1section \<open>The Reals as Dedekind Sections of Positive Rationals\<close> 2 3text \<open>Fundamentals of Abstract Analysis [Gleason- p. 121] provides some of the definitions.\<close> 4 5(* Title: HOL/ex/Dedekind_Real.thy 6 Author: Jacques D. Fleuriot, University of Cambridge 7 Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4; 2019 8*) 9 10theory Dedekind_Real 11imports Complex_Main 12begin 13 14text\<open>Could be moved to \<open>Groups\<close>\<close> 15lemma add_eq_exists: "\<exists>x. a+x = (b::'a::ab_group_add)" 16 by (rule_tac x="b-a" in exI, simp) 17 18subsection \<open>Dedekind cuts or sections\<close> 19 20definition 21 cut :: "rat set \<Rightarrow> bool" where 22 "cut A \<equiv> {} \<subset> A \<and> A \<subset> {0<..} \<and> 23 (\<forall>y \<in> A. ((\<forall>z. 0<z \<and> z < y \<longrightarrow> z \<in> A) \<and> (\<exists>u \<in> A. y < u)))" 24 25lemma cut_of_rat: 26 assumes q: "0 < q" shows "cut {r::rat. 0 < r \<and> r < q}" (is "cut ?A") 27proof - 28 from q have pos: "?A \<subset> {0<..}" by force 29 have nonempty: "{} \<subset> ?A" 30 proof 31 show "{} \<subseteq> ?A" by simp 32 show "{} \<noteq> ?A" 33 using field_lbound_gt_zero q by auto 34 qed 35 show ?thesis 36 by (simp add: cut_def pos nonempty, 37 blast dest: dense intro: order_less_trans) 38qed 39 40 41typedef preal = "Collect cut" 42 by (blast intro: cut_of_rat [OF zero_less_one]) 43 44lemma Abs_preal_induct [induct type: preal]: 45 "(\<And>x. cut x \<Longrightarrow> P (Abs_preal x)) \<Longrightarrow> P x" 46 using Abs_preal_induct [of P x] by simp 47 48lemma cut_Rep_preal [simp]: "cut (Rep_preal x)" 49 using Rep_preal [of x] by simp 50 51definition 52 psup :: "preal set \<Rightarrow> preal" where 53 "psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)" 54 55definition 56 add_set :: "[rat set,rat set] \<Rightarrow> rat set" where 57 "add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}" 58 59definition 60 diff_set :: "[rat set,rat set] \<Rightarrow> rat set" where 61 "diff_set A B = {w. \<exists>x. 0 < w \<and> 0 < x \<and> x \<notin> B \<and> x + w \<in> A}" 62 63definition 64 mult_set :: "[rat set,rat set] \<Rightarrow> rat set" where 65 "mult_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}" 66 67definition 68 inverse_set :: "rat set \<Rightarrow> rat set" where 69 "inverse_set A \<equiv> {x. \<exists>y. 0 < x \<and> x < y \<and> inverse y \<notin> A}" 70 71instantiation preal :: "{ord, plus, minus, times, inverse, one}" 72begin 73 74definition 75 preal_less_def: 76 "r < s \<equiv> Rep_preal r < Rep_preal s" 77 78definition 79 preal_le_def: 80 "r \<le> s \<equiv> Rep_preal r \<subseteq> Rep_preal s" 81 82definition 83 preal_add_def: 84 "r + s \<equiv> Abs_preal (add_set (Rep_preal r) (Rep_preal s))" 85 86definition 87 preal_diff_def: 88 "r - s \<equiv> Abs_preal (diff_set (Rep_preal r) (Rep_preal s))" 89 90definition 91 preal_mult_def: 92 "r * s \<equiv> Abs_preal (mult_set (Rep_preal r) (Rep_preal s))" 93 94definition 95 preal_inverse_def: 96 "inverse r \<equiv> Abs_preal (inverse_set (Rep_preal r))" 97 98definition "r div s = r * inverse (s::preal)" 99 100definition 101 preal_one_def: 102 "1 \<equiv> Abs_preal {x. 0 < x \<and> x < 1}" 103 104instance .. 105 106end 107 108 109text\<open>Reduces equality on abstractions to equality on representatives\<close> 110declare Abs_preal_inject [simp] 111declare Abs_preal_inverse [simp] 112 113lemma rat_mem_preal: "0 < q \<Longrightarrow> cut {r::rat. 0 < r \<and> r < q}" 114by (simp add: cut_of_rat) 115 116lemma preal_nonempty: "cut A \<Longrightarrow> \<exists>x\<in>A. 0 < x" 117 unfolding cut_def [abs_def] by blast 118 119lemma preal_Ex_mem: "cut A \<Longrightarrow> \<exists>x. x \<in> A" 120 using preal_nonempty by blast 121 122lemma preal_exists_bound: "cut A \<Longrightarrow> \<exists>x. 0 < x \<and> x \<notin> A" 123 using Dedekind_Real.cut_def by fastforce 124 125lemma preal_exists_greater: "\<lbrakk>cut A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>u \<in> A. y < u" 126 unfolding cut_def [abs_def] by blast 127 128lemma preal_downwards_closed: "\<lbrakk>cut A; y \<in> A; 0 < z; z < y\<rbrakk> \<Longrightarrow> z \<in> A" 129 unfolding cut_def [abs_def] by blast 130 131text\<open>Relaxing the final premise\<close> 132lemma preal_downwards_closed': "\<lbrakk>cut A; y \<in> A; 0 < z; z \<le> y\<rbrakk> \<Longrightarrow> z \<in> A" 133 using less_eq_rat_def preal_downwards_closed by blast 134 135text\<open>A positive fraction not in a positive real is an upper bound. 136 Gleason p. 122 - Remark (1)\<close> 137 138lemma not_in_preal_ub: 139 assumes A: "cut A" 140 and notx: "x \<notin> A" 141 and y: "y \<in> A" 142 and pos: "0 < x" 143 shows "y < x" 144proof (cases rule: linorder_cases) 145 assume "x<y" 146 with notx show ?thesis 147 by (simp add: preal_downwards_closed [OF A y] pos) 148next 149 assume "x=y" 150 with notx and y show ?thesis by simp 151next 152 assume "y<x" 153 thus ?thesis . 154qed 155 156text \<open>preal lemmas instantiated to \<^term>\<open>Rep_preal X\<close>\<close> 157 158lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X" 159thm preal_Ex_mem 160by (rule preal_Ex_mem [OF cut_Rep_preal]) 161 162lemma Rep_preal_exists_bound: "\<exists>x>0. x \<notin> Rep_preal X" 163by (rule preal_exists_bound [OF cut_Rep_preal]) 164 165lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF cut_Rep_preal] 166 167 168subsection\<open>Properties of Ordering\<close> 169 170instance preal :: order 171proof 172 fix w :: preal 173 show "w \<le> w" by (simp add: preal_le_def) 174next 175 fix i j k :: preal 176 assume "i \<le> j" and "j \<le> k" 177 then show "i \<le> k" by (simp add: preal_le_def) 178next 179 fix z w :: preal 180 assume "z \<le> w" and "w \<le> z" 181 then show "z = w" by (simp add: preal_le_def Rep_preal_inject) 182next 183 fix z w :: preal 184 show "z < w \<longleftrightarrow> z \<le> w \<and> \<not> w \<le> z" 185 by (auto simp: preal_le_def preal_less_def Rep_preal_inject) 186qed 187 188lemma preal_imp_pos: "\<lbrakk>cut A; r \<in> A\<rbrakk> \<Longrightarrow> 0 < r" 189 by (auto simp: cut_def) 190 191instance preal :: linorder 192proof 193 fix x y :: preal 194 show "x \<le> y \<or> y \<le> x" 195 unfolding preal_le_def 196 by (meson cut_Rep_preal not_in_preal_ub preal_downwards_closed preal_imp_pos subsetI) 197qed 198 199instantiation preal :: distrib_lattice 200begin 201 202definition 203 "(inf :: preal \<Rightarrow> preal \<Rightarrow> preal) = min" 204 205definition 206 "(sup :: preal \<Rightarrow> preal \<Rightarrow> preal) = max" 207 208instance 209 by intro_classes 210 (auto simp: inf_preal_def sup_preal_def max_min_distrib2) 211 212end 213 214subsection\<open>Properties of Addition\<close> 215 216lemma preal_add_commute: "(x::preal) + y = y + x" 217 unfolding preal_add_def add_set_def 218 by (metis (no_types, hide_lams) add.commute) 219 220text\<open>Lemmas for proving that addition of two positive reals gives 221 a positive real\<close> 222 223lemma mem_add_set: 224 assumes "cut A" "cut B" 225 shows "cut (add_set A B)" 226proof - 227 have "{} \<subset> add_set A B" 228 using assms by (force simp: add_set_def dest: preal_nonempty) 229 moreover 230 obtain q where "q > 0" "q \<notin> add_set A B" 231 proof - 232 obtain a b where "a > 0" "a \<notin> A" "b > 0" "b \<notin> B" "\<And>x. x \<in> A \<Longrightarrow> x < a" "\<And>y. y \<in> B \<Longrightarrow> y < b" 233 by (meson assms preal_exists_bound not_in_preal_ub) 234 with assms have "a+b \<notin> add_set A B" 235 by (fastforce simp add: add_set_def) 236 then show thesis 237 using \<open>0 < a\<close> \<open>0 < b\<close> add_pos_pos that by blast 238 qed 239 then have "add_set A B \<subset> {0<..