ` 399` ```qed "diff_commute"; ``` paulson@3352 ` 400` paulson@5429 ` 401` ```Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)"; ``` paulson@3352 ` 402` ```by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); ``` paulson@3352 ` 403` ```by (ALLGOALS Asm_simp_tac); ``` paulson@5414 ` 404` ```by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1); ``` paulson@3352 ` 405` ```qed_spec_mp "diff_diff_right"; ``` paulson@3352 ` 406` paulson@5429 ` 407` ```Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)"; ``` paulson@3352 ` 408` ```by (res_inst_tac [("m","j"),("n","k")] diff_induct 1); ``` paulson@3352 ` 409` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3352 ` 410` ```qed_spec_mp "diff_add_assoc"; ``` paulson@3352 ` 411` paulson@5429 ` 412` ```Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)"; ``` paulson@4732 ` 413` ```by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1); ``` paulson@4732 ` 414` ```qed_spec_mp "diff_add_assoc2"; ``` paulson@4732 ` 415` paulson@5429 ` 416` ```Goal "(n+m) - n = (m::nat)"; ``` paulson@3339 ` 417` ```by (induct_tac "n" 1); ``` paulson@3234 ` 418` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 419` ```qed "diff_add_inverse"; ``` paulson@3234 ` 420` ```Addsimps [diff_add_inverse]; ``` paulson@3234 ` 421` paulson@5429 ` 422` ```Goal "(m+n) - n = (m::nat)"; ``` wenzelm@4089 ` 423` ```by (simp_tac (simpset() addsimps [diff_add_assoc]) 1); ``` paulson@3234 ` 424` ```qed "diff_add_inverse2"; ``` paulson@3234 ` 425` ```Addsimps [diff_add_inverse2]; ``` paulson@3234 ` 426` paulson@5429 ` 427` ```Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)"; ``` paulson@3724 ` 428` ```by Safe_tac; ``` paulson@3381 ` 429` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3366 ` 430` ```qed "le_imp_diff_is_add"; ``` paulson@3366 ` 431` paulson@5356 ` 432` ```Goal "(m-n = 0) = (m <= n)"; ``` paulson@3234 ` 433` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@5497 ` 434` ```by (ALLGOALS Asm_simp_tac); ``` paulson@5356 ` 435` ```qed "diff_is_0_eq"; ``` paulson@5356 ` 436` ```Addsimps [diff_is_0_eq RS iffD2]; ``` paulson@3234 ` 437` paulson@5316 ` 438` ```Goal "m-n = 0 --> n-m = 0 --> m=n"; ``` paulson@3234 ` 439` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3234 ` 440` ```by (REPEAT(Simp_tac 1 THEN TRY(atac 1))); ``` paulson@3234 ` 441` ```qed_spec_mp "diffs0_imp_equal"; ``` paulson@3234 ` 442` paulson@5333 ` 443` ```Goal "(0 ? k. 0 (!n. P(Suc(n))--> P(n)) --> P(k-i)"; ``` paulson@3234 ` 463` ```by (res_inst_tac [("m","k"),("n","i")] diff_induct 1); ``` paulson@3718 ` 464` ```by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac)); ``` paulson@3234 ` 465` ```qed "zero_induct_lemma"; ``` paulson@3234 ` 466` paulson@5316 ` 467` ```val prems = Goal "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)"; ``` paulson@3234 ` 468` ```by (rtac (diff_self_eq_0 RS subst) 1); ``` paulson@3234 ` 469` ```by (rtac (zero_induct_lemma RS mp RS mp) 1); ``` paulson@3234 ` 470` ```by (REPEAT (ares_tac ([impI,allI]@prems) 1)); ``` paulson@3234 ` 471` ```qed "zero_induct"; ``` paulson@3234 ` 472` paulson@5429 ` 473` ```Goal "(k+m) - (k+n) = m - (n::nat)"; ``` paulson@3339 ` 474` ```by (induct_tac "k" 1); ``` paulson@3234 ` 475` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 476` ```qed "diff_cancel"; ``` paulson@3234 ` 477` ```Addsimps [diff_cancel]; ``` paulson@3234 ` 478` paulson@5429 ` 479` ```Goal "(m+k) - (n+k) = m - (n::nat)"; ``` paulson@3234 ` 480` ```val add_commute_k = read_instantiate [("n","k")] add_commute; ``` paulson@5537 ` 481` ```by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1); ``` paulson@3234 ` 482` ```qed "diff_cancel2"; ``` paulson@3234 ` 483` ```Addsimps [diff_cancel2]; ``` paulson@3234 ` 484` paulson@5414 ` 485` ```(*From Clemens Ballarin, proof by lcp*) ``` paulson@5429 ` 486` ```Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)"; ``` paulson@5414 ` 487` ```by (REPEAT (etac rev_mp 1)); ``` paulson@5414 ` 488` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@5414 ` 489` ```by (ALLGOALS Asm_simp_tac); ``` paulson@5414 ` 490` ```(*a confluence problem*) ``` paulson@5414 ` 491` ```by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1); ``` paulson@3234 ` 492` ```qed "diff_right_cancel"; ``` paulson@3234 ` 493` paulson@5429 ` 494` ```Goal "n - (n+m) = 0"; ``` paulson@3339 ` 495` ```by (induct_tac "n" 1); ``` paulson@3234 ` 496` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 497` ```qed "diff_add_0"; ``` paulson@3234 ` 498` ```Addsimps [diff_add_0]; ``` paulson@3234 ` 499` paulson@5409 ` 500` paulson@3234 ` 501` ```(** Difference distributes over multiplication **) ``` paulson@3234 ` 502` wenzelm@5069 ` 503` ```Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)"; ``` paulson@3234 ` 504` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@3234 ` 505` ```by (ALLGOALS Asm_simp_tac); ``` paulson@3234 ` 506` ```qed "diff_mult_distrib" ; ``` paulson@3234 ` 507` wenzelm@5069 ` 508` ```Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)"; ``` paulson@3234 ` 509` ```val mult_commute_k = read_instantiate [("m","k")] mult_commute; ``` wenzelm@4089 ` 510` ```by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1); ``` paulson@3234 ` 511` ```qed "diff_mult_distrib2" ; ``` paulson@3234 ` 512` ```(*NOT added as rewrites, since sometimes they are used from right-to-left*) ``` paulson@3234 ` 513` paulson@3234 ` 514` paulson@1713 ` 515` ```(*** Monotonicity of Multiplication ***) ``` paulson@1713 ` 516` paulson@5429 ` 517` ```Goal "i <= (j::nat) ==> i*k<=j*k"; ``` paulson@3339 ` 518` ```by (induct_tac "k" 1); ``` wenzelm@4089 ` 519` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono]))); ``` paulson@1713 ` 520` ```qed "mult_le_mono1"; ``` paulson@1713 ` 521` paulson@1713 ` 522` ```(*<=monotonicity, BOTH arguments*) ``` paulson@5429 ` 523` ```Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l"; ``` paulson@2007 ` 524` ```by (etac (mult_le_mono1 RS le_trans) 1); ``` paulson@1713 ` 525` ```by (rtac le_trans 1); ``` paulson@2007 ` 526` ```by (stac mult_commute 2); ``` paulson@2007 ` 527` ```by (etac mult_le_mono1 2); ``` wenzelm@4089 ` 528` ```by (simp_tac (simpset() addsimps [mult_commute]) 1); ``` paulson@1713 ` 529` ```qed "mult_le_mono"; ``` paulson@1713 ` 530` paulson@1713 ` 531` ```(*strict, in 1st argument; proof is by induction on k>0*) ``` paulson@5429 ` 532` ```Goal "[| i k*i < k*j"; ``` paulson@5078 ` 533` ```by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1); ``` paulson@1713 ` 534` ```by (Asm_simp_tac 1); ``` paulson@3339 ` 535` ```by (induct_tac "x" 1); ``` wenzelm@4089 ` 536` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono]))); ``` paulson@1713 ` 537` ```qed "mult_less_mono2"; ``` paulson@1713 ` 538` paulson@5429 ` 539` ```Goal "[| i i*k < j*k"; ``` paulson@3457 ` 540` ```by (dtac mult_less_mono2 1); ``` wenzelm@4089 ` 541` ```by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute]))); ``` paulson@3234 ` 542` ```qed "mult_less_mono1"; ``` paulson@3234 ` 543` wenzelm@5069 ` 544` ```Goal "(0 < m*n) = (0 (m*k < n*k) = (m (k*m < k*n) = (m (m*k = n*k) = (m=n)"; ``` paulson@3234 ` 583` ```by (cut_facts_tac [less_linear] 1); ``` paulson@3724 ` 584` ```by Safe_tac; ``` paulson@3457 ` 585` ```by (assume_tac 2); ``` paulson@3234 ` 586` ```by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac)); ``` paulson@3234 ` 587` ```by (ALLGOALS Asm_full_simp_tac); ``` paulson@3234 ` 588` ```qed "mult_cancel2"; ``` paulson@3234 ` 589` paulson@5143 ` 590` ```Goal "0 (k*m = k*n) = (m=n)"; ``` paulson@3457 ` 591` ```by (dtac mult_cancel2 1); ``` wenzelm@4089 ` 592` ```by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1); ``` paulson@3234 ` 593` ```qed "mult_cancel1"; ``` paulson@3234 ` 594` ```Addsimps [mult_cancel1, mult_cancel2]; ``` paulson@3234 ` 595` wenzelm@5069 ` 596` ```Goal "(Suc k * m = Suc k * n) = (m = n)"; ``` wenzelm@4423 ` 597` ```by (rtac mult_cancel1 1); ``` wenzelm@4297 ` 598` ```by (Simp_tac 1); ``` wenzelm@4297 ` 599` ```qed "Suc_mult_cancel1"; ``` wenzelm@4297 ` 600` paulson@3234 ` 601` paulson@1795 ` 602` ```(** Lemma for gcd **) ``` paulson@1795 ` 603` paulson@5143 ` 604` ```Goal "m = m*n ==> n=1 | m=0"; ``` paulson@1795 ` 605` ```by (dtac sym 1); ``` paulson@1795 ` 606` ```by (rtac disjCI 1); ``` paulson@1795 ` 607` ```by (rtac nat_less_cases 1 THEN assume_tac 2); ``` wenzelm@4089 ` 608` ```by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1); ``` nipkow@4356 ` 609` ```by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1); ``` paulson@1795 ` 610` ```qed "mult_eq_self_implies_10"; ``` paulson@1795 ` 611` paulson@1795 ` 612` paulson@4736 ` 613` ```(*** Subtraction laws -- mostly from Clemens Ballarin ***) ``` paulson@3234 ` 614` paulson@5429 ` 615` ```Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c"; ``` paulson@3234 ` 616` ```by (subgoal_tac "c+(a-c) < c+(b-c)" 1); ``` paulson@3381 ` 617` ```by (Full_simp_tac 1); ``` paulson@3234 ` 618` ```by (subgoal_tac "c <= b" 1); ``` wenzelm@4089 ` 619` ```by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2); ``` paulson@3381 ` 620` ```by (Asm_simp_tac 