src/HOL/Real/RealDef.thy
 author huffman Wed Jun 20 05:18:39 2007 +0200 (2007-06-20) changeset 23431 25ca91279a9b parent 23289 0cf371d943b1 child 23438 dd824e86fa8a permissions -rw-r--r--
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
 paulson@5588 ` 1` ```(* Title : Real/RealDef.thy ``` paulson@7219 ` 2` ``` ID : \$Id\$ ``` paulson@5588 ` 3` ``` Author : Jacques D. Fleuriot ``` paulson@5588 ` 4` ``` Copyright : 1998 University of Cambridge ``` paulson@14387 ` 5` ``` Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 ``` avigad@16819 ` 6` ``` Additional contributions by Jeremy Avigad ``` paulson@14269 ` 7` ```*) ``` paulson@14269 ` 8` paulson@14387 ` 9` ```header{*Defining the Reals from the Positive Reals*} ``` paulson@14387 ` 10` nipkow@15131 ` 11` ```theory RealDef ``` nipkow@15140 ` 12` ```imports PReal ``` haftmann@16417 ` 13` ```uses ("real_arith.ML") ``` nipkow@15131 ` 14` ```begin ``` paulson@5588 ` 15` wenzelm@19765 ` 16` ```definition ``` wenzelm@21404 ` 17` ``` realrel :: "((preal * preal) * (preal * preal)) set" where ``` wenzelm@19765 ` 18` ``` "realrel = {p. \x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" ``` paulson@14269 ` 19` paulson@14484 ` 20` ```typedef (Real) real = "UNIV//realrel" ``` paulson@14269 ` 21` ``` by (auto simp add: quotient_def) ``` paulson@5588 ` 22` wenzelm@14691 ` 23` ```instance real :: "{ord, zero, one, plus, times, minus, inverse}" .. ``` paulson@14269 ` 24` wenzelm@19765 ` 25` ```definition ``` paulson@14484 ` 26` paulson@14484 ` 27` ``` (** these don't use the overloaded "real" function: users don't see them **) ``` paulson@14484 ` 28` wenzelm@21404 ` 29` ``` real_of_preal :: "preal => real" where ``` huffman@23288 ` 30` ``` "real_of_preal m = Abs_Real(realrel``{(m + 1, 1)})" ``` paulson@14484 ` 31` paulson@14269 ` 32` ```consts ``` paulson@14269 ` 33` ``` (*overloaded constant for injecting other types into "real"*) ``` paulson@14269 ` 34` ``` real :: "'a => real" ``` paulson@5588 ` 35` paulson@5588 ` 36` paulson@14269 ` 37` ```defs (overloaded) ``` paulson@5588 ` 38` paulson@14269 ` 39` ``` real_zero_def: ``` huffman@23287 ` 40` ``` "0 == Abs_Real(realrel``{(1, 1)})" ``` paulson@12018 ` 41` paulson@14269 ` 42` ``` real_one_def: ``` huffman@23287 ` 43` ``` "1 == Abs_Real(realrel``{(1 + 1, 1)})" ``` paulson@5588 ` 44` paulson@14269 ` 45` ``` real_minus_def: ``` paulson@14484 ` 46` ``` "- r == contents (\(x,y) \ Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" ``` paulson@14484 ` 47` paulson@14484 ` 48` ``` real_add_def: ``` paulson@14484 ` 49` ``` "z + w == ``` paulson@14484 ` 50` ``` contents (\(x,y) \ Rep_Real(z). \(u,v) \ Rep_Real(w). ``` paulson@14484 ` 51` ``` { Abs_Real(realrel``{(x+u, y+v)}) })" ``` bauerg@10606 ` 52` paulson@14269 ` 53` ``` real_diff_def: ``` paulson@14484 ` 54` ``` "r - (s::real) == r + - s" ``` paulson@14484 ` 55` paulson@14484 ` 56` ``` real_mult_def: ``` paulson@14484 ` 57` ``` "z * w == ``` paulson@14484 ` 58` ``` contents (\(x,y) \ Rep_Real(z). \(u,v) \ Rep_Real(w). ``` paulson@14484 ` 59` ``` { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" ``` paulson@5588 ` 60` paulson@14269 ` 61` ``` real_inverse_def: ``` huffman@23287 ` 62` ``` "inverse (R::real) == (THE S. (R = 0 & S = 0) | S * R = 1)" ``` paulson@5588 ` 63` paulson@14269 ` 64` ``` real_divide_def: ``` bauerg@10606 ` 65` ``` "R / (S::real) == R * inverse S" ``` paulson@14269 ` 66` paulson@14484 ` 67` ``` real_le_def: ``` paulson@14484 ` 68` ``` "z \ (w::real) == ``` paulson@14484 ` 69` ``` \x y u v. x+v \ u+y & (x,y) \ Rep_Real z & (u,v) \ Rep_Real w" ``` paulson@5588 ` 70` paulson@14365 ` 71` ``` real_less_def: "(x < (y::real)) == (x \ y & x \ y)" ``` paulson@14365 ` 72` huffman@22962 ` 73` ``` real_abs_def: "abs (r::real) == (if r < 0 then - r else r)" ``` paulson@14334 ` 74` paulson@14334 ` 75` huffman@23287 ` 76` ```subsection {* Equivalence relation over positive reals *} ``` paulson@14269 ` 77` paulson@14270 ` 78` ```lemma preal_trans_lemma: ``` paulson@14365 ` 79` ``` assumes "x + y1 = x1 + y" ``` paulson@14365 ` 80` ``` and "x + y2 = x2 + y" ``` paulson@14365 ` 81` ``` shows "x1 + y2 = x2 + (y1::preal)" ``` paulson@14365 ` 82` ```proof - ``` huffman@23287 ` 83` ``` have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac) ``` paulson@14365 ` 84` ``` also have "... = (x2 + y) + x1" by (simp add: prems) ``` huffman@23287 ` 85` ``` also have "... = x2 + (x1 + y)" by (simp add: add_ac) ``` paulson@14365 ` 86` ``` also have "... = x2 + (x + y1)" by (simp add: prems) ``` huffman@23287 ` 87` ``` also have "... = (x2 + y1) + x" by (simp add: add_ac) ``` paulson@14365 ` 88` ``` finally have "(x1 + y2) + x = (x2 + y1) + x" . ``` huffman@23287 ` 89` ``` thus ?thesis by (rule add_right_imp_eq) ``` paulson@14365 ` 90` ```qed ``` paulson@14365 ` 91` paulson@14269 ` 92` paulson@14484 ` 93` ```lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \ realrel) = (x1 + y2 = x2 + y1)" ``` paulson@14484 ` 94` ```by (simp add: realrel_def) ``` paulson@14269 ` 95` paulson@14269 ` 96` ```lemma equiv_realrel: "equiv UNIV realrel" ``` paulson@14365 ` 97` ```apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def) ``` paulson@14365 ` 98` ```apply (blast dest: preal_trans_lemma) ``` paulson@14269 ` 99` ```done ``` paulson@14269 ` 100` paulson@14497 ` 101` ```text{*Reduces equality of equivalence classes to the @{term realrel} relation: ``` paulson@14497 ` 102` ``` @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \ realrel)"} *} ``` paulson@14269 ` 103` ```lemmas equiv_realrel_iff = ``` paulson@14269 ` 104` ``` eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] ``` paulson@14269 ` 105` paulson@14269 ` 106` ```declare equiv_realrel_iff [simp] ``` paulson@14269 ` 107` paulson@14497 ` 108` paulson@14484 ` 109` ```lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" ``` paulson@14484 ` 110` ```by (simp add: Real_def realrel_def quotient_def, blast) ``` paulson@14269 ` 111` huffman@22958 ` 112` ```declare Abs_Real_inject [simp] ``` paulson@14484 ` 113` ```declare Abs_Real_inverse [simp] ``` paulson@14269 ` 114` paulson@14269 ` 115` paulson@14484 ` 116` ```text{*Case analysis on the representation of a real number as an equivalence ``` paulson@14484 ` 117` ``` class of pairs of positive reals.