}" 240 unfolding add_set_def 241 using preal_imp_pos [OF \<open>cut A\<close>] preal_imp_pos [OF \<open>cut B\<close>] by fastforce 242 moreover have "z \<in> add_set A B" 243 if u: "u \<in> add_set A B" and "0 < z" "z < u" for u z 244 using u unfolding add_set_def 245 proof (clarify) 246 fix x::rat and y::rat 247 assume ueq: "u = x + y" and x: "x \<in> A" and y:"y \<in> B" 248 have xpos [simp]: "x > 0" and ypos [simp]: "y > 0" 249 using assms preal_imp_pos x y by blast+ 250 have xypos [simp]: "x+y > 0" by (simp add: pos_add_strict) 251 let ?f = "z/(x+y)" 252 have fless: "?f < 1" 253 using divide_less_eq_1_pos \<open>z < u\<close> ueq xypos by blast 254 show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'" 255 proof (intro bexI) 256 show "z = x*?f + y*?f" 257 by (simp add: distrib_right [symmetric] divide_inverse ac_simps order_less_imp_not_eq2) 258 next 259 show "y * ?f \<in> B" 260 proof (rule preal_downwards_closed [OF \<open>cut B\<close> y]) 261 show "0 < y * ?f" 262 by (simp add: \<open>0 < z\<close>) 263 next 264 show "y * ?f < y" 265 by (insert mult_strict_left_mono [OF fless ypos], simp) 266 qed 267 next 268 show "x * ?f \<in> A" 269 proof (rule preal_downwards_closed [OF \<open>cut A\<close> x]) 270 show "0 < x * ?f" 271 by (simp add: \<open>0 < z\<close>) 272 next 273 show "x * ?f < x" 274 by (insert mult_strict_left_mono [OF fless xpos], simp) 275 qed 276 qed 277 qed 278 moreover 279 have "\<And>y. y \<in> add_set A B \<Longrightarrow> \<exists>u \<in> add_set A B. y < u" 280 unfolding add_set_def using preal_exists_greater assms by fastforce 281 ultimately show ?thesis 282 by (simp add: Dedekind_Real.cut_def) 283qed 284 285lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)" 286 apply (simp add: preal_add_def mem_add_set) 287 apply (force simp: add_set_def ac_simps) 288 done 289 290instance preal :: ab_semigroup_add 291proof 292 fix a b c :: preal 293 show "(a + b) + c = a + (b + c)" by (rule preal_add_assoc) 294 show "a + b = b + a" by (rule preal_add_commute) 295qed 296 297 298subsection\<open>Properties of Multiplication\<close> 299 300text\<open>Proofs essentially same as for addition\<close> 301 302lemma preal_mult_commute: "(x::preal) * y = y * x" 303 unfolding preal_mult_def mult_set_def 304 by (metis (no_types, hide_lams) mult.commute) 305 306text\<open>Multiplication of two positive reals gives a positive real.\<close> 307 308lemma mem_mult_set: 309 assumes "cut A" "cut B" 310 shows "cut (mult_set A B)" 311proof - 312 have "{} \<subset> mult_set A B" 313 using assms 314 by (force simp: mult_set_def dest: preal_nonempty) 315 moreover 316 obtain q where "q > 0" "q \<notin> mult_set A B" 317 proof - 318 obtain x y where x [simp]: "0 < x" "x \<notin> A" and y [simp]: "0 < y" "y \<notin> B" 319 using preal_exists_bound assms by blast 320 show thesis 321 proof 322 show "0 < x*y" by simp 323 show "x * y \<notin> mult_set A B" 324 proof - 325 { 326 fix u::rat and v::rat 327 assume u: "u \<in> A" and v: "v \<in> B" and xy: "x*y = u*v" 328 moreover have "u<x" and "v<y" using assms x y u v by (blast dest: not_in_preal_ub)+ 329 moreover have "0\<le>v" 330 using less_imp_le preal_imp_pos assms x y u v by blast 331 moreover have "u*v < x*y" 332 using assms x \<open>u < x\<close> \<open>v < y\<close> \<open>0 \<le> v\<close> by (blast intro: mult_strict_mono) 333 ultimately have False by force 334 } 335 thus ?thesis by (auto simp: mult_set_def) 336 qed 337 qed 338 qed 339 then have "mult_set A B \<subset> {0<..}" 340 unfolding mult_set_def 341 using preal_imp_pos [OF \<open>cut A\<close>] preal_imp_pos [OF \<open>cut B\<close>] by fastforce 342 moreover have "z \<in> mult_set A B" 343 if u: "u \<in> mult_set A B" and "0 < z" "z < u" for u z 344 using u unfolding mult_set_def 345 proof (clarify) 346 fix x::rat and y::rat 347 assume ueq: "u = x * y" and x: "x \<in> A" and y: "y \<in> B" 348 have [simp]: "y > 0" 349 using \<open>cut B\<close> preal_imp_pos y by blast 350 show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'" 351 proof 352 have "z = (z/y)*y" 353 by (simp add: divide_inverse mult.commute [of y] mult.assoc order_less_imp_not_eq2) 354 then show "\<exists>y'\<in>B. z = (z/y) * y'" 355 using y by blast 356 next 357 show "z/y \<in> A" 358 proof (rule preal_downwards_closed [OF \<open>cut A\<close> x]) 359 show "0 < z/y" 360 by (simp add: \<open>0 < z\<close>) 361 show "z/y < x" 362 using \<open>0 < y\<close> pos_divide_less_eq \<open>z < u\<close> ueq by blast 363 qed 364 qed 365 qed 366 moreover have "\<And>y. y \<in> mult_set A B \<Longrightarrow> \<exists>u \<in> mult_set A B. y < u" 367 apply (simp add: mult_set_def) 368 by (metis preal_exists_greater mult_strict_right_mono preal_imp_pos assms) 369 ultimately show ?thesis 370 by (simp add: Dedekind_Real.cut_def) 371qed 372 373lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)" 374 apply (simp add: preal_mult_def mem_mult_set Rep_preal) 375 apply (force simp: mult_set_def ac_simps) 376 done 377 378instance preal :: ab_semigroup_mult 379proof 380 fix a b c :: preal 381 show "(a * b) * c = a * (b * c)" by (rule preal_mult_assoc) 382 show "a * b = b * a" by (rule preal_mult_commute) 383qed 384 385 386text\<open>Positive real 1 is the multiplicative identity element\<close> 387 388lemma preal_mult_1: "(1::preal) * z = z" 389proof (induct z) 390 fix A :: "rat set" 391 assume A: "cut A" 392 have "{w. \<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A") 393 proof 394 show "?lhs \<subseteq> A" 395 proof clarify 396 fix x::rat and u::rat and v::rat 397 assume upos: "0<u" and "u<1" and v: "v \<in> A" 398 have vpos: "0<v" by (rule preal_imp_pos [OF A v]) 399 hence "u*v < 1*v" by (simp only: mult_strict_right_mono upos \<open>u < 1\<close> v) 400 thus "u * v \<in> A" 401 by (force intro: preal_downwards_closed [OF A v] mult_pos_pos upos vpos) 402 qed 403 next 404 show "A \<subseteq> ?lhs" 405 proof clarify 406 fix x::rat 407 assume x: "x \<in> A" 408 have xpos: "0<x" by (rule preal_imp_pos [OF A x]) 409 from preal_exists_greater [OF A x] 410 obtain v where v: "v \<in> A" and xlessv: "x < v" .. 411 have vpos: "0<v" by (rule preal_imp_pos [OF A v]) 412 show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)" 413 proof (intro exI conjI) 414 show "0 < x/v" 415 by (simp add: zero_less_divide_iff xpos vpos) 416 show "x / v < 1" 417 by (simp add: pos_divide_less_eq vpos xlessv) 418 have "x = (x/v)*v" 419 by (simp add: divide_inverse mult.assoc vpos order_less_imp_not_eq2) 420 then show "\<exists>v'\<in>A. x = (x / v) * v'" 421 using v by blast 422 qed 423 qed 424 qed 425 thus "1 * Abs_preal A = Abs_preal A" 426 by (simp add: preal_one_def preal_mult_def mult_set_def rat_mem_preal A) 427qed 428 429instance preal :: comm_monoid_mult 430 by intro_classes (rule preal_mult_1) 431 432 433subsection\<open>Distribution of Multiplication across Addition\<close> 434 435lemma mem_Rep_preal_add_iff: 436 "(z \<in> Rep_preal(r+s)) = (\<exists>x \<in> Rep_preal r. \<exists>y \<in> Rep_preal s. z = x + y)" 437 apply (simp add: preal_add_def mem_add_set Rep_preal) 438 apply (simp add: add_set_def) 439 done 440 441lemma mem_Rep_preal_mult_iff: 442 "(z \<in> Rep_preal(r*s)) = (\<exists>x \<in> Rep_preal r. \<exists>y \<in> Rep_preal s. z = x * y)" 443 apply (simp add: preal_mult_def mem_mult_set Rep_preal) 444 apply (simp add: mult_set_def) 445 done 446 447lemma distrib_subset1: 448 "Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)" 449 by (force simp: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff distrib_left) 450 451lemma preal_add_mult_distrib_mean: 452 assumes a: "a \<in> Rep_preal w" 453 and b: "b \<in> Rep_preal w" 454 and d: "d \<in> Rep_preal x" 455 and e: "e \<in> Rep_preal y" 456 shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)" 457proof 458 let ?c = "(a*d + b*e)/(d+e)" 459 have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e" 