src/HOL/Real/RealDef.thy
 author webertj Wed Jul 26 19:23:04 2006 +0200 (2006-07-26) changeset 20217 25b068a99d2b parent 19765 dfe940911617 child 20485 3078fd2eec7b permissions -rw-r--r--
linear arithmetic splits certain operators (e.g. min, max, abs)
 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 ``` paulson@5588 ` 17` ``` realrel :: "((preal * preal) * (preal * preal)) set" ``` 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` paulson@14484 ` 29` ``` real_of_preal :: "preal => real" ``` wenzelm@19765 ` 30` ``` "real_of_preal m = Abs_Real(realrel``{(m + preal_of_rat 1, preal_of_rat 1)})" ``` paulson@14484 ` 31` paulson@14269 ` 32` ```consts ``` paulson@14378 ` 33` ``` (*Overloaded constant denoting the Real subset of enclosing ``` paulson@14269 ` 34` ``` types such as hypreal and complex*) ``` paulson@14269 ` 35` ``` Reals :: "'a set" ``` paulson@14269 ` 36` paulson@14269 ` 37` ``` (*overloaded constant for injecting other types into "real"*) ``` paulson@14269 ` 38` ``` real :: "'a => real" ``` paulson@5588 ` 39` wenzelm@19765 ` 40` ```const_syntax (xsymbols) ``` wenzelm@19765 ` 41` ``` Reals ("\") ``` wenzelm@14691 ` 42` paulson@5588 ` 43` paulson@14269 ` 44` ```defs (overloaded) ``` paulson@5588 ` 45` paulson@14269 ` 46` ``` real_zero_def: ``` paulson@14484 ` 47` ``` "0 == Abs_Real(realrel``{(preal_of_rat 1, preal_of_rat 1)})" ``` paulson@12018 ` 48` paulson@14269 ` 49` ``` real_one_def: ``` paulson@14484 ` 50` ``` "1 == Abs_Real(realrel`` ``` paulson@14365 ` 51` ``` {(preal_of_rat 1 + preal_of_rat 1, ``` paulson@14365 ` 52` ``` preal_of_rat 1)})" ``` paulson@5588 ` 53` paulson@14269 ` 54` ``` real_minus_def: ``` paulson@14484 ` 55` ``` "- r == contents (\(x,y) \ Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })" ``` paulson@14484 ` 56` paulson@14484 ` 57` ``` real_add_def: ``` paulson@14484 ` 58` ``` "z + w == ``` paulson@14484 ` 59` ``` contents (\(x,y) \ Rep_Real(z). \(u,v) \ Rep_Real(w). ``` paulson@14484 ` 60` ``` { Abs_Real(realrel``{(x+u, y+v)}) })" ``` bauerg@10606 ` 61` paulson@14269 ` 62` ``` real_diff_def: ``` paulson@14484 ` 63` ``` "r - (s::real) == r + - s" ``` paulson@14484 ` 64` paulson@14484 ` 65` ``` real_mult_def: ``` paulson@14484 ` 66` ``` "z * w == ``` paulson@14484 ` 67` ``` contents (\(x,y) \ Rep_Real(z). \(u,v) \ Rep_Real(w). ``` paulson@14484 ` 68` ``` { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" ``` paulson@5588 ` 69` paulson@14269 ` 70` ``` real_inverse_def: ``` wenzelm@11713 ` 71` ``` "inverse (R::real) == (SOME S. (R = 0 & S = 0) | S * R = 1)" ``` paulson@5588 ` 72` paulson@14269 ` 73` ``` real_divide_def: ``` bauerg@10606 ` 74` ``` "R / (S::real) == R * inverse S" ``` paulson@14269 ` 75` paulson@14484 ` 76` ``` real_le_def: ``` paulson@14484 ` 77` ``` "z \ (w::real) == ``` paulson@14484 ` 78` ``` \x y u v. x+v \ u+y & (x,y) \ Rep_Real z & (u,v) \ Rep_Real w" ``` paulson@5588 ` 79` paulson@14365 ` 80` ``` real_less_def: "(x < (y::real)) == (x \ y & x \ y)" ``` paulson@14365 ` 81` paulson@14334 ` 82` ``` real_abs_def: "abs (r::real) == (if 0 \ r then r else -r)" ``` paulson@14334 ` 83` paulson@14334 ` 84` paulson@14365 ` 85` paulson@14329 ` 86` ```subsection{*Proving that realrel is an equivalence relation*} ``` paulson@14269 ` 87` paulson@14270 ` 88` ```lemma preal_trans_lemma