*} ``` paulson@14484 ` 118` ```lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: ``` paulson@14484 ` 119` ``` "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" ``` paulson@14484 ` 120` ```apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) ``` paulson@14484 ` 121` ```apply (drule arg_cong [where f=Abs_Real]) ``` paulson@14484 ` 122` ```apply (auto simp add: Rep_Real_inverse) ``` paulson@14269 ` 123` ```done ``` paulson@14269 ` 124` paulson@14269 ` 125` huffman@23287 ` 126` ```subsection {* Addition and Subtraction *} ``` paulson@14269 ` 127` paulson@14269 ` 128` ```lemma real_add_congruent2_lemma: ``` paulson@14269 ` 129` ``` "[|a + ba = aa + b; ab + bc = ac + bb|] ``` paulson@14269 ` 130` ``` ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" ``` huffman@23287 ` 131` ```apply (simp add: add_assoc) ``` huffman@23287 ` 132` ```apply (rule add_left_commute [of ab, THEN ssubst]) ``` huffman@23287 ` 133` ```apply (simp add: add_assoc [symmetric]) ``` huffman@23287 ` 134` ```apply (simp add: add_ac) ``` paulson@14269 ` 135` ```done ``` paulson@14269 ` 136` paulson@14269 ` 137` ```lemma real_add: ``` paulson@14497 ` 138` ``` "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = ``` paulson@14497 ` 139` ``` Abs_Real (realrel``{(x+u, y+v)})" ``` paulson@14497 ` 140` ```proof - ``` paulson@15169 ` 141` ``` have "(\z w. (\(x,y). (\(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) ``` paulson@15169 ` 142` ``` respects2 realrel" ``` paulson@14497 ` 143` ``` by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) ``` paulson@14497 ` 144` ``` thus ?thesis ``` paulson@14497 ` 145` ``` by (simp add: real_add_def UN_UN_split_split_eq ``` paulson@14658 ` 146` ``` UN_equiv_class2 [OF equiv_realrel equiv_realrel]) ``` paulson@14497 ` 147` ```qed ``` paulson@14269 ` 148` paulson@14484 ` 149` ```lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" ``` paulson@14484 ` 150` ```proof - ``` paulson@15169 ` 151` ``` have "(\(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" ``` huffman@23288 ` 152` ``` by (simp add: congruent_def add_commute) ``` paulson@14484 ` 153` ``` thus ?thesis ``` paulson@14484 ` 154` ``` by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) ``` paulson@14484 ` 155` ```qed ``` paulson@14334 ` 156` huffman@23287 ` 157` ```instance real :: ab_group_add ``` huffman@23287 ` 158` ```proof ``` huffman@23287 ` 159` ``` fix x y z :: real ``` huffman@23287 ` 160` ``` show "(x + y) + z = x + (y + z)" ``` huffman@23287 ` 161` ``` by (cases x, cases y, cases z, simp add: real_add add_assoc) ``` huffman@23287 ` 162` ``` show "x + y = y + x" ``` huffman@23287 ` 163` ``` by (cases x, cases y, simp add: real_add add_commute) ``` huffman@23287 ` 164` ``` show "0 + x = x" ``` huffman@23287 ` 165` ``` by (cases x, simp add: real_add real_zero_def add_ac) ``` huffman@23287 ` 166` ``` show "- x + x = 0" ``` huffman@23287 ` 167` ``` by (cases x, simp add: real_minus real_add real_zero_def add_commute) ``` huffman@23287 ` 168` ``` show "x - y = x + - y" ``` huffman@23287 ` 169` ``` by (simp add: real_diff_def) ``` huffman@23287 ` 170` ```qed ``` paulson@14269 ` 171` paulson@14269 ` 172` huffman@23287 ` 173` ```subsection {* Multiplication *} ``` paulson@14269 ` 174` paulson@14329 ` 175` ```lemma real_mult_congruent2_lemma: ``` paulson@14329 ` 176` ``` "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> ``` paulson@14484 ` 177` ``` x * x1 + y * y1 + (x * y2 + y * x2) = ``` paulson@14484 ` 178` ``` x * x2 + y * y2 + (x * y1 + y * x1)" ``` huffman@23287 ` 179` ```apply (simp add: add_left_commute add_assoc [symmetric]) ``` huffman@23288 ` 180` ```apply (simp add: add_assoc right_distrib [symmetric]) ``` huffman@23288 ` 181` ```apply (simp add: add_commute) ``` paulson@14269 ` 182` ```done ``` paulson@14269 ` 183` paulson@14269 ` 184` ```lemma real_mult_congruent2: ``` paulson@15169 ` 185` ``` "(%p1 p2. ``` paulson@14484 ` 186` ``` (%(x1,y1). (%(x2,y2). ``` paulson@15169 ` 187` ``` { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) ``` paulson@15169 ` 188` ``` respects2 realrel" ``` paulson@14658 ` 189` ```apply (rule congruent2_commuteI [OF equiv_realrel], clarify) ``` huffman@23288 ` 190` ```apply (simp add: mult_commute add_commute) ``` paulson@14269 ` 191` ```apply (auto simp add: real_mult_congruent2_lemma) ``` paulson@14269 ` 192` ```done ``` paulson@14269 ` 193` paulson@14269 ` 194` ```lemma real_mult: ``` paulson@14484 ` 195` ``` "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = ``` paulson@14484 ` 196` ``` Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" ``` paulson@14484 ` 197` ```by (simp add: real_mult_def UN_UN_split_split_eq ``` paulson@14658 ` 198` ``` UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) ``` paulson@14269 ` 199` paulson@14269 ` 200` ```lemma real_mult_commute: "(z::real) * w = w * z" ``` huffman@23288 ` 201` ```by (cases z, cases w, simp add: real_mult add_ac mult_ac) ``` paulson@14269 ` 202` paulson@14269 ` 203` ```lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" ``` paulson@14484 ` 204` ```apply (cases z1, cases z2, cases z3) ``` huffman@23288 ` 205` ```apply (simp add: real_mult right_distrib add_ac mult_ac) ``` paulson@14269 ` 206` ```done ``` paulson@14269 ` 207` paulson@14269 ` 208` ```lemma real_mult_1: "(1::real) * z = z" ``` paulson@14484 ` 209` ```apply (cases z) ``` huffman@23288 ` 210` ```apply (simp add: real_mult real_one_def right_distrib ``` huffman@23288 ` 211` ``` mult_1_right mult_ac add_ac) ``` paulson@14269 ` 212` ```done ``` paulson@14269 ` 213` paulson@14269 ` 214` ```lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" ``` paulson@14484 ` 215` ```apply (cases z1, cases z2, cases w) ``` huffman@23288 ` 216` ```apply (simp add: real_add real_mult right_distrib add_ac mult_ac) ``` paulson@14269 ` 217` ```done ``` paulson@14269 ` 218` paulson@14329 ` 219` ```text{*one and zero are distinct*} ``` paulson@14365 ` 220` ```lemma real_zero_not_eq_one: "0 \ (1::real)" ``` paulson@14484 ` 221` ```proof - ``` huffman@23287 ` 222` ``` have "(1::preal) < 1 + 1" ``` huffman@23287 ` 223` ``` by (simp add: preal_self_less_add_left) ``` paulson@14484 ` 224` ``` thus ?thesis ``` huffman@23288 ` 225` ``` by (simp add: real_zero_def real_one_def) ``` paulson@14484 ` 226` ```qed ``` paulson@14269 ` 227` huffman@23287 ` 228` ```instance real :: comm_ring_1 ``` huffman@23287 ` 229` ```proof ``` huffman@23287 ` 230` ``` fix x y z :: real ``` huffman@23287 ` 231` ``` show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) ``` huffman@23287 ` 232` ``` show "x * y = y * x" by (rule real_mult_commute) ``` huffman@23287 ` 233` ``` show "1 * x = x" by (rule real_mult_1) ``` huffman@23287 ` 234` ``` show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib) ``` huffman@23287 ` 235` ``` show "0 \ (1::real)" by (rule real_zero_not_eq_one) ``` huffman@23287 ` 236` ```qed ``` huffman@23287 ` 237` huffman@23287 ` 238` ```subsection {* Inverse and Division *} ``` paulson@14365 ` 239` paulson@14484 ` 240` ```lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" ``` huffman@23288 ` 241` ```by (simp add: real_zero_def add_commute) ``` paulson@14269 ` 242` paulson@14365 ` 243` ```text{*Instead of using an existential quantifier and constructing the inverse ``` paulson@14365 ` 244` ```within the proof, we could define the inverse explicitly.