460 by (blast intro: preal_imp_pos [OF cut_Rep_preal] a b d e pos_add_strict)+ 461 have cpos: "0 < ?c" 462 by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict) 463 show "a * d + b * e = ?c * (d + e)" 464 by (simp add: divide_inverse mult.assoc order_less_imp_not_eq2) 465 show "?c \<in> Rep_preal w" 466 proof (cases rule: linorder_le_cases) 467 assume "a \<le> b" 468 hence "?c \<le> b" 469 by (simp add: pos_divide_le_eq distrib_left mult_right_mono 470 order_less_imp_le) 471 thus ?thesis by (rule preal_downwards_closed' [OF cut_Rep_preal b cpos]) 472 next 473 assume "b \<le> a" 474 hence "?c \<le> a" 475 by (simp add: pos_divide_le_eq distrib_left mult_right_mono 476 order_less_imp_le) 477 thus ?thesis by (rule preal_downwards_closed' [OF cut_Rep_preal a cpos]) 478 qed 479qed 480 481lemma distrib_subset2: 482 "Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))" 483 apply (clarsimp simp: mem_Rep_preal_add_iff mem_Rep_preal_mult_iff) 484 using mem_Rep_preal_add_iff preal_add_mult_distrib_mean by blast 485 486lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)" 487 by (metis Rep_preal_inverse distrib_subset1 distrib_subset2 subset_antisym) 488 489lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)" 490 by (simp add: preal_mult_commute preal_add_mult_distrib2) 491 492instance preal :: comm_semiring 493 by intro_classes (rule preal_add_mult_distrib) 494 495 496subsection\<open>Existence of Inverse, a Positive Real\<close> 497 498lemma mem_inverse_set: 499 assumes "cut A" shows "cut (inverse_set A)" 500proof - 501 have "\<exists>x y. 0 < x \<and> x < y \<and> inverse y \<notin> A" 502 proof - 503 from preal_exists_bound [OF \<open>cut A\<close>] 504 obtain x where [simp]: "0<x" "x \<notin> A" by blast 505 show ?thesis 506 proof (intro exI conjI) 507 show "0 < inverse (x+1)" 508 by (simp add: order_less_trans [OF _ less_add_one]) 509 show "inverse(x+1) < inverse x" 510 by (simp add: less_imp_inverse_less less_add_one) 511 show "inverse (inverse x) \<notin> A" 512 by (simp add: order_less_imp_not_eq2) 513 qed 514 qed 515 then have "{} \<subset> inverse_set A" 516 using inverse_set_def by fastforce 517 moreover obtain q where "q > 0" "q \<notin> inverse_set A" 518 proof - 519 from preal_nonempty [OF \<open>cut A\<close>] 520 obtain x where x: "x \<in> A" and xpos [simp]: "0<x" .. 521 show ?thesis 522 proof 523 show "0 < inverse x" by simp 524 show "inverse x \<notin> inverse_set A" 525 proof - 526 { fix y::rat 527 assume ygt: "inverse x < y" 528 have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt]) 529 have iyless: "inverse y < x" 530 by (simp add: inverse_less_imp_less [of x] ygt) 531 have "inverse y \<in> A" 532 by (simp add: preal_downwards_closed [OF \<open>cut A\<close> x] iyless)} 533 thus ?thesis by (auto simp: inverse_set_def) 534 qed 535 qed 536 qed 537 moreover have "inverse_set A \<subset> {0<..}" 538 using calculation inverse_set_def by blast 539 moreover have "z \<in> inverse_set A" 540 if u: "u \<in> inverse_set A" and "0 < z" "z < u" for u z 541 using u that less_trans unfolding inverse_set_def by auto 542 moreover have "\<And>y. y \<in> inverse_set A \<Longrightarrow> \<exists>u \<in> inverse_set A. y < u" 543 by (simp add: inverse_set_def) (meson dense less_trans) 544 ultimately show ?thesis 545 by (simp add: Dedekind_Real.cut_def) 546qed 547 548 549subsection\<open>Gleason's Lemma 9-3.4, page 122\<close> 550 551lemma Gleason9_34_exists: 552 assumes A: "cut A" 553 and "\<forall>x\<in>A. x + u \<in> A" 554 and "0 \<le> z" 555 shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A" 556proof (cases z rule: int_cases) 557 case (nonneg n) 558 show ?thesis 559 proof (simp add: nonneg, induct n) 560 case 0 561 from preal_nonempty [OF A] 562 show ?case by force 563 next 564 case (Suc k) 565 then obtain b where b: "b \<in> A" "b + of_nat k * u \<in> A" .. 566 hence "b + of_int (int k)*u + u \<in> A" by (simp add: assms) 567 thus ?case by (force simp: algebra_simps b) 568 qed 569next 570 case (neg n) 571 with assms show ?thesis by simp 572qed 573 574lemma Gleason9_34_contra: 575 assumes A: "cut A" 576 shows "\<lbrakk>\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A\<rbrakk> \<Longrightarrow> False" 577proof (induct u, induct y) 578 fix a::int and b::int 579 fix c::int and d::int 580 assume bpos [simp]: "0 < b" 581 and dpos [simp]: "0 < d" 582 and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A" 583 and upos: "0 < Fract c d" 584 and ypos: "0 < Fract a b" 585 and notin: "Fract a b \<notin> A" 586 have cpos [simp]: "0 < c" 587 by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos) 588 have apos [simp]: "0 < a" 589 by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos) 590 let ?k = "a*d" 591 have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)" 592 proof - 593 have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))" 594 by (simp add: order_less_imp_not_eq2 ac_simps) 595 moreover 596 have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)" 597 by (rule mult_mono, 598 simp_all add: int_one_le_iff_zero_less zero_less_mult_iff 599 order_less_imp_le) 600 ultimately 601 show ?thesis by simp 602 qed 603 have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff) 604 from Gleason9_34_exists [OF A closed k] 605 obtain z where z: "z \<in> A" 606 and mem: "z + of_int ?k * Fract c d \<in> A" .. 607 have less: "z + of_int ?k * Fract c d < Fract a b" 608 by (rule not_in_preal_ub [OF A notin mem ypos]) 609 have "0<z" by (rule preal_imp_pos [OF A z]) 610 with frle and less show False by (simp add: Fract_of_int_eq) 611qed 612 613 614lemma Gleason9_34: 615 assumes "cut A" "0 < u" 616 shows "\<exists>r \<in> A. r + u \<notin> A" 617 using assms Gleason9_34_contra preal_exists_bound by blast 618 619 620 621subsection\<open>Gleason's Lemma 9-3.6\<close> 622 623lemma lemma_gleason9_36: 624 assumes A: "cut A" 625 and x: "1 < x" 626 shows "\<exists>r \<in> A. r*x \<notin> A" 627proof - 628 from preal_nonempty [OF A] 629 obtain y where y: "y \<in> A" and ypos: "0<y" .. 630 show ?thesis 631 proof (rule classical) 632 assume "~(\<exists>r\<in>A. r * x \<notin> A)" 633 with y have ymem: "y * x \<in> A" by blast 634 from ypos mult_strict_left_mono [OF x] 635 have yless: "y < y*x" by simp 636 let ?d = "y*x - y" 637 from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto 638 from Gleason9_34 [OF A dpos] 639 obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" .. 640 have rpos: "0<r" by (rule preal_imp_pos [OF A r]) 641 with dpos have rdpos: "0 < r + ?d" by arith 642 have "~ (r + ?d \<le> y + ?d)" 643 proof 644 assume le: "r + ?d \<le> y + ?d" 645 from ymem have yd: "y + ?d \<in> A" by (simp add: eq) 646 have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le]) 647 with notin show False by simp 648 qed 649 hence "y < r" by simp 650 with ypos have dless: "?d < (r * ?d)/y" 651 using dpos less_divide_eq_1 by fastforce 652 have "r + ?d < r*x" 653 proof - 654 have "r + ?d < r + (r * ?d)/y" by (simp add: dless) 655 also from ypos have "\<dots> = (r/y) * (y + ?d)" 656 by (simp only: algebra_simps divide_inverse, simp) 657 also have "\<dots> = r*x" using ypos 658 by simp 659 finally show "r + ?d < r*x" . 660 qed 661 with r notin rdpos 662 show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest: preal_downwards_closed [OF A]) 663 qed 664qed 665 666subsection\<open>Existence of Inverse: Part 2\<close> 667 668lemma mem_Rep_preal_inverse_iff: 669 "(z \<in> Rep_preal(inverse r)) \<longleftrightarrow> (0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal r))" 670 apply (simp add: preal_inverse_def mem_inverse_set Rep_preal) 671 apply (simp add: inverse_set_def) 672 done 673 674lemma Rep_preal_one: 675 "Rep_preal 1 = {x. 0 < x \<and> x < 1}" 676by (simp add: preal_one_def rat_mem_preal) 677 678lemma subset_inverse_mult_lemma: 679 assumes xpos: "0 < x" and xless: "x < 1" 680 shows "\<exists>v u y. 0 < v \<and> v < y \<and> inverse y \<notin> Rep_preal R \<and> 681 u \<in> Rep_preal R \<and> x = v * u" 682proof - 683 from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff) 684 from lemma_gleason9_36 [OF cut_Rep_preal this] 685 obtain t where t: "t \<in> Rep_preal R" 686 and notin: "t * (inverse x) \<notin> Rep_preal R" .. 