1); ``` paulson@3234 ` 621` ```qed "diff_less_mono"; ``` paulson@3234 ` 622` paulson@5429 ` 623` ```Goal "a+b < (c::nat) ==> a < c-b"; ``` paulson@3457 ` 624` ```by (dtac diff_less_mono 1); ``` paulson@3457 ` 625` ```by (rtac le_add2 1); ``` paulson@3234 ` 626` ```by (Asm_full_simp_tac 1); ``` paulson@3234 ` 627` ```qed "add_less_imp_less_diff"; ``` paulson@3234 ` 628` nipkow@5427 ` 629` ```Goal "(i < j-k) = (i+k < (j::nat))"; ``` paulson@5497 ` 630` ```by (rtac iffI 1); ``` paulson@5497 ` 631` ``` by (case_tac "k <= j" 1); ``` paulson@5497 ` 632` ``` by (dtac le_add_diff_inverse2 1); ``` paulson@5497 ` 633` ``` by (dres_inst_tac [("k","k")] add_less_mono1 1); ``` paulson@5497 ` 634` ``` by (Asm_full_simp_tac 1); ``` paulson@5497 ` 635` ``` by (rotate_tac 1 1); ``` paulson@5497 ` 636` ``` by (asm_full_simp_tac (simpset() addSolver cut_trans_tac) 1); ``` paulson@5497 ` 637` ```by (etac add_less_imp_less_diff 1); ``` nipkow@5427 ` 638` ```qed "less_diff_conv"; ``` nipkow@5427 ` 639` paulson@5497 ` 640` ```Goal "(j-k <= (i::nat)) = (j <= i+k)"; ``` paulson@5497 ` 641` ```by (simp_tac (simpset() addsimps [less_diff_conv, le_def]) 1); ``` paulson@5485 ` 642` ```qed "le_diff_conv"; ``` paulson@5485 ` 643` paulson@5497 ` 644` ```Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))"; ``` paulson@5497 ` 645` ```by (asm_full_simp_tac ``` paulson@5497 ` 646` ``` (simpset() delsimps [less_Suc_eq_le] ``` paulson@5497 ` 647` ``` addsimps [less_Suc_eq_le RS sym, less_diff_conv, ``` paulson@5497 ` 648` ``` Suc_diff_le RS sym]) 1); ``` paulson@5497 ` 649` ```qed "le_diff_conv2"; ``` paulson@5497 ` 650` paulson@5143 ` 651` ```Goal "Suc i <= n ==> Suc (n - Suc i) = n - i"; ``` paulson@5497 ` 652` ```by (asm_full_simp_tac (simpset() addsimps [Suc_diff_le RS sym]) 1); ``` paulson@3234 ` 653` ```qed "Suc_diff_Suc"; ``` paulson@3234 ` 654` paulson@5429 ` 655` ```Goal "i <= (n::nat) ==> n - (n - i) = i"; ``` paulson@3903 ` 656` ```by (etac rev_mp 1); ``` paulson@3903 ` 657` ```by (res_inst_tac [("m","n"),("n","i")] diff_induct 1); ``` wenzelm@4089 ` 658` ```by (ALLGOALS (asm_simp_tac (simpset() addsimps [Suc_diff_le]))); ``` paulson@3234 ` 659` ```qed "diff_diff_cancel"; ``` paulson@3381 ` 660` ```Addsimps [diff_diff_cancel]; ``` paulson@3234 ` 661` paulson@5429 ` 662` ```Goal "k <= (n::nat) ==> m <= n + m - k"; ``` paulson@3457 ` 663` ```by (etac rev_mp 1); ``` paulson@3234 ` 664` ```by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1); ``` paulson@3234 ` 665` ```by (Simp_tac 1); ``` paulson@5497 ` 666` ```by (simp_tac (simpset() addsimps [le_add2, less_imp_le]) 1); ``` paulson@3234 ` 667` ```by (Simp_tac 1); ``` paulson@3234 ` 668` ```qed "le_add_diff"; ``` paulson@3234 ` 669` paulson@5429 ` 670` ```Goal "0 j j+k-i < k"; ``` paulson@4736 ` 671` ```by (res_inst_tac [("m","j"),("n","i")] diff_induct 1); ``` paulson@4736 ` 672` ```by (ALLGOALS Asm_simp_tac); ``` paulson@4736 ` 673` ```qed_spec_mp "add_diff_less"; ``` paulson@4736 ` 674` paulson@3234 ` 675` paulson@5356 ` 676` ```Goal "m-1 < n ==> m <= n"; ``` paulson@5356 ` 677` ```by (exhaust_tac "m" 1); ``` paulson@5356 ` 678` ```by (auto_tac (claset(), simpset() addsimps [Suc_le_eq])); ``` paulson@5356 ` 679` ```qed "pred_less_imp_le"; ``` paulson@5356 ` 680` paulson@5356 ` 681` ```Goal "j<=i ==> i - j < Suc i - j"; ``` paulson@5356 ` 682` ```by (REPEAT (etac rev_mp 1)); ``` paulson@5356 ` 683` ```by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); ``` paulson@5356 ` 684` ```by Auto_tac; ``` paulson@5356 ` 685` ```qed "diff_less_Suc_diff"; ``` paulson@5356 ` 686` paulson@5356 ` 687` ```Goal "i - j <= Suc i - j"; ``` paulson@5356 ` 688` ```by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); ``` paulson@5356 ` 689` ```by Auto_tac; ``` paulson@5356 ` 690` ```qed "diff_le_Suc_diff"; ``` paulson@5356 ` 691` ```AddIffs [diff_le_Suc_diff]; ``` paulson@5356 ` 692` paulson@5356 ` 693` ```Goal "n - Suc i <= n - i"; ``` paulson@5356 ` 694` ```by (case_tac "i (m <= n-1) = (m (m-1 < n) = (m<=n)"; ``` paulson@5409 ` 706` ```by (exhaust_tac "m" 1); ``` paulson@5409 ` 707` ```by (auto_tac (claset(), simpset() addsimps [Suc_le_eq])); ``` paulson@5409 ` 708` ```qed "less_pred_eq"; ``` paulson@5409 ` 709` paulson@5414 ` 710` ```(*In ordinary notation: if 0 m - n < m"; ``` paulson@5414 ` 712` ```by (subgoal_tac "0 ~ m m - n < m" 1); ``` paulson@5414 ` 713` ```by (Blast_tac 1); ``` paulson@5414 ` 714` ```by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); ``` paulson@5414 ` 715` ```by (ALLGOALS(asm_simp_tac(simpset() addsimps [diff_less_Suc]))); ``` paulson@5414 ` 716` ```qed "diff_less"; ``` paulson@5414 ` 717` paulson@5414 ` 718` ```Goal "[| 0 m - n < m"; ``` paulson@5414 ` 719` ```by (asm_simp_tac (simpset() addsimps [diff_less, not_less_iff_le]) 1); ``` paulson@5414 ` 720` ```qed "le_diff_less"; ``` paulson@5414 ` 721` paulson@5356 ` 722` paulson@4732 ` 723` nipkow@3484 ` 724` ```(** (Anti)Monotonicity of subtraction -- by Stefan Merz **) ``` nipkow@3484 ` 725` nipkow@3484 ` 726` ```(* Monotonicity of subtraction in first argument *) ``` paulson@5429 ` 727` ```Goal "m <= (n::nat) --> (m-l) <= (n-l)"; ``` nipkow@3484 ` 728` ```by (induct_tac "n" 1); ``` nipkow@3484 ` 729` ```by (Simp_tac 1); ``` wenzelm@4089 ` 730` ```by (simp_tac (simpset() addsimps [le_Suc_eq]) 1); ``` paulson@4732 ` 731` ```by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1); ``` nipkow@3484 ` 732` ```qed_spec_mp "diff_le_mono"; ``` nipkow@3484 ` 733` paulson@5429 ` 734` ```Goal "m <= (n::nat) ==> (l-n) <= (l-m)"; ``` nipkow@3484 ` 735` ```by (induct_tac "l" 1); ``` nipkow@3484 ` 736` ```by (Simp_tac 1); ``` berghofe@5183 ` 737` ```by (case_tac "n <= na" 1); ``` berghofe@5183 ` 738` ```by (subgoal_tac "m <= na" 1); ``` wenzelm@4089 ` 739` ```by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1); ``` wenzelm@4089 ` 740` ```by (fast_tac (claset() addEs [le_trans]) 1); ``` nipkow@3484 ` 741` ```by (dtac not_leE 1); ``` paulson@5414 ` 742` ```by (asm_simp_tac (simpset() addsimps [if_Suc_diff_le]) 1); ``` nipkow@3484 ` 743` ```qed_spec_mp "diff_le_mono2"; ```