*} ``` paulson@14365 ` 245` paulson@14365 ` 246` ```lemma real_mult_inverse_left_ex: "x \ 0 ==> \y. y*x = (1::real)" ``` paulson@14484 ` 247` ```apply (simp add: real_zero_def real_one_def, cases x) ``` paulson@14269 ` 248` ```apply (cut_tac x = xa and y = y in linorder_less_linear) ``` paulson@14365 ` 249` ```apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) ``` paulson@14334 ` 250` ```apply (rule_tac ``` huffman@23287 ` 251` ``` x = "Abs_Real (realrel``{(1, inverse (D) + 1)})" ``` paulson@14334 ` 252` ``` in exI) ``` paulson@14334 ` 253` ```apply (rule_tac [2] ``` huffman@23287 ` 254` ``` x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" ``` paulson@14334 ` 255` ``` in exI) ``` huffman@23288 ` 256` ```apply (auto simp add: real_mult ring_distrib ``` huffman@23288 ` 257` ``` preal_mult_inverse_right add_ac mult_ac) ``` paulson@14269 ` 258` ```done ``` paulson@14269 ` 259` paulson@14365 ` 260` ```lemma real_mult_inverse_left: "x \ 0 ==> inverse(x)*x = (1::real)" ``` paulson@14484 ` 261` ```apply (simp add: real_inverse_def) ``` huffman@23287 ` 262` ```apply (drule real_mult_inverse_left_ex, safe) ``` huffman@23287 ` 263` ```apply (rule theI, assumption, rename_tac z) ``` huffman@23287 ` 264` ```apply (subgoal_tac "(z * x) * y = z * (x * y)") ``` huffman@23287 ` 265` ```apply (simp add: mult_commute) ``` huffman@23287 ` 266` ```apply (rule mult_assoc) ``` paulson@14269 ` 267` ```done ``` paulson@14334 ` 268` paulson@14341 ` 269` paulson@14341 ` 270` ```subsection{*The Real Numbers form a Field*} ``` paulson@14341 ` 271` paulson@14334 ` 272` ```instance real :: field ``` paulson@14334 ` 273` ```proof ``` paulson@14334 ` 274` ``` fix x y z :: real ``` paulson@14365 ` 275` ``` show "x \ 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) ``` paulson@14430 ` 276` ``` show "x / y = x * inverse y" by (simp add: real_divide_def) ``` paulson@14334 ` 277` ```qed ``` paulson@14334 ` 278` paulson@14334 ` 279` paulson@14341 ` 280` ```text{*Inverse of zero! Useful to simplify certain equations*} ``` paulson@14269 ` 281` paulson@14334 ` 282` ```lemma INVERSE_ZERO: "inverse 0 = (0::real)" ``` paulson@14484 ` 283` ```by (simp add: real_inverse_def) ``` paulson@14334 ` 284` paulson@14334 ` 285` ```instance real :: division_by_zero ``` paulson@14334 ` 286` ```proof ``` paulson@14334 ` 287` ``` show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) ``` paulson@14334 ` 288` ```qed ``` paulson@14334 ` 289` paulson@14269 ` 290` paulson@14365 ` 291` ```subsection{*The @{text "\"} Ordering*} ``` paulson@14269 ` 292` paulson@14365 ` 293` ```lemma real_le_refl: "w \ (w::real)" ``` paulson@14484 ` 294` ```by (cases w, force simp add: real_le_def) ``` paulson@14269 ` 295` paulson@14378 ` 296` ```text{*The arithmetic decision procedure is not set up for type preal. ``` paulson@14378 ` 297` ``` This lemma is currently unused, but it could simplify the proofs of the ``` paulson@14378 ` 298` ``` following two lemmas.*} ``` paulson@14378 ` 299` ```lemma preal_eq_le_imp_le: ``` paulson@14378 ` 300` ``` assumes eq: "a+b = c+d" and le: "c \ a" ``` paulson@14378 ` 301` ``` shows "b \ (d::preal)" ``` paulson@14378 ` 302` ```proof - ``` huffman@23288 ` 303` ``` have "c+d \ a+d" by (simp add: prems) ``` paulson@14378 ` 304` ``` hence "a+b \ a+d" by (simp add: prems) ``` huffman@23288 ` 305` ``` thus "b \ d" by simp ``` paulson@14378 ` 306` ```qed ``` paulson@14378 ` 307` paulson@14378 ` 308` ```lemma real_le_lemma: ``` paulson@14378 ` 309` ``` assumes l: "u1 + v2 \ u2 + v1" ``` paulson@14378 ` 310` ``` and "x1 + v1 = u1 + y1" ``` paulson@14378 ` 311` ``` and "x2 + v2 = u2 + y2" ``` paulson@14378 ` 312` ``` shows "x1 + y2 \ x2 + (y1::preal)" ``` paulson@14365 ` 313` ```proof - ``` paulson@14378 ` 314` ``` have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) ``` huffman@23288 ` 315` ``` hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac) ``` huffman@23288 ` 316` ``` also have "... \ (x2+y1) + (u2+v1)" by (simp add: prems) ``` huffman@23288 ` 317` ``` finally show ?thesis by simp ``` huffman@23288 ` 318` ```qed ``` paulson@14378 ` 319` paulson@14378 ` 320` ```lemma real_le: ``` paulson@14484 ` 321` ``` "(Abs_Real(realrel``{(x1,y1)}) \ Abs_Real(realrel``{(x2,y2)})) = ``` paulson@14484 ` 322` ``` (x1 + y2 \ x2 + y1)" ``` huffman@23288 ` 323` ```apply (simp add: real_le_def) ``` paulson@14387 ` 324` ```apply (auto intro: real_le_lemma) ``` paulson@14378 ` 325` ```done ``` paulson@14378 ` 326` paulson@14378 ` 327` ```lemma real_le_anti_sym: "[| z \ w; w \ z |] ==> z = (w::real)" ``` nipkow@15542 ` 328` ```by (cases z, cases w, simp add: real_le) ``` paulson@14378 ` 329` paulson@14378 ` 330` ```lemma real_trans_lemma: ``` paulson@14378 ` 331` ``` assumes "x + v \ u + y" ``` paulson@14378 ` 332` ``` and "u + v' \ u' + v" ``` paulson@14378 ` 333` ``` and "x2 + v2 = u2 + y2" ``` paulson@14378 ` 334` ``` shows "x + v' \ u' + (y::preal)" ``` paulson@14378 ` 335` ```proof - ``` huffman@23288 ` 336` ``` have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac) ``` huffman@23288 ` 337` ``` also have "... \ (u+y) + (u+v')" by (simp add: prems) ``` huffman@23288 ` 338` ``` also have "... \ (u+y) + (u'+v)" by (simp add: prems) ``` huffman@23288 ` 339` ``` also have "... = (u'+y) + (u+v)" by (simp add: add_ac) ``` huffman@23288 ` 340` ``` finally show ?thesis by simp ``` nipkow@15542 ` 341` ```qed ``` paulson@14269 ` 342` paulson@14365 ` 343` ```lemma real_le_trans: "[| i \ j; j \ k |] ==> i \ (k::real)" ``` paulson@14484 ` 344` ```apply (cases i, cases j, cases k) ``` paulson@14484 ` 345` ```apply (simp add: real_le) ``` huffman@23288 ` 346` ```apply (blast intro: real_trans_lemma) ``` paulson@14334 ` 347` ```done ``` paulson@14334 ` 348` paulson@14334 ` 349` ```(* Axiom 'order_less_le' of class 'order': *) ``` paulson@14334 ` 350` ```lemma real_less_le: "((w::real) < z) = (w \ z & w \ z)" ``` paulson@14365 ` 351` ```by (simp add: real_less_def) ``` paulson@14365 ` 352` paulson@14365 ` 353` ```instance real :: order ``` paulson@14365 ` 354` ```proof qed ``` paulson@14365 ` 355` ``` (assumption | ``` paulson@14365 ` 356` ``` rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+ ``` paulson@14365 ` 357` paulson@14378 ` 358` ```(* Axiom 'linorder_linear' of class 'linorder': *) ``` paulson@14378 ` 359` ```lemma real_le_linear: "(z::real) \ w | w \ z" ``` huffman@23288 ` 360` ```apply (cases z, cases w) ``` huffman@23288 ` 361` ```apply (auto simp add: real_le real_zero_def add_ac) ``` paulson@14334 ` 362` ```done ``` paulson@14334 ` 363` paulson@14334 ` 364` paulson@14334 ` 365` ```instance real :: linorder ``` paulson@14334 ` 366` ``` by (intro_classes, rule real_le_linear) ``` paulson@14334 ` 367` paulson@14334 ` 368` paulson@14378 ` 369` ```lemma real_le_eq_diff: "(x \ y) = (x-y \ (0::real))" ``` paulson@14484 ` 370` ```apply (cases x, cases y) ``` paulson@14378 ` 371` ```apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus ``` huffman@23288 ` 372` ``` add_ac) ``` huffman@23288 ` 373` ```apply (simp_all add: add_assoc [symmetric]) ``` nipkow@15542 ` 374` ```done ``` paulson@14378 ` 375` paulson@14484 ` 376` ```lemma real_add_left_mono: ``` paulson@14484 ` 377` ``` assumes le: "x \ y" shows "z + x \ z + (y::real)" ``` paulson@14484 ` 378` ```proof - ``` paulson@14484 ` 379` ``` have "z + x - (z + y) = (z + -z) + (x - y)" ``` paulson@14484 ` 380` ``` by (simp add: diff_minus add_ac) ``` paulson@14484 ` 381` ``` with le show ?thesis ``` obua@14754 ` 382` ``` by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) ``` paulson@14484 ` 383` ```qed ``` paulson@14334 ` 384` paulson@14365 ` 385` ```lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)" ``` paulson@14365 ` 386` ```by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) ``` paulson@14365 ` 387` paulson@14365 ` 388` ```lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))" ``` paulson@14365 ` 389` ```by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) ``` paulson@14334 ` 390` paulson@14334 ` 391` ```lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" ``` paulson@14484 ` 392` ```apply (cases x, cases y) ``` paulson@14378 ` 393` ```apply (simp add: linorder_not_le [where 'a = real, symmetric] ``` paulson@14378 ` 394` ``` linorder_not_le [where 'a = preal] ``` paulson@14378 ` 395` ``` real_zero_def real_le real_mult) ``` paulson@14365 ` 396` ``` --{*Reduce to the (simpler) @{text "\"} relation *} ``` wenzelm@16973 ` 397` ```apply (auto dest!: less_add_left_Ex ``` huffman@23288 ` 398` ``` simp add: add_ac mult_ac ``` huffman@23288 ` 399` ``` right_distrib preal_self_less_add_left) ``` paulson@14334 ` 400` ```done ``` paulson@14334 ` 401` paulson@14334 ` 402` ```lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" ``` paulson@14334 ` 403` ```apply (rule real_sum_gt_zero_less) ``` paulson@14334 ` 404` ```apply (drule real_less_sum_gt_zero [of x y]) ``` paulson@14334 ` 405` ```apply (drule real_mult_order, assumption) ``` paulson@14334 ` 406` ```apply (simp add: right_distrib) ``` paulson@14334 ` 407` ```done ``` paulson@14334 ` 408` haftmann@22456 ` 409` ```instance real :: distrib_lattice ``` haftmann@22456 ` 410` ``` "inf x y \ min x y" ``` haftmann@22456 ` 411` ``` "sup x y \ max x y" ``` haftmann@22456 ` 412` ``` by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1) ``` haftmann@22456 ` 413` paulson@14378 ` 414` paulson@14334 ` 415` ```subsection{*The Reals Form an Ordered Field*} ``` paulson@14334 ` 416` paulson@14334 ` 417` ```instance real :: ordered_field ``` paulson@14334 ` 418` ```proof ``` paulson@14334 ` 419` ``` fix x y z :: real ``` paulson@14334 ` 420` ``` show "x \ y ==> z + x \ z + y" by (rule real_add_left_mono) ``` huffman@22962 ` 421` ``` show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2) ``` huffman@22962 ` 422` ``` show "\x\ = (if x < 0 then -x else x)" by (simp only: real_abs_def) ``` paulson@14334 ` 423` ```qed ``` paulson@14334 ` 424` paulson@14365 ` 425` ```text{*The function @{term real_of_preal} requires many proofs, but it seems ``` paulson@14365 ` 426` ```to be essential for proving completeness of the reals from that of the ``` paulson@14365 ` 427` ```positive reals.*} ``` paulson@14365 ` 428` paulson@14365 ` 429` ```lemma real_of_preal_add: ``` paulson@14365 ` 430` ``` "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" ``` huffman@23288 ` 431` ```by (simp add: real_of_preal_def real_add left_distrib add_ac) ``` paulson@14365 ` 432` paulson@14365 ` 433` ```lemma real_of_preal_mult: ``` paulson@14365 ` 434` ``` "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" ``` huffman@23288 ` 435` ```by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac) ``` paulson@14365 ` 436` paulson@14365 ` 437` paulson@14365 ` 438` ```text{*Gleason prop 9-4.4 p 127*} ``` paulson@14365 ` 439` ```lemma real_of_preal_trichotomy: ``` paulson@14365 ` 440` ``` "\m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)" ``` paulson@14484 ` 441` ```apply (simp add: real_of_preal_def real_zero_def, cases x) ``` huffman@23288 ` 442` ```apply (auto simp add: real_minus add_ac) ``` paulson@14365 ` 443` ```apply (cut_tac x = x and y = y in linorder_less_linear) ``` huffman@23288 ` 444` ```apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric]) ``` paulson@14365 ` 445` ```done ``` paulson@14365 ` 446` paulson@14365 ` 447` ```lemma real_of_preal_leD: ``` paulson@14365 ` 448` ``` "real_of_preal m1 \ real_of_preal m2 ==> m1 \ m2" ``` huffman@23288 ` 449` ```by (simp add: real_of_preal_def real_le) ``` paulson@14365 ` 450` paulson@14365 ` 451` ```lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2" ``` paulson@14365 ` 452` ```by (auto simp add: real_of_preal_leD linorder_not_le [symmetric]) ``` paulson@14365 ` 453` paulson@14365 ` 454` ```lemma real_of_preal_lessD: ``` paulson@14365 ` 455` ``` "real_of_preal m1 < real_of_preal m2 ==> m1 < m2" ``` huffman@23288 ` 456` ```by (simp add: real_of_preal_def real_le linorder_not_le [symmetric]) ``` paulson@14365 ` 457` paulson@14365 ` 458` ```lemma real_of_preal_less_iff [simp]: ``` paulson@14365 ` 459` ``` "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" ``` paulson@14365 ` 460` ```by (blast intro: real_of_preal_lessI real_of_preal_lessD) ``` paulson@14365 ` 461` paulson@14365 ` 462` ```lemma real_of_preal_le_iff: ``` paulson@14365 ` 463` ``` "(real_of_preal m1 \ real_of_preal m2) = (m1 \ m2)" ``` huffman@23288 ` 464` ```by (simp add: linorder_not_less [symmetric]) ``` paulson@14365 ` 465` paulson@14365 ` 466` ```lemma real_of_preal_zero_less: "0 < real_of_preal m" ``` huffman@23288 ` 467` ```apply (insert preal_self_less_add_left [of 1 m]) ``` huffman@23288 ` 468` ```apply (auto simp add: real_zero_def real_of_preal_def ``` huffman@23288 ` 469` ``` real_less_def real_le_def add_ac) ``` huffman@23288 ` 470` ```apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI) ``` huffman@23288 ` 471` ```apply (simp add: add_ac) ``` paulson@14365 ` 472` ```done ``` paulson@14365 ` 473` paulson@14365 ` 474` ```lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0" ``` paulson@14365 ` 475` ```by (simp add: real_of_preal_zero_less) ``` paulson@14365 ` 476` paulson@14365 ` 477` ```lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m" ``` paulson@14484 ` 478` ```proof - ``` paulson@14484 ` 479` ``` from real_of_preal_minus_less_zero ``` paulson@14484 ` 480` ``` show ?thesis by (blast dest: order_less_trans) ``` paulson@14484 ` 481` ```qed ``` paulson@14365 ` 482` paulson@14365 ` 483` paulson@14365 ` 484` ```subsection{*Theorems About the Ordering*} ``` paulson@14365 ` 485` paulson@14365 ` 486` ```lemma real_gt_zero_preal_Ex: "(0 < x) = (\y. x = real_of_preal y)" ``` paulson@14365 ` 487` ```apply (auto simp add: real_of_preal_zero_less) ``` paulson@14365 ` 488` ```apply (cut_tac x = x in real_of_preal_trichotomy) ``` paulson@14365 ` 489` ```apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE]) ``` paulson@14365 ` 490` ```done ``` paulson@14365 ` 491` paulson@14365 ` 492` ```lemma real_gt_preal_preal_Ex: ``` paulson@14365 ` 493` ``` "real_of_preal z < x ==> \y. x = real_of_preal y" ``` paulson@14365 ` 494` ```by (blast dest!: real_of_preal_zero_less [THEN order_less_trans] ``` paulson@14365 ` 495` ``` intro: real_gt_zero_preal_Ex [THEN iffD1]) ``` paulson@14365 ` 496` paulson@14365 ` 497` ```lemma real_ge_preal_preal_Ex: ``` paulson@14365 ` 498` ``` "real_of_preal z \ x ==> \y. x = real_of_preal y" ``` paulson@14365 ` 499` ```by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) ``` paulson@14365 ` 500` paulson@14365 ` 501` ```lemma real_less_all_preal: "y \ 0 ==> \x. y < real_of_preal x" ``` paulson@14365 ` 502` ```by (auto elim: order_le_imp_less_or_eq [THEN disjE] ``` paulson@14365 ` 503` ``` intro: real_of_preal_zero_less [THEN [2] order_less_trans] ``` paulson@14365 ` 504` ``` simp add: real_of_preal_zero_less) ``` paulson@14365 ` 505` paulson@14365 ` 506` ```lemma real_less_all_real2: "~ 0 < y ==> \x. y < real_of_preal x" ``` paulson@14365 ` 507` ```by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) ``` paulson@14365 ` 508` paulson@14334 ` 509` paulson@14334 ` 510` ```subsection{*More Lemmas*} ``` paulson@14334 ` 511` paulson@14334 ` 512` ```lemma real_mult_left_cancel: "(c::real) \ 0 ==> (c*a=c*b) = (a=b)" ``` paulson@14334 ` 513` ```by auto ``` paulson@14334 ` 514` paulson@14334 ` 515` ```lemma real_mult_right_cancel: "(c::real) \ 0 ==> (a*c=b*c) = (a=b)" ``` paulson@14334 ` 516` ```by auto ``` paulson@14334 ` 517` paulson@14334 ` 518` ```lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" ``` paulson@14334 ` 519` ``` by (force elim: order_less_asym ``` paulson@14334 ` 520` ``` simp add: Ring_and_Field.mult_less_cancel_right) ``` paulson@14334 ` 521` paulson@14334 ` 522` ```lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \ y*z) = (x\y)" ``` paulson@14365 ` 523` ```apply (simp add: mult_le_cancel_right) ``` huffman@23289 ` 524` ```apply (blast intro: elim: order_less_asym) ``` paulson@14365 ` 525` ```done ``` paulson@14334 ` 526` paulson@14334 ` 527` ```lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \ z*y) = (x\y)" ``` nipkow@15923 ` 528` ```by(simp add:mult_commute) ``` paulson@14334 ` 529` huffman@23289 ` 530` ```(* FIXME: redundant, but used by Integration/Integral.thy in AFP *) ``` paulson@14334 ` 531` ```lemma real_le_add_order: "[| 0 \ x; 0 \ y |] ==> (0::real) \ x + y" ``` huffman@22958 ` 532` ```by (rule add_nonneg_nonneg) ``` paulson@14334 ` 533` paulson@14365 ` 534` ```lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" ``` huffman@23289 ` 535` ```by (simp add: one_less_inverse_iff) (* TODO: generalize/move *) ``` paulson@14334 ` 536` paulson@14334 ` 537` paulson@14365 ` 538` ```subsection{*Embedding the Integers into the Reals*} ``` paulson@14365 ` 539` paulson@14378 ` 540` ```defs (overloaded) ``` paulson@14378 ` 541` ``` real_of_nat_def: "real z == of_nat z" ``` paulson@14378 ` 542` ``` real_of_int_def: "real z == of_int z" ``` paulson@14365 ` 543` avigad@16819 ` 544` ```lemma real_eq_of_nat: "real = of_nat" ``` avigad@16819 ` 545` ``` apply (rule ext) ``` avigad@16819 ` 546` ``` apply (unfold real_of_nat_def) ``` avigad@16819 ` 547` ``` apply (rule refl) ``` avigad@16819 ` 548` ``` done ``` avigad@16819 ` 549` avigad@16819 ` 550` ```lemma real_eq_of_int: "real = of_int" ``` avigad@16819 ` 551` ``` apply (rule ext) ``` avigad@16819 ` 552` ``` apply (unfold real_of_int_def) ``` avigad@16819 ` 553` ``` apply (rule refl) ``` avigad@16819 ` 554` ``` done ``` avigad@16819 ` 555` paulson@14365 ` 556` ```lemma real_of_int_zero [simp]: "real (0::int) = 0" ``` paulson@14378 ` 557` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 558` paulson@14365 ` 559` ```lemma real_of_one [simp]: "real (1::int) = (1::real)" ``` paulson@14378 ` 560` ```by (simp add: real_of_int_def) ``` paulson@14334 ` 561` avigad@16819 ` 562` ```lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y" ``` paulson@14378 ` 563` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 564` avigad@16819 ` 565` ```lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)" ``` paulson@14378 ` 566` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 567` avigad@16819 ` 568` ```lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y" ``` avigad@16819 ` 569` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 570` avigad@16819 ` 571` ```lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y" ``` paulson@14378 ` 572` ```by (simp add: real_of_int_def) ``` paulson@14334 ` 573` avigad@16819 ` 574` ```lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" ``` avigad@16819 ` 575` ``` apply (subst real_eq_of_int)+ ``` avigad@16819 ` 576` ``` apply (rule of_int_setsum) ``` avigad@16819 ` 577` ```done ``` avigad@16819 ` 578` avigad@16819 ` 579` ```lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = ``` avigad@16819 ` 580` ``` (PROD x:A. real(f x))" ``` avigad@16819 ` 581` ``` apply (subst real_eq_of_int)+ ``` avigad@16819 ` 582` ``` apply (rule of_int_setprod) ``` avigad@16819 ` 583` ```done ``` paulson@14365 ` 584` paulson@14365 ` 585` ```lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))" ``` paulson@14378 ` 586` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 587` paulson@14365 ` 588` ```lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)" ``` paulson@14378 ` 589` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 590` paulson@14365 ` 