687 have rpos: "0<t" by (rule preal_imp_pos [OF cut_Rep_preal t]) 688 from preal_exists_greater [OF cut_Rep_preal t] 689 obtain u where u: "u \<in> Rep_preal R" and rless: "t < u" .. 690 have upos: "0<u" by (rule preal_imp_pos [OF cut_Rep_preal u]) 691 show ?thesis 692 proof (intro exI conjI) 693 show "0 < x/u" using xpos upos 694 by (simp add: zero_less_divide_iff) 695 show "x/u < x/t" using xpos upos rpos 696 by (simp add: divide_inverse mult_less_cancel_left rless) 697 show "inverse (x / t) \<notin> Rep_preal R" using notin 698 by (simp add: divide_inverse mult.commute) 699 show "u \<in> Rep_preal R" by (rule u) 700 show "x = x / u * u" using upos 701 by (simp add: divide_inverse mult.commute) 702 qed 703qed 704 705lemma subset_inverse_mult: 706 "Rep_preal 1 \<subseteq> Rep_preal(inverse r * r)" 707 by (force simp: Rep_preal_one mem_Rep_preal_inverse_iff mem_Rep_preal_mult_iff dest: subset_inverse_mult_lemma) 708 709lemma inverse_mult_subset: "Rep_preal(inverse r * r) \<subseteq> Rep_preal 1" 710 proof - 711 have "0 < u * v" if "v \<in> Rep_preal r" "0 < u" "u < t" for u v t :: rat 712 using that by (simp add: zero_less_mult_iff preal_imp_pos [OF cut_Rep_preal]) 713 moreover have "t * q < 1" 714 if "q \<in> Rep_preal r" "0 < t" "t < y" "inverse y \<notin> Rep_preal r" 715 for t q y :: rat 716 proof - 717 have "q < inverse y" 718 using not_in_Rep_preal_ub that by auto 719 hence "t * q < t/y" 720 using that by (simp add: divide_inverse mult_less_cancel_left) 721 also have "\<dots> \<le> 1" 722 using that by (simp add: pos_divide_le_eq) 723 finally show ?thesis . 724 qed 725 ultimately show ?thesis 726 by (auto simp: Rep_preal_one mem_Rep_preal_inverse_iff mem_Rep_preal_mult_iff) 727qed 728 729lemma preal_mult_inverse: "inverse r * r = (1::preal)" 730 by (meson Rep_preal_inject inverse_mult_subset subset_antisym subset_inverse_mult) 731 732lemma preal_mult_inverse_right: "r * inverse r = (1::preal)" 733 using preal_mult_commute preal_mult_inverse by auto 734 735 736text\<open>Theorems needing \<open>Gleason9_34\<close>\<close> 737 738lemma Rep_preal_self_subset: "Rep_preal (r) \<subseteq> Rep_preal(r + s)" 739proof 740 fix x 741 assume x: "x \<in> Rep_preal r" 742 obtain y where y: "y \<in> Rep_preal s" and "y > 0" 743 using Rep_preal preal_nonempty by blast 744 have ry: "x+y \<in> Rep_preal(r + s)" using x y 745 by (auto simp: mem_Rep_preal_add_iff) 746 then show "x \<in> Rep_preal(r + s)" 747 by (meson \<open>0 < y\<close> add_less_same_cancel1 not_in_Rep_preal_ub order.asym preal_imp_pos [OF cut_Rep_preal x]) 748qed 749 750lemma Rep_preal_sum_not_subset: "~ Rep_preal (r + s) \<subseteq> Rep_preal(r)" 751proof - 752 obtain y where y: "y \<in> Rep_preal s" and "y > 0" 753 using Rep_preal preal_nonempty by blast 754 obtain x where "x \<in> Rep_preal r" and notin: "x + y \<notin> Rep_preal r" 755 using Dedekind_Real.Rep_preal Gleason9_34 \<open>0 < y\<close> by blast 756 then have "x + y \<in> Rep_preal (r + s)" using y 757 by (auto simp: mem_Rep_preal_add_iff) 758 thus ?thesis using notin by blast 759qed 760 761text\<open>at last, Gleason prop. 9-3.5(iii) page 123\<close> 762proposition preal_self_less_add_left: "(r::preal) < r + s" 763 by (meson Rep_preal_sum_not_subset not_less preal_le_def) 764 765 766subsection\<open>Subtraction for Positive Reals\<close> 767 768text\<open>gleason prop. 9-3.5(iv), page 123: proving \<^prop>\<open>a < b \<Longrightarrow> \<exists>d. a + d = b\<close>. 769We define the claimed \<^term>\<open>D\<close> and show that it is a positive real\<close> 770 771lemma mem_diff_set: 772 assumes "r < s" 773 shows "cut (diff_set (Rep_preal s) (Rep_preal r))" 774proof - 775 obtain p where "Rep_preal r \<subseteq> Rep_preal s" "p \<in> Rep_preal s" "p \<notin> Rep_preal r" 776 using assms unfolding preal_less_def by auto 777 then have "{} \<subset> diff_set (Rep_preal s) (Rep_preal r)" 778 apply (simp add: diff_set_def psubset_eq) 779 by (metis cut_Rep_preal add_eq_exists less_add_same_cancel1 preal_exists_greater preal_imp_pos) 780 moreover 781 obtain q where "q > 0" "q \<notin> Rep_preal s" 782 using Rep_preal_exists_bound by blast 783 then have qnot: "q \<notin> diff_set (Rep_preal s) (Rep_preal r)" 784 by (auto simp: diff_set_def dest: cut_Rep_preal [THEN preal_downwards_closed]) 785 moreover have "diff_set (Rep_preal s) (Rep_preal r) \<subset> {0<..}" (is "?lhs < ?rhs") 786 using \<open>0 < q\<close> diff_set_def qnot by blast 787 moreover have "z \<in> diff_set (Rep_preal s) (Rep_preal r)" 788 if u: "u \<in> diff_set (Rep_preal s) (Rep_preal r)" and "0 < z" "z < u" for u z 789 using u that less_trans Rep_preal unfolding diff_set_def Dedekind_Real.cut_def by auto 790 moreover have "\<exists>u \<in> diff_set (Rep_preal s) (Rep_preal r). y < u" 791 if y: "y \<in> diff_set (Rep_preal s) (Rep_preal r)" for y 792 proof - 793 obtain a b where "0 < a" "0 < b" "a \<notin> Rep_preal r" "a + y + b \<in> Rep_preal s" 794 using y 795 by (simp add: diff_set_def) (metis cut_Rep_preal add_eq_exists less_add_same_cancel1 preal_exists_greater) 796 then have "a + (y + b) \<in> Rep_preal s" 797 by (simp add: add.assoc) 798 then have "y + b \<in> diff_set (Rep_preal s) (Rep_preal r)" 799 using \<open>0 < a\<close> \<open>0 < b\<close> \<open>a \<notin> Rep_preal r\<close> y 800 by (auto simp: diff_set_def) 801 then show ?thesis 802 using \<open>0 < b\<close> less_add_same_cancel1 by blast 803 qed 804 ultimately show ?thesis 805 by (simp add: Dedekind_Real.cut_def) 806qed 807 808lemma mem_Rep_preal_diff_iff: 809 "r < s \<Longrightarrow> 810 (z \<in> Rep_preal (s - r)) \<longleftrightarrow> 811 (\<exists>x. 0 < x \<and> 0 < z \<and> x \<notin> Rep_preal r \<and> x + z \<in> Rep_preal s)" 812 apply (simp add: preal_diff_def mem_diff_set Rep_preal) 813 apply (force simp: diff_set_def) 814 done 815 816proposition less_add_left: 817 fixes r::preal 818 assumes "r < s" 819 shows "r + (s-r) = s" 820proof - 821 have "a + b \<in> Rep_preal s" 822 if "a \<in> Rep_preal r" "c + b \<in> Rep_preal s" "c \<notin> Rep_preal r" 823 and "0 < b" "0 < c" for a b c 824 by (meson cut_Rep_preal add_less_imp_less_right add_pos_pos not_in_Rep_preal_ub preal_downwards_closed preal_imp_pos that) 825 then have "r + (s-r) \<le> s" 826 using assms mem_Rep_preal_add_iff mem_Rep_preal_diff_iff preal_le_def by auto 827 have "x \<in> Rep_preal (r + (s - r))" if "x \<in> Rep_preal s" for x 828 proof (cases "x \<in> Rep_preal r") 829 case True 830 then show ?thesis 831 using Rep_preal_self_subset by blast 832 next 833 case False 834 have "\<exists>u v z. 0 < v \<and> 0 < z \<and> u \<in> Rep_preal r \<and> z \<notin> Rep_preal r \<and> z + v \<in> Rep_preal s \<and> x = u + v" 835 if x: "x \<in> Rep_preal s" 836 proof - 837 have xpos: "x > 0" 838 using Rep_preal preal_imp_pos that by blast 839 obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal s" 840 by (metis cut_Rep_preal x add_eq_exists less_add_same_cancel1 preal_exists_greater) 841 from Gleason9_34 [OF cut_Rep_preal epos] 842 obtain u where r: "u \<in> Rep_preal r" and notin: "u + e \<notin> Rep_preal r" .. 843 with x False xpos have rless: "u < x" by (blast intro: not_in_Rep_preal_ub) 844 from add_eq_exists [of u x] 845 obtain y where eq: "x = u+y" by auto 846 show ?thesis 847 proof (intro exI conjI) 848 show "u + e \<notin> Rep_preal r" by (rule notin) 849 show "u + e + y \<in> Rep_preal s" using xe eq by (simp add: ac_simps) 850 show "0 < u + e" 851 using epos preal_imp_pos [OF cut_Rep_preal r] by simp 852 qed (use r rless eq in auto) 853 qed 854 then show ?thesis 855 using assms mem_Rep_preal_add_iff mem_Rep_preal_diff_iff that by blast 856 qed 857 then have "s \<le> r + (s-r)" 858 by (auto simp: preal_le_def) 859 then show ?thesis 860 by (simp add: \<open>r + (s - r) \<le> s\<close> antisym) 861qed 862 863lemma preal_add_less2_mono1: "r < (s::preal) \<Longrightarrow> r + t < s + t" 864 by (metis add.assoc add.commute less_add_left preal_self_less_add_left) 865 866lemma preal_add_less2_mono2: "r < (s::preal) \<Longrightarrow> t + r < t + s" 867 by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of t]) 868 869lemma preal_add_right_less_cancel: "r + t < s + t \<Longrightarrow> r < (s::preal)" 870 by (metis linorder_cases order.asym preal_add_less2_mono1) 871 872lemma preal_add_left_less_cancel: "t + r < t + s \<Longrightarrow> r < (s::preal)" 873 by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of t]) 874 875lemma preal_add_less_cancel_left [simp]: "(t + (r::preal) < t + s) \<longleftrightarrow> (r < s)" 876 by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel) 877 878lemma preal_add_less_cancel_right [simp]: "((r::preal) + t < s + t) = (r < s)" 879 using preal_add_less_cancel_left [symmetric, of r s t] by (simp add: ac_simps) 880 881lemma preal_add_le_cancel_left [simp]: "(t + (r::preal) \<le> t + s) = (r \<le> s)" 882 by (simp add: linorder_not_less [symmetric]) 883 884lemma preal_add_le_cancel_right [simp]: "((r::preal) + t \<le> s + t) = (r \<le> s)" 885 using preal_add_le_cancel_left [symmetric, of r s t] by (simp add: ac_simps) 886 887lemma preal_add_right_cancel: "(r::preal) + t = s + t \<Longrightarrow> r = s" 888 by (metis less_irrefl linorder_cases preal_add_less_cancel_right) 889 890lemma preal_add_left_cancel: "c + a = c + b \<Longrightarrow> a = (b::preal)" 891 by (auto intro: preal_add_right_cancel simp add: preal_add_commute) 892 893instance preal :: linordered_ab_semigroup_add 894proof 895 fix a b c :: preal 896 show "a \<le> b \<Longrightarrow> c + a \<le> c + b" by (simp only: preal_add_le_cancel_left) 897qed 898 899 900subsection\<open>Completeness of type \<^typ>\<open>preal\<close>\<close> 901 902text\<open>Prove that supremum is a cut\<close> 903 904text\<open>Part 1 of Dedekind sections definition\<close> 905 906lemma preal_sup: 907 assumes le: "\<And>X. X \<in> P \<Longrightarrow> X \<le> Y" and "P \<noteq> {}" 908 shows "cut (\<Union>X \<in> P. Rep_preal(X))" 909proof - 910 have "{} \<subset> (\<Union>X \<in> P. Rep_preal(X))" 911 using \<open>P \<noteq> {}\<close> mem_Rep_preal_Ex by fastforce 912 moreover 913 obtain q where "q > 0" and "q \<notin> (\<Union>X \<in> P. Rep_preal(X))" 914 using Rep_preal_exists_bound [of Y] le by (auto simp: preal_le_def) 915 then have "(\<Union>X \<in> P. Rep_preal(X)) \<subset> {0<..}" 916 using cut_Rep_preal preal_imp_pos by force 917 moreover 918 have "\<And>u z. \<lbrakk>u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u\<rbrakk> \<Longrightarrow> z \<in> (\<Union>X \<in> P. Rep_preal(X))" 919 by (auto elim: cut_Rep_preal [THEN preal_downwards_closed]) 920 moreover 921 have "\<And>y. y \<in> (\<Union>X \<in> P. Rep_preal(X)) \<Longrightarrow> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u" 922 by (blast dest: cut_Rep_preal [THEN preal_exists_greater]) 923 ultimately show ?thesis 924 by (simp add: Dedekind_Real.cut_def) 925qed 926 927lemma preal_psup_le: 928 "\<lbrakk>\<And>X. X \<in> P \<Longrightarrow> X \<le> Y; x \<in> P\<rbrakk> \<Longrightarrow> x \<le> psup P" 929 using preal_sup [of P Y] unfolding preal_le_def psup_def by fastforce 930 931lemma psup_le_ub: "\<lbrakk>\<And>X. X \<in> P \<Longrightarrow> X \<le> Y; P \<noteq> {}\<rbrakk> \<Longrightarrow> psup P \<le> Y" 932 using preal_sup [of P Y] by (simp add: SUP_least preal_le_def psup_def) 933 934text\<open>Supremum property\<close> 935proposition preal_complete: 936 assumes le: "\<And>X. X \<in> P \<Longrightarrow> X \<le> Y" and "P \<noteq> {}" 937 shows "(\<exists>X \<in> P. Z < X) \<longleftrightarrow> (Z < psup P)" (is "?lhs = ?rhs") 938proof 939 assume ?lhs 940 then show ?rhs 941 using preal_sup [OF assms] preal_less_def psup_def by auto 942next 943 assume ?rhs 944 then show ?lhs 945 by (meson \<open>P \<noteq> {}\<close> not_less psup_le_ub) 946qed 947 948subsection \<open>Defining the Reals from the Positive Reals\<close> 949 950text \<open>Here we do quotients the old-fashioned way\<close> 951 952definition 953 realrel :: "((preal * preal) * (preal * preal)) set" where 954 "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) \<and> x1+y2 = x2+y1}" 955 956definition "Real = UNIV//realrel" 957 958typedef real = Real 959 morphisms Rep_Real Abs_Real 960 unfolding Real_def by (auto simp: quotient_def) 961 962text \<open>This doesn't involve the overloaded "real" function: users don't see it\<close> 963definition 964 real_of_preal :: "preal \<Rightarrow> real" where 965 "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})" 966 967instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}" 968begin 969 970definition 971 real_zero_def: "0 = Abs_Real(realrel``{(1, 1)})" 972 973definition 974 real_one_def: "1 = Abs_Real(realrel``{(1 + 1, 1)})" 975 976definition 977 real_add_def: "z + w = 978 the_elem (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 979 { Abs_Real(realrel``{(x+u, y+v)}) })" 980 981definition 982 real_minus_def: "- r = the_elem (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" 983 984definition 985 real_diff_def: "r - (s::real) = r + - s" 986 987definition 988 real_mult_def: 989 "z * w = 990 the_elem (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w). 991 { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" 992 993definition 994 real_inverse_def: "inverse (r::real) = (THE s. (r = 0 \<and> s = 0) \<or> s * r = 1)" 995 996definition 997 real_divide_def: "r div (s::real) = r * inverse s" 998 999definition 1000 real_le_def: "z \<le> (w::real) \<longleftrightarrow> 1001 (\<exists>x y u v. x+v \<le> u+y \<and> (x,y) \<in> Rep_Real z \<and> (u,v) \<in> Rep_Real w)" 1002 1003definition 1004 real_less_def: "x < (y::real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y" 1005 1006definition 1007 real_abs_def: "\<bar>r::real\<bar> = (if r < 0 then - r else r)" 1008 1009definition 1010 real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)" 1011 1012instance .. 1013 1014end 1015 1016subsection \<open>Equivalence relation over positive reals\<close> 1017 1018lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)" 1019 by (simp add: realrel_def) 1020 1021lemma preal_trans_lemma: 1022 assumes "x + y1 = x1 + y" and "x + y2 = x2 + y" 1023 shows "x1 + y2 = x2 + (y1::preal)" 1024 by (metis add.left_commute assms preal_add_left_cancel) 1025 1026lemma equiv_realrel: "equiv UNIV realrel" 1027 by (auto simp: equiv_def refl_on_def sym_def trans_def realrel_def intro: dest: preal_trans_lemma) 1028 1029text\<open>Reduces equality of equivalence classes to the \<^term>\<open>realrel\<close> relation: 1030 \<^term>\<open>(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)\<close>\<close> 1031lemmas equiv_realrel_iff [simp] = 1032 eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] 1033 1034lemma realrel_in_real [simp]: "realrel``{(x,y)} \<in> Real" 1035 by (simp add: Real_def realrel_def quotient_def, blast) 1036 1037declare Abs_Real_inject [simp] Abs_Real_inverse [simp] 1038 1039 1040text\<open>Case analysis on the representation of a real number as an equivalence 1041 class of pairs of positive reals.