591` ```lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)" ``` paulson@14378 ` 592` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 593` paulson@14365 ` 594` ```lemma real_of_int_le_iff [simp]: "(real (x::int) \ real y) = (x \ y)" ``` paulson@14378 ` 595` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 596` avigad@16819 ` 597` ```lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)" ``` avigad@16819 ` 598` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 599` avigad@16819 ` 600` ```lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)" ``` avigad@16819 ` 601` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 602` avigad@16819 ` 603` ```lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)" ``` avigad@16819 ` 604` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 605` avigad@16819 ` 606` ```lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)" ``` avigad@16819 ` 607` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 608` avigad@16888 ` 609` ```lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" ``` avigad@16888 ` 610` ```by (auto simp add: abs_if) ``` avigad@16888 ` 611` avigad@16819 ` 612` ```lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" ``` avigad@16819 ` 613` ``` apply (subgoal_tac "real n + 1 = real (n + 1)") ``` avigad@16819 ` 614` ``` apply (simp del: real_of_int_add) ``` avigad@16819 ` 615` ``` apply auto ``` avigad@16819 ` 616` ```done ``` avigad@16819 ` 617` avigad@16819 ` 618` ```lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" ``` avigad@16819 ` 619` ``` apply (subgoal_tac "real m + 1 = real (m + 1)") ``` avigad@16819 ` 620` ``` apply (simp del: real_of_int_add) ``` avigad@16819 ` 621` ``` apply simp ``` avigad@16819 ` 622` ```done ``` avigad@16819 ` 623` avigad@16819 ` 624` ```lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = ``` avigad@16819 ` 625` ``` real (x div d) + (real (x mod d)) / (real d)" ``` avigad@16819 ` 626` ```proof - ``` avigad@16819 ` 627` ``` assume "d ~= 0" ``` avigad@16819 ` 628` ``` have "x = (x div d) * d + x mod d" ``` avigad@16819 ` 629` ``` by auto ``` avigad@16819 ` 630` ``` then have "real x = real (x div d) * real d + real(x mod d)" ``` avigad@16819 ` 631` ``` by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) ``` avigad@16819 ` 632` ``` then have "real x / real d = ... / real d" ``` avigad@16819 ` 633` ``` by simp ``` avigad@16819 ` 634` ``` then show ?thesis ``` avigad@16819 ` 635` ``` by (auto simp add: add_divide_distrib ring_eq_simps prems) ``` avigad@16819 ` 636` ```qed ``` avigad@16819 ` 637` avigad@16819 ` 638` ```lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> ``` avigad@16819 ` 639` ``` real(n div d) = real n / real d" ``` avigad@16819 ` 640` ``` apply (frule real_of_int_div_aux [of d n]) ``` avigad@16819 ` 641` ``` apply simp ``` avigad@16819 ` 642` ``` apply (simp add: zdvd_iff_zmod_eq_0) ``` avigad@16819 ` 643` ```done ``` avigad@16819 ` 644` avigad@16819 ` 645` ```lemma real_of_int_div2: ``` avigad@16819 ` 646` ``` "0 <= real (n::int) / real (x) - real (n div x)" ``` avigad@16819 ` 647` ``` apply (case_tac "x = 0") ``` avigad@16819 ` 648` ``` apply simp ``` avigad@16819 ` 649` ``` apply (case_tac "0 < x") ``` avigad@16819 ` 650` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 651` ``` apply (subst real_of_int_div_aux) ``` avigad@16819 ` 652` ``` apply simp ``` avigad@16819 ` 653` ``` apply simp ``` avigad@16819 ` 654` ``` apply (subst zero_le_divide_iff) ``` avigad@16819 ` 655` ``` apply auto ``` avigad@16819 ` 656` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 657` ``` apply (subst real_of_int_div_aux) ``` avigad@16819 ` 658` ``` apply simp ``` avigad@16819 ` 659` ``` apply simp ``` avigad@16819 ` 660` ``` apply (subst zero_le_divide_iff) ``` avigad@16819 ` 661` ``` apply auto ``` avigad@16819 ` 662` ```done ``` avigad@16819 ` 663` avigad@16819 ` 664` ```lemma real_of_int_div3: ``` avigad@16819 ` 665` ``` "real (n::int) / real (x) - real (n div x) <= 1" ``` avigad@16819 ` 666` ``` apply(case_tac "x = 0") ``` avigad@16819 ` 667` ``` apply simp ``` avigad@16819 ` 668` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 669` ``` apply (subst real_of_int_div_aux) ``` avigad@16819 ` 670` ``` apply assumption ``` avigad@16819 ` 671` ``` apply simp ``` avigad@16819 ` 672` ``` apply (subst divide_le_eq) ``` avigad@16819 ` 673` ``` apply clarsimp ``` avigad@16819 ` 674` ``` apply (rule conjI) ``` avigad@16819 ` 675` ``` apply (rule impI) ``` avigad@16819 ` 676` ``` apply (rule order_less_imp_le) ``` avigad@16819 ` 677` ``` apply simp ``` avigad@16819 ` 678` ``` apply (rule impI) ``` avigad@16819 ` 679` ``` apply (rule order_less_imp_le) ``` avigad@16819 ` 680` ``` apply simp ``` avigad@16819 ` 681` ```done ``` avigad@16819 ` 682` avigad@16819 ` 683` ```lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" ``` avigad@16819 ` 684` ``` by (insert real_of_int_div2 [of n x], simp) ``` paulson@14365 ` 685` paulson@14365 ` 686` ```subsection{*Embedding the Naturals into the Reals*} ``` paulson@14365 ` 687` paulson@14334 ` 688` ```lemma real_of_nat_zero [simp]: "real (0::nat) = 0" ``` paulson@14365 ` 689` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 690` paulson@14334 ` 691` ```lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" ``` paulson@14365 ` 692` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 693` paulson@14365 ` 694` ```lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n" ``` paulson@14378 ` 695` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 696` paulson@14334 ` 697` ```(*Not for addsimps: often the LHS is used to represent a positive natural*) ``` paulson@14334 ` 698` ```lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" ``` paulson@14378 ` 699` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 700` paulson@14334 ` 701` ```lemma real_of_nat_less_iff [iff]: ``` paulson@14334 ` 702` ``` "(real (n::nat) < real m) = (n < m)" ``` paulson@14365 ` 703` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 704` paulson@14334 ` 705` ```lemma real_of_nat_le_iff [iff]: "(real (n::nat) \ real m) = (n \ m)" ``` paulson@14378 ` 706` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 707` paulson@14334 ` 708` ```lemma real_of_nat_ge_zero [iff]: "0 \ real (n::nat)" ``` paulson@14378 ` 709` ```by (simp add: real_of_nat_def zero_le_imp_of_nat) ``` paulson@14334 ` 710` paulson@14365 ` 711` ```lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)" ``` paulson@14378 ` 712` ```by (simp add: real_of_nat_def del: of_nat_Suc) ``` paulson@14365 ` 713` paulson@14334 ` 714` ```lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" ``` huffman@23431 ` 715` ```by (simp add: real_of_nat_def of_nat_mult) ``` paulson@14334 ` 716` avigad@16819 ` 717` ```lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = ``` avigad@16819 ` 718` ``` (SUM x:A. real(f x))" ``` avigad@16819 ` 719` ``` apply (subst real_eq_of_nat)+ ``` avigad@16819 ` 720` ``` apply (rule of_nat_setsum) ``` avigad@16819 ` 721` ```done ``` avigad@16819 ` 722` avigad@16819 ` 723` ```lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = ``` avigad@16819 ` 724` ``` (PROD x:A. real(f x))" ``` avigad@16819 ` 725` ``` apply (subst real_eq_of_nat)+ ``` avigad@16819 ` 726` ``` apply (rule of_nat_setprod) ``` avigad@16819 ` 727` ```done ``` avigad@16819 ` 728` avigad@16819 ` 729` ```lemma real_of_card: "real (card A) = setsum (%x.1) A" ``` avigad@16819 ` 730` ``` apply (subst card_eq_setsum) ``` avigad@16819 ` 731` ``` apply (subst real_of_nat_setsum) ``` avigad@16819 ` 732` ``` apply simp ``` avigad@16819 ` 733` ```done ``` avigad@16819 ` 734` paulson@14334 ` 735` ```lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" ``` paulson@14378 ` 736` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 737` paulson@14387 ` 738` ```lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)" ``` paulson@14378 ` 739` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 740` paulson@14365 ` 741` ```lemma real_of_nat_diff: "n \ m ==> real (m - n) = real (m::nat) - real n" ``` paulson@14378 ` 742` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 743` paulson@14365 ` 744` ```lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)" ``` paulson@14378 ` 745` ```by (simp add: add: real_of_nat_def) ``` paulson@14365 ` 746` paulson@14365 ` 747` ```lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \ 0) = (n = 0)" ``` paulson@14378 ` 748` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 749` paulson@14365 ` 750` ```lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0" ``` paulson@14378 ` 751` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 752` paulson@14365 ` 753` ```lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \ real (n::nat)) = (0 \ n)" ``` paulson@14378 ` 754` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 755` avigad@16819 ` 756` ```lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" ``` avigad@16819 ` 757` ``` apply (subgoal_tac "real n + 1 = real (Suc n)") ``` avigad@16819 ` 758` ``` apply simp ``` avigad@16819 ` 759` ``` apply (auto simp add: real_of_nat_Suc) ``` avigad@16819 ` 760` ```done ``` avigad@16819 ` 761` avigad@16819 ` 762` ```lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" ``` avigad@16819 ` 763` ``` apply (subgoal_tac "real m + 1 = real (Suc m)") ``` avigad@16819 ` 764` ``` apply (simp add: less_Suc_eq_le) ``` avigad@16819 ` 765` ``` apply (simp add: real_of_nat_Suc) ``` avigad@16819 ` 766` ```done ``` avigad@16819 ` 767` avigad@16819 ` 768` ```lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = ``` avigad@16819 ` 769` ``` real (x div d) + (real (x mod d)) / (real d)" ``` avigad@16819 ` 770` ```proof - ``` avigad@16819 ` 771` ``` assume "0 < d" ``` avigad@16819 ` 772` ``` have "x = (x div d) * d + x mod d" ``` avigad@16819 ` 773` ``` by auto ``` avigad@16819 ` 774` ``` then have "real x = real (x div d) * real d + real(x mod d)" ``` avigad@16819 ` 775` ``` by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) ``` avigad@16819 ` 776` ``` then have "real x / real d = \ / real d" ``` avigad@16819 ` 777` ``` by simp ``` avigad@16819 ` 778` ``` then show ?thesis ``` avigad@16819 ` 779` ``` by (auto simp add: add_divide_distrib ring_eq_simps prems) ``` avigad@16819 ` 780` ```qed ``` avigad@16819 ` 781` avigad@16819 ` 782` ```lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> ``` avigad@16819 ` 783` ``` real(n div d) = real n / real d" ``` avigad@16819 ` 784` ``` apply (frule real_of_nat_div_aux [of d n]) ``` avigad@16819 ` 785` ``` apply simp ``` avigad@16819 ` 786` ``` apply (subst dvd_eq_mod_eq_0 [THEN sym]) ``` avigad@16819 ` 787` ``` apply assumption ``` avigad@16819 ` 788` ```done ``` avigad@16819 ` 789` avigad@16819 ` 790` ```lemma real_of_nat_div2: ``` avigad@16819 ` 791` ``` "0 <= real (n::nat) / real (x) - real (n div x)" ``` avigad@16819 ` 792` ``` apply(case_tac "x = 0") ``` avigad@16819 ` 793` ``` apply simp ``` avigad@16819 ` 794` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 795` ``` apply (subst real_of_nat_div_aux) ``` avigad@16819 ` 796` ``` apply assumption ``` avigad@16819 ` 797` ``` apply simp ``` avigad@16819 ` 798` ``` apply (subst zero_le_divide_iff) ``` avigad@16819 ` 799` ``` apply simp ``` avigad@16819 ` 800` ```done ``` avigad@16819 ` 801` avigad@16819 ` 802` ```lemma real_of_nat_div3: ``` avigad@16819 ` 803` ``` "real (n::nat) / real (x) - real (n div x) <= 1" ``` avigad@16819 ` 804` ``` apply(case_tac "x = 0") ``` avigad@16819 ` 805` ``` apply simp ``` avigad@16819 ` 806` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 807` ``` apply (subst real_of_nat_div_aux) ``` avigad@16819 ` 808` ``` apply assumption ``` avigad@16819 ` 809` ``` apply simp ``` avigad@16819 ` 810` ```done ``` avigad@16819 ` 811` avigad@16819 ` 812` ```lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" ``` avigad@16819 ` 813` ``` by (insert real_of_nat_div2 [of n x], simp) ``` avigad@16819 ` 814` paulson@14365 ` 815` ```lemma real_of_int_real_of_nat: "real (int n) = real n" ``` paulson@14378 ` 816` ```by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat) ``` paulson@14378 ` 817` paulson@14426 ` 818` ```lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" ``` paulson@14426 ` 819` ```by (simp add: real_of_int_def real_of_nat_def) ``` paulson@14334 ` 820` avigad@16819 ` 821` ```lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" ``` avigad@16819 ` 822` ``` apply (subgoal_tac "real(int(nat x)) = real(nat x)") ``` avigad@16819 ` 823` ``` apply force ``` avigad@16819 ` 824` ``` apply (simp only: real_of_int_real_of_nat) ``` avigad@16819 ` 825` ```done ``` paulson@14387 ` 826` paulson@14387 ` 827` ```subsection{*Numerals and Arithmetic*} ``` paulson@14387 ` 828` paulson@14387 ` 829` ```instance real :: number .. ``` paulson@14387 ` 830` paulson@15013 ` 831` ```defs (overloaded) ``` haftmann@20485 ` 832` ``` real_number_of_def: "(number_of w :: real) == of_int w" ``` paulson@15013 ` 833` ``` --{*the type constraint is essential!