\<close> 1042lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 1043 "(\<And>x y. z = Abs_Real(realrel``{(x,y)}) \<Longrightarrow> P) \<Longrightarrow> P" 1044 by (metis Rep_Real_inverse prod.exhaust Rep_Real [of z, unfolded Real_def, THEN quotientE]) 1045 1046subsection \<open>Addition and Subtraction\<close> 1047 1048lemma real_add: 1049 "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = 1050 Abs_Real (realrel``{(x+u, y+v)})" 1051proof - 1052 have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) 1053 respects2 realrel" 1054 by (clarsimp simp: congruent2_def) (metis add.left_commute preal_add_assoc) 1055 thus ?thesis 1056 by (simp add: real_add_def UN_UN_split_split_eq UN_equiv_class2 [OF equiv_realrel equiv_realrel]) 1057qed 1058 1059lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" 1060proof - 1061 have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" 1062 by (auto simp: congruent_def add.commute) 1063 thus ?thesis 1064 by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) 1065qed 1066 1067instance real :: ab_group_add 1068proof 1069 fix x y z :: real 1070 show "(x + y) + z = x + (y + z)" 1071 by (cases x, cases y, cases z, simp add: real_add add.assoc) 1072 show "x + y = y + x" 1073 by (cases x, cases y, simp add: real_add add.commute) 1074 show "0 + x = x" 1075 by (cases x, simp add: real_add real_zero_def ac_simps) 1076 show "- x + x = 0" 1077 by (cases x, simp add: real_minus real_add real_zero_def add.commute) 1078 show "x - y = x + - y" 1079 by (simp add: real_diff_def) 1080qed 1081 1082 1083subsection \<open>Multiplication\<close> 1084 1085lemma real_mult_congruent2_lemma: 1086 "!!(x1::preal). \<lbrakk>x1 + y2 = x2 + y1\<rbrakk> \<Longrightarrow> 1087 x * x1 + y * y1 + (x * y2 + y * x2) = 1088 x * x2 + y * y2 + (x * y1 + y * x1)" 1089 by (metis (no_types, hide_lams) add.left_commute preal_add_commute preal_add_mult_distrib2) 1090 1091lemma real_mult_congruent2: 1092 "(\<lambda>p1 p2. 1093 (\<lambda>(x1,y1). (\<lambda>(x2,y2). 1094 { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) 1095 respects2 realrel" 1096 apply (rule congruent2_commuteI [OF equiv_realrel]) 1097 by (auto simp: mult.commute add.commute combine_common_factor preal_add_assoc preal_add_commute) 1098 1099lemma real_mult: 1100 "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = 1101 Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" 1102 by (simp add: real_mult_def UN_UN_split_split_eq 1103 UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) 1104 1105lemma real_mult_commute: "(z::real) * w = w * z" 1106by (cases z, cases w, simp add: real_mult ac_simps) 1107 1108lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" 1109 by (cases z1, cases z2, cases z3) (simp add: real_mult algebra_simps) 1110 1111lemma real_mult_1: "(1::real) * z = z" 1112 by (cases z) (simp add: real_mult real_one_def algebra_simps) 1113 1114lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" 1115 by (cases z1, cases z2, cases w) (simp add: real_add real_mult algebra_simps) 1116 1117text\<open>one and zero are distinct\<close> 1118lemma real_zero_not_eq_one: "0 \<noteq> (1::real)" 1119proof - 1120 have "(1::preal) < 1 + 1" 1121 by (simp add: preal_self_less_add_left) 1122 then show ?thesis 1123 by (simp add: real_zero_def real_one_def neq_iff) 1124qed 1125 1126instance real :: comm_ring_1 1127proof 1128 fix x y z :: real 1129 show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) 1130 show "x * y = y * x" by (rule real_mult_commute) 1131 show "1 * x = x" by (rule real_mult_1) 1132 show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib) 1133 show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one) 1134qed 1135 1136subsection \<open>Inverse and Division\<close> 1137 1138lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" 1139 by (simp add: real_zero_def add.commute) 1140 1141lemma real_mult_inverse_left_ex: 1142 assumes "x \<noteq> 0" obtains y::real where "y*x = 1" 1143proof (cases x) 1144 case (Abs_Real u v) 1145 show ?thesis 1146 proof (cases u v rule: linorder_cases) 1147 case less 1148 then have "v * inverse (v - u) = 1 + u * inverse (v - u)" 1149 using less_add_left [of u v] 1150 by (metis preal_add_commute preal_add_mult_distrib preal_mult_inverse_right) 1151 then have "Abs_Real (realrel``{(1, inverse (v-u) + 1)}) * x - 1 = 0" 1152 by (simp add: Abs_Real real_mult preal_mult_inverse_right real_one_def) (simp add: algebra_simps) 1153 with that show thesis by auto 1154 next 1155 case equal 1156 then show ?thesis 1157 using Abs_Real assms real_zero_iff by blast 1158 next 1159 case greater 1160 then have "u * inverse (u - v) = 1 + v * inverse (u - v)" 1161 using less_add_left [of v u] by (metis add.commute distrib_right preal_mult_inverse_right) 1162 then have "Abs_Real (realrel``{(inverse (u-v) + 1, 1)}) * x - 1 = 0" 1163 by (simp add: Abs_Real real_mult preal_mult_inverse_right real_one_def) (simp add: algebra_simps) 1164 with that show thesis by auto 1165 qed 1166qed 1167 1168 1169lemma real_mult_inverse_left: 1170 fixes x :: real 1171 assumes "x \<noteq> 0" shows "inverse x * x = 1" 1172proof - 1173 obtain y where "y*x = 1" 1174 using assms real_mult_inverse_left_ex by blast 1175 then have "(THE s. s * x = 1) * x = 1" 1176 proof (rule theI) 1177 show "y' = y" if "y' * x = 1" for y' 1178 by (metis \<open>y * x = 1\<close> mult.left_commute mult.right_neutral that) 1179 qed 1180 then show ?thesis 1181 using assms real_inverse_def by auto 1182qed 1183 1184 1185subsection\<open>The Real Numbers form a Field\<close> 1186 1187instance real :: field 1188proof 1189 fix x y z :: real 1190 show "x \<noteq> 0 \<Longrightarrow> inverse x * x = 1" by (rule real_mult_inverse_left) 1191 show "x / y = x * inverse y" by (simp add: real_divide_def) 1192 show "inverse 0 = (0::real)" by (simp add: real_inverse_def) 1193qed 1194 1195 1196subsection\<open>The \<open>\<le>\<close> Ordering\<close> 1197 1198lemma real_le_refl: "w \<le> (w::real)" 1199 by (cases w, force simp: real_le_def) 1200 1201text\<open>The arithmetic decision procedure is not set up for type preal. 1202 This lemma is currently unused, but it could simplify the proofs of the 1203 following two lemmas.\<close> 1204lemma preal_eq_le_imp_le: 1205 assumes eq: "a+b = c+d" and le: "c \<le> a" 1206 shows "b \<le> (d::preal)" 1207proof - 1208 from le have "c+d \<le> a+d" by simp 1209 hence "a+b \<le> a+d" by (simp add: eq) 1210 thus "b \<le> d" by simp 1211qed 1212 1213lemma real_le_lemma: 1214 assumes l: "u1 + v2 \<le> u2 + v1" 1215 and "x1 + v1 = u1 + y1" 1216 and "x2 + v2 = u2 + y2" 1217 shows "x1 + y2 \<le> x2 + (y1::preal)" 1218proof - 1219 have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: assms) 1220 hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: ac_simps) 1221 also have "\<dots> \<le> (x2+y1) + (u2+v1)" by (simp add: assms) 1222 finally show ?thesis by simp 1223qed 1224 1225lemma real_le: 1226 "Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)}) \<longleftrightarrow> x1 + y2 \<le> x2 + y1" 1227 unfolding real_le_def by (auto intro: real_le_lemma) 1228 1229lemma real_le_antisym: "\<lbrakk>z \<le> w; w \<le> z\<rbrakk> \<Longrightarrow> z = (w::real)" 1230 by (cases z, cases w, simp add: real_le) 1231 1232lemma real_trans_lemma: 1233 assumes "x + v \<le> u + y" 1234 and "u + v' \<le> u' + v" 1235 and "x2 + v2 = u2 + y2" 1236 shows "x + v' \<le> u' + (y::preal)" 1237proof - 1238 have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: ac_simps) 1239 also have "\<dots> \<le> (u+y) + (u+v')" by (simp add: assms) 1240 also have "\<dots> \<le> (u+y) + (u'+v)" by (simp add: assms) 1241 also have "\<dots> = (u'+y) + (u+v)" by (simp add: ac_simps) 1242 finally show ?thesis by simp 1243qed 1244 1245lemma real_le_trans: "\<lbrakk>i \<le> j; j \<le> k\<rbrakk> \<Longrightarrow> i \<le> (k::real)" 1246 by (cases i, cases j, cases k) (auto simp: real_le intro: real_trans_lemma) 1247 1248instance real :: order 1249proof 1250 show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u" for u v::real 1251 by (auto simp: real_less_def intro: real_le_antisym) 1252qed (auto intro: real_le_refl real_le_trans real_le_antisym) 1253 1254instance real :: linorder 1255proof 1256 show "x \<le> y \<or> y \<le> x" for x y :: real 1257 by (meson eq_refl le_cases real_le_def) 1258qed 1259 1260instantiation real :: distrib_lattice 1261begin 1262 1263definition 1264 "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min" 1265 1266definition 1267 "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max" 1268 1269instance 1270 by standard (auto simp: inf_real_def sup_real_def max_min_distrib2) 1271 1272end 1273 1274subsection\<open>The Reals Form an Ordered Field\<close> 1275 1276lemma real_le_eq_diff: "(x \<le> y) \<longleftrightarrow> (x-y \<le> (0::real))" 1277 by (cases x, cases y) (simp add: real_le real_zero_def real_diff_def real_add real_minus preal_add_commute) 1278 1279lemma real_add_left_mono: 1280 assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)" 1281proof - 1282 have "z + x - (z + y) = (z + -z) + (x - y)" 1283 by (simp add: algebra_simps) 1284 with le show ?thesis 1285 by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"]) 1286qed 1287 1288lemma real_sum_gt_zero_less: "(0 < s + (-w::real)) \<Longrightarrow> (w < s)" 1289 by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of s]) 1290 1291lemma real_less_sum_gt_zero: "(w < s) \<Longrightarrow> (0 < s + (-w::real))" 1292 by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of s]) 1293 1294lemma real_mult_order: 1295 fixes x y::real 1296 assumes "0 < x" "0 < y" 1297 shows "0 < x * y" 1298 proof (cases x, cases y) 1299 show "0 < x * y" 1300 if x: "x = Abs_Real (Dedekind_Real.realrel `` {(x1, x2)})" 1301 and y: "y = Abs_Real (Dedekind_Real.realrel `` {(y1, y2)})" 1302 for x1 x2 y1 y2 1303 proof - 1304 have "x2 < x1" "y2 < y1" 1305 using assms not_le real_zero_def real_le x y 1306 by (metis preal_add_le_cancel_left real_zero_iff)+ 1307 then obtain xd yd where "x1 = x2 + xd" "y1 = y2 + yd" 1308 using less_add_left by metis 1309 then have "\<not> (x * y \<le> 0)" 1310 apply (simp add: x y real_mult real_zero_def real_le) 1311 apply (simp add: not_le algebra_simps preal_self_less_add_left) 1312 done 1313 then show ?thesis 1314 by auto 1315 qed 1316qed 1317 1318lemma real_mult_less_mono2: "\<lbrakk>(0::real) < z; x < y\<rbrakk> \<Longrightarrow> z * x < z * y" 1319 by (metis add_uminus_conv_diff real_less_sum_gt_zero real_mult_order real_sum_gt_zero_less right_diff_distrib') 1320 1321 1322instance real :: linordered_field 1323proof 1324 fix x y z :: real 1325 show "x \<le> y \<Longrightarrow> z + x \<le> z + y" by (rule real_add_left_mono) 1326 show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def) 1327 show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)" 1328 by (simp only: real_sgn_def) 1329 show "z * x < z * y" if "x < y" "0 < z" 1330 by (simp add: real_mult_less_mono2 that) 1331qed 1332 1333 1334subsection \<open>Completeness of the reals\<close> 1335 1336text\<open>The function \<^term>\<open>real_of_preal\<close> requires many proofs, but it seems 1337to be essential for proving completeness of the reals from that of the 1338positive reals.\<close> 1339 1340lemma real_of_preal_add: 1341 "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" 1342 by (simp add: real_of_preal_def real_add algebra_simps) 1343 1344lemma real_of_preal_mult: 1345 "real_of_preal ((x::preal) * y) = real_of_preal x * real_of_preal y" 1346 by (simp add: real_of_preal_def real_mult algebra_simps) 1347 1348text\<open>Gleason prop 9-4.4 p 127\<close> 1349lemma real_of_preal_trichotomy: 1350 "\<exists>m. (x::real) = real_of_preal m \<or> x = 0 \<or> x = -(real_of_preal m)" 1351proof (cases x) 1352 case (Abs_Real u v) 1353 show ?thesis 1354 proof (cases u v rule: linorder_cases) 1355 case less 1356 then show ?thesis 1357 using less_add_left 1358 apply (simp add: Abs_Real real_of_preal_def real_minus real_zero_def) 1359 by (metis preal_add_assoc preal_add_commute) 1360 next 1361 case equal 1362 then show ?thesis 1363 using Abs_Real real_zero_iff by blast 1364 next 1365 case greater 1366 then show ?thesis 1367 using less_add_left 1368 apply (simp add: Abs_Real real_of_preal_def real_minus real_zero_def) 1369 by (metis preal_add_assoc preal_add_commute) 1370 qed 1371qed 1372 1373lemma real_of_preal_less_iff [simp]: 1374 "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" 1375 by (metis not_less preal_add_less_cancel_right real_le real_of_preal_def) 1376 1377lemma real_of_preal_le_iff [simp]: 1378 "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)" 1379 by (simp add: linorder_not_less [symmetric]) 1380 1381lemma real_of_preal_zero_less [simp]: "0 < real_of_preal m" 1382 by (metis less_add_same_cancel2 preal_self_less_add_left real_of_preal_add real_of_preal_less_iff) 1383 1384 1385subsection\<open>Theorems About the Ordering\<close> 1386 1387lemma real_gt_zero_preal_Ex: "(0 < x) \<longleftrightarrow> (\<exists>y. x = real_of_preal y)" 1388 using order.asym real_of_preal_trichotomy by fastforce 1389 1390subsection \<open>Completeness of Positive Reals\<close> 1391 1392text \<open> 1393 Supremum property for the set of positive reals 1394 1395 Let \<open>P\<close> be a non-empty set of positive reals, with an upper 1396 bound \<open>y\<close>. Then \<open>P\<close> has a least upper bound 1397 (written \<open>S\<close>). 1398 1399 FIXME: Can the premise be weakened to \<open>\<forall>x \<in> P. x\<le> y\<close>? 1400\<close> 1401 1402lemma posreal_complete: 1403 assumes positive_P: "\<forall>x \<in> P. (0::real) < x" 1404 and not_empty_P: "\<exists>x. x \<in> P" 1405 and upper_bound_Ex: "\<exists>y. \<forall>x \<in> P. x<y" 1406 shows "\<exists>s. \<forall>y. (\<exists>x \<in> P. y < x) = (y < s)" 1407proof (rule exI, rule allI) 1408 fix y 1409 let ?pP = "{w. real_of_preal w \<in> P}" 1410 1411 show "(\<exists>x\<in>P. y < x) = (y < real_of_preal (psup ?pP))" 1412 proof (cases "0 < y") 1413 assume neg_y: "\<not> 0 < y" 1414 show ?thesis 1415 proof 1416 assume "\<exists>x\<in>P. y < x" 1417 thus "y < real_of_preal (psup ?pP)" 1418 by (metis dual_order.strict_trans neg_y not_less_iff_gr_or_eq real_of_preal_zero_less) 1419 next 1420 assume "y < real_of_preal (psup ?pP)" 1421 obtain "x" where x_in_P: "x \<in> P" using not_empty_P .. 1422 thus "\<exists>x \<in> P. y < x" using x_in_P 1423 using neg_y not_less_iff_gr_or_eq positive_P by fastforce 1424 qed 1425 next 1426 assume pos_y: "0 < y" 1427 then obtain py where y_is_py: "y = real_of_preal py" 1428 by (auto simp: real_gt_zero_preal_Ex) 1429 1430 obtain a where "a \<in> P" using not_empty_P .. 1431 with positive_P have a_pos: "0 < a" .. 1432 then obtain pa where "a = real_of_preal pa" 1433 by (auto simp: real_gt_zero_preal_Ex) 1434 hence "pa \<in> ?pP" using \<open>a \<in> P\<close> by auto 1435 hence pP_not_empty: "?pP \<noteq> {}" by auto 1436 1437 obtain sup where sup: "\<forall>x \<in> P. x < sup" 1438 using upper_bound_Ex .. 1439 from this and \<open>a \<in> P\<close> have "a < sup" .. 1440 hence "0 < sup" using a_pos by arith 1441 then obtain possup where "sup = real_of_preal possup" 1442 by (auto simp: real_gt_zero_preal_Ex) 1443 hence "\<forall>X \<in> ?pP. X \<le> possup" 1444 using sup by auto 1445 with pP_not_empty have psup: "\<And>Z. (\<exists>X \<in> ?pP. Z < X) = (Z < psup ?pP)" 1446 by (meson preal_complete) 1447 show ?thesis 1448 proof 1449 assume "\<exists>x \<in> P. y < x" 1450 then obtain x where x_in_P: "x \<in> P" and y_less_x: "y < x" .. 