*} ``` paulson@14387 ` 834` paulson@14387 ` 835` ```instance real :: number_ring ``` paulson@15013 ` 836` ```by (intro_classes, simp add: real_number_of_def) ``` paulson@14387 ` 837` paulson@14387 ` 838` ```text{*Collapse applications of @{term real} to @{term number_of}*} ``` paulson@14387 ` 839` ```lemma real_number_of [simp]: "real (number_of v :: int) = number_of v" ``` paulson@14387 ` 840` ```by (simp add: real_of_int_def of_int_number_of_eq) ``` paulson@14387 ` 841` paulson@14387 ` 842` ```lemma real_of_nat_number_of [simp]: ``` paulson@14387 ` 843` ``` "real (number_of v :: nat) = ``` paulson@14387 ` 844` ``` (if neg (number_of v :: int) then 0 ``` paulson@14387 ` 845` ``` else (number_of v :: real))" ``` paulson@14387 ` 846` ```by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of) ``` paulson@14387 ` 847` ``` ``` paulson@14387 ` 848` paulson@14387 ` 849` ```use "real_arith.ML" ``` paulson@14387 ` 850` paulson@14387 ` 851` ```setup real_arith_setup ``` paulson@14387 ` 852` kleing@19023 ` 853` paulson@14387 ` 854` ```subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} ``` paulson@14387 ` 855` paulson@14387 ` 856` ```text{*Needed in this non-standard form by Hyperreal/Transcendental*} ``` paulson@14387 ` 857` ```lemma real_0_le_divide_iff: ``` paulson@14387 ` 858` ``` "((0::real) \ x/y) = ((x \ 0 | 0 \ y) & (0 \ x | y \ 0))" ``` paulson@14387 ` 859` ```by (simp add: real_divide_def zero_le_mult_iff, auto) ``` paulson@14387 ` 860` paulson@14387 ` 861` ```lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" ``` paulson@14387 ` 862` ```by arith ``` paulson@14387 ` 863` paulson@15085 ` 864` ```lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)" ``` paulson@14387 ` 865` ```by auto ``` paulson@14387 ` 866` paulson@15085 ` 867` ```lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)" ``` paulson@14387 ` 868` ```by auto ``` paulson@14387 ` 869` paulson@15085 ` 870` ```lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)" ``` paulson@14387 ` 871` ```by auto ``` paulson@14387 ` 872` paulson@15085 ` 873` ```lemma real_add_le_0_iff: "(x+y \ (0::real)) = (y \ -x)" ``` paulson@14387 ` 874` ```by auto ``` paulson@14387 ` 875` paulson@15085 ` 876` ```lemma real_0_le_add_iff: "((0::real) \ x+y) = (-x \ y)" ``` paulson@14387 ` 877` ```by auto ``` paulson@14387 ` 878` paulson@14387 ` 879` paulson@14387 ` 880` ```(* ``` paulson@14387 ` 881` ```FIXME: we should have this, as for type int, but many proofs would break. ``` paulson@14387 ` 882` ```It replaces x+-y by x-y. ``` paulson@15086 ` 883` ```declare real_diff_def [symmetric, simp] ``` paulson@14387 ` 884` ```*) ``` paulson@14387 ` 885` paulson@14387 ` 886` paulson@14387 ` 887` ```subsubsection{*Density of the Reals*} ``` paulson@14387 ` 888` paulson@14387 ` 889` ```lemma real_lbound_gt_zero: ``` paulson@14387 ` 890` ``` "[| (0::real) < d1; 0 < d2 |] ==> \e. 0 < e & e < d1 & e < d2" ``` paulson@14387 ` 891` ```apply (rule_tac x = " (min d1 d2) /2" in exI) ``` paulson@14387 ` 892` ```apply (simp add: min_def) ``` paulson@14387 ` 893` ```done ``` paulson@14387 ` 894` paulson@14387 ` 895` paulson@14387 ` 896` ```text{*Similar results are proved in @{text Ring_and_Field}*} ``` paulson@14387 ` 897` ```lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)" ``` paulson@14387 ` 898` ``` by auto ``` paulson@14387 ` 899` paulson@14387 ` 900` ```lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y" ``` paulson@14387 ` 901` ``` by auto ``` paulson@14387 ` 902` paulson@14387 ` 903` paulson@14387 ` 904` ```subsection{*Absolute Value Function for the Reals*} ``` paulson@14387 ` 905` paulson@14387 ` 906` ```lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))" ``` paulson@15003 ` 907` ```by (simp add: abs_if) ``` paulson@14387 ` 908` huffman@23289 ` 909` ```(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *) ``` paulson@14387 ` 910` ```lemma abs_le_interval_iff: "(abs x \ r) = (-r\x & x\(r::real))" ``` obua@14738 ` 911` ```by (force simp add: OrderedGroup.abs_le_iff) ``` paulson@14387 ` 912` paulson@14387 ` 913` ```lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" ``` paulson@15003 ` 914` ```by (simp add: abs_if) ``` paulson@14387 ` 915` paulson@14387 ` 916` ```lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" ``` huffman@22958 ` 917` ```by (rule abs_of_nonneg [OF real_of_nat_ge_zero]) ``` paulson@14387 ` 918` paulson@14387 ` 919` ```lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x" ``` webertj@20217 ` 920` ```by simp ``` paulson@14387 ` 921` ``` ``` paulson@14387 ` 922` ```lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \ abs(x + -l) + abs(y + -m)" ``` webertj@20217 ` 923` ```by simp ``` paulson@14387 ` 924` nipkow@23031 ` 925` ```subsection{*Code generation using Isabelle's rats*} ``` nipkow@23031 ` 926` nipkow@23031 ` 927` ```types_code ``` nipkow@23031 ` 928` ``` real ("Rat.rat") ``` nipkow@23031 ` 929` ```attach (term_of) {* ``` nipkow@23031 ` 930` ```fun term_of_real x = ``` nipkow@23031 ` 931` ``` let ``` nipkow@23031 ` 932` ``` val rT = HOLogic.realT ``` nipkow@23031 ` 933` ``` val (p, q) = Rat.quotient_of_rat x ``` nipkow@23031 ` 934` ``` in if q = 1 then HOLogic.mk_number rT p ``` nipkow@23031 ` 935` ``` else Const("HOL.divide",[rT,rT] ---> rT) \$ ``` nipkow@23031 ` 936` ``` (HOLogic.mk_number rT p) \$ (HOLogic.mk_number rT q) ``` nipkow@23031 ` 937` ```end; ``` nipkow@23031 ` 938` ```*} ``` nipkow@23031 ` 939` ```attach (test) {* ``` nipkow@23031 ` 940` ```fun gen_real i = ``` nipkow@23031 ` 941` ```let val p = random_range 0 i; val q = random_range 0 i; ``` nipkow@23031 ` 942` ``` val r = if q=0 then Rat.rat_of_int i else Rat.rat_of_quotient(p,q) ``` nipkow@23031 ` 943` ```in if one_of [true,false] then r else Rat.neg r end; ``` nipkow@23031 ` 944` ```*} ``` nipkow@23031 ` 945` nipkow@23031 ` 946` ```consts_code ``` nipkow@23031 ` 947` ``` "0 :: real" ("Rat.zero") ``` nipkow@23031 ` 948` ``` "1 :: real" ("Rat.one") ``` nipkow@23031 ` 949` ``` "uminus :: real \ real" ("Rat.neg") ``` nipkow@23031 ` 950` ``` "op + :: real \ real \ real" ("Rat.add") ``` nipkow@23031 ` 951` ``` "op * :: real \ real \ real" ("Rat.mult") ``` nipkow@23031 ` 952` ``` "inverse :: real \ real" ("Rat.inv") ``` nipkow@23031 ` 953` ``` "op \ :: real \ real \ bool" ("Rat.le") ``` nipkow@23031 ` 954` ``` "op < :: real \ real \ bool" ("(Rat.ord (_, _) = LESS)") ``` nipkow@23031 ` 955` ``` "op = :: real \ real \ bool" ("curry Rat.eq") ``` nipkow@23031 ` 956` ``` "real :: int \ real" ("Rat.rat'_of'_int") ``` nipkow@23031 ` 957` ``` "real :: nat \ real" ("(Rat.rat'_of'_int o {*int*})") ``` nipkow@23031 ` 958` nipkow@23031 ` 959` nipkow@23031 ` 960` ```lemma [code, code unfold]: ``` nipkow@23031 ` 961` ``` "number_of k = real (number_of k :: int)" ``` nipkow@23031 ` 962` ``` by simp ``` nipkow@23031 ` 963` paulson@5588 ` 964` ```end ```