1451 hence "0 < x" using pos_y by arith 1452 then obtain px where x_is_px: "x = real_of_preal px" 1453 by (auto simp: real_gt_zero_preal_Ex) 1454 1455 have py_less_X: "\<exists>X \<in> ?pP. py < X" 1456 proof 1457 show "py < px" using y_is_py and x_is_px and y_less_x 1458 by simp 1459 show "px \<in> ?pP" using x_in_P and x_is_px by simp 1460 qed 1461 1462 have "(\<exists>X \<in> ?pP. py < X) \<Longrightarrow> (py < psup ?pP)" 1463 using psup by simp 1464 hence "py < psup ?pP" using py_less_X by simp 1465 thus "y < real_of_preal (psup {w. real_of_preal w \<in> P})" 1466 using y_is_py and pos_y by simp 1467 next 1468 assume y_less_psup: "y < real_of_preal (psup ?pP)" 1469 1470 hence "py < psup ?pP" using y_is_py 1471 by simp 1472 then obtain "X" where py_less_X: "py < X" and X_in_pP: "X \<in> ?pP" 1473 using psup by auto 1474 then obtain x where x_is_X: "x = real_of_preal X" 1475 by (simp add: real_gt_zero_preal_Ex) 1476 hence "y < x" using py_less_X and y_is_py 1477 by simp 1478 moreover have "x \<in> P" 1479 using x_is_X and X_in_pP by simp 1480 ultimately show "\<exists> x \<in> P. y < x" .. 1481 qed 1482 qed 1483qed 1484 1485 1486subsection \<open>Completeness\<close> 1487 1488lemma reals_complete: 1489 fixes S :: "real set" 1490 assumes notempty_S: "\<exists>X. X \<in> S" 1491 and exists_Ub: "bdd_above S" 1492 shows "\<exists>x. (\<forall>s\<in>S. s \<le> x) \<and> (\<forall>y. (\<forall>s\<in>S. s \<le> y) \<longrightarrow> x \<le> y)" 1493proof - 1494 obtain X where X_in_S: "X \<in> S" using notempty_S .. 1495 obtain Y where Y_isUb: "\<forall>s\<in>S. s \<le> Y" 1496 using exists_Ub by (auto simp: bdd_above_def) 1497 let ?SHIFT = "{z. \<exists>x \<in>S. z = x + (-X) + 1} \<inter> {x. 0 < x}" 1498 1499 { 1500 fix x 1501 assume S_le_x: "\<forall>s\<in>S. s \<le> x" 1502 { 1503 fix s 1504 assume "s \<in> {z. \<exists>x\<in>S. z = x + - X + 1}" 1505 hence "\<exists> x \<in> S. s = x + -X + 1" .. 1506 then obtain x1 where x1: "x1 \<in> S" "s = x1 + (-X) + 1" .. 1507 then have "x1 \<le> x" using S_le_x by simp 1508 with x1 have "s \<le> x + - X + 1" by arith 1509 } 1510 then have "\<forall>s\<in>?SHIFT. s \<le> x + (-X) + 1" 1511 by auto 1512 } note S_Ub_is_SHIFT_Ub = this 1513 1514 have *: "\<forall>s\<in>?SHIFT. s \<le> Y + (-X) + 1" using Y_isUb by (rule S_Ub_is_SHIFT_Ub) 1515 have "\<forall>s\<in>?SHIFT. s < Y + (-X) + 2" 1516 proof 1517 fix s assume "s\<in>?SHIFT" 1518 with * have "s \<le> Y + (-X) + 1" by simp 1519 also have "\<dots> < Y + (-X) + 2" by simp 1520 finally show "s < Y + (-X) + 2" . 1521 qed 1522 moreover have "\<forall>y \<in> ?SHIFT. 0 < y" by auto 1523 moreover have shifted_not_empty: "\<exists>u. u \<in> ?SHIFT" 1524 using X_in_S and Y_isUb by auto 1525 ultimately obtain t where t_is_Lub: "\<forall>y. (\<exists>x\<in>?SHIFT. y < x) = (y < t)" 1526 using posreal_complete [of ?SHIFT] unfolding bdd_above_def by blast 1527 1528 show ?thesis 1529 proof 1530 show "(\<forall>s\<in>S. s \<le> (t + X + (-1))) \<and> (\<forall>y. (\<forall>s\<in>S. s \<le> y) \<longrightarrow> (t + X + (-1)) \<le> y)" 1531 proof safe 1532 fix x 1533 assume "\<forall>s\<in>S. s \<le> x" 1534 hence "\<forall>s\<in>?SHIFT. s \<le> x + (-X) + 1" 1535 using S_Ub_is_SHIFT_Ub by simp 1536 then have "\<not> x + (-X) + 1 < t" 1537 by (subst t_is_Lub[rule_format, symmetric]) (simp add: not_less) 1538 thus "t + X + -1 \<le> x" by arith 1539 next 1540 fix y 1541 assume y_in_S: "y \<in> S" 1542 obtain "u" where u_in_shift: "u \<in> ?SHIFT" using shifted_not_empty .. 1543 hence "\<exists> x \<in> S. u = x + - X + 1" by simp 1544 then obtain "x" where x_and_u: "u = x + - X + 1" .. 1545 have u_le_t: "u \<le> t" 1546 proof (rule dense_le) 1547 fix x assume "x < u" then have "x < t" 1548 using u_in_shift t_is_Lub by auto 1549 then show "x \<le> t" by simp 1550 qed 1551 1552 show "y \<le> t + X + -1" 1553 proof cases 1554 assume "y \<le> x" 1555 moreover have "x = u + X + - 1" using x_and_u by arith 1556 moreover have "u + X + - 1 \<le> t + X + -1" using u_le_t by arith 1557 ultimately show "y \<le> t + X + -1" by arith 1558 next 1559 assume "~(y \<le> x)" 1560 hence x_less_y: "x < y" by arith 1561 1562 have "x + (-X) + 1 \<in> ?SHIFT" using x_and_u and u_in_shift by simp 1563 hence "0 < x + (-X) + 1" by simp 1564 hence "0 < y + (-X) + 1" using x_less_y by arith 1565 hence *: "y + (-X) + 1 \<in> ?SHIFT" using y_in_S by simp 1566 have "y + (-X) + 1 \<le> t" 1567 proof (rule dense_le) 1568 fix x assume "x < y + (-X) + 1" then have "x < t" 1569 using * t_is_Lub by auto 1570 then show "x \<le> t" by simp 1571 qed 1572 thus ?thesis by simp 1573 qed 1574 qed 1575 qed 1576qed 1577 1578subsection \<open>The Archimedean Property of the Reals\<close> 1579 1580theorem reals_Archimedean: 1581 fixes x :: real 1582 assumes x_pos: "0 < x" 1583 shows "\<exists>n. inverse (of_nat (Suc n)) < x" 1584proof (rule ccontr) 1585 assume contr: "\<not> ?thesis" 1586 have "\<forall>n. x * of_nat (Suc n) \<le> 1" 1587 proof 1588 fix n 1589 from contr have "x \<le> inverse (of_nat (Suc n))" 1590 by (simp add: linorder_not_less) 1591 hence "x \<le> (1 / (of_nat (Suc n)))" 1592 by (simp add: inverse_eq_divide) 1593 moreover have "(0::real) \<le> of_nat (Suc n)" 1594 by (rule of_nat_0_le_iff) 1595 ultimately have "x * of_nat (Suc n) \<le> (1 / of_nat (Suc n)) * of_nat (Suc n)" 1596 by (rule mult_right_mono) 1597 thus "x * of_nat (Suc n) \<le> 1" by (simp del: of_nat_Suc) 1598 qed 1599 hence 2: "bdd_above {z. \<exists>n. z = x * (of_nat (Suc n))}" 1600 by (auto intro!: bdd_aboveI[of _ 1]) 1601 have 1: "\<exists>X. X \<in> {z. \<exists>n. z = x* (of_nat (Suc n))}" by auto 1602 obtain t where 1603 upper: "\<And>z. z \<in> {z. \<exists>n. z = x * of_nat (Suc n)} \<Longrightarrow> z \<le> t" and 1604 least: "\<And>y. (\<And>a. a \<in> {z. \<exists>n. z = x * of_nat (Suc n)} \<Longrightarrow> a \<le> y) \<Longrightarrow> t \<le> y" 1605 using reals_complete[OF 1 2] by auto 1606 1607 have "t \<le> t + - x" 1608 proof (rule least) 1609 fix a assume a: "a \<in> {z. \<exists>n. z = x * (of_nat (Suc n))}" 1610 have "\<forall>n::nat. x * of_nat n \<le> t + - x" 1611 proof 1612 fix n 1613 have "x * of_nat (Suc n) \<le> t" 1614 by (simp add: upper) 1615 hence "x * (of_nat n) + x \<le> t" 1616 by (simp add: distrib_left) 1617 thus "x * (of_nat n) \<le> t + - x" by arith 1618 qed hence "\<forall>m. x * of_nat (Suc m) \<le> t + - x" by (simp del: of_nat_Suc) 1619 with a show "a \<le> t + - x" 1620 by auto 1621 qed 1622 thus False using x_pos by arith 1623qed 1624 1625text \<open> 1626 There must be other proofs, e.g. \<open>Suc\<close> of the largest 1627 integer in the cut representing \<open>x\<close>. 1628\<close> 1629 1630lemma reals_Archimedean2: "\<exists>n. (x::real) < of_nat (n::nat)" 1631proof cases 1632 assume "x \<le> 0" 1633 hence "x < of_nat (1::nat)" by simp 1634 thus ?thesis .. 1635next 1636 assume "\<not> x \<le> 0" 1637 hence x_greater_zero: "0 < x" by simp 1638 hence "0 < inverse x" by simp 1639 then obtain n where "inverse (of_nat (Suc n)) < inverse x" 1640 using reals_Archimedean by blast 1641 hence "inverse (of_nat (Suc n)) * x < inverse x * x" 1642 using x_greater_zero by (rule mult_strict_right_mono) 1643 hence "inverse (of_nat (Suc n)) * x < 1" 1644 using x_greater_zero by simp 1645 hence "of_nat (Suc n) * (inverse (of_nat (Suc n)) * x) < of_nat (Suc n) * 1" 1646 by (rule mult_strict_left_mono) (simp del: of_nat_Suc) 1647 hence "x < of_nat (Suc n)" 1648 by (simp add: algebra_simps del: of_nat_Suc) 1649 thus "\<exists>(n::nat). x < of_nat n" .. 1650qed 1651 1652instance real :: archimedean_field 1653proof 1654 fix r :: real 1655 obtain n :: nat where "r < of_nat n" 1656 using reals_Archimedean2 .. 1657 then have "r \<le> of_int (int n)" 1658 by simp 1659 then show "\<exists>z. r \<le> of_int z" .. 1660qed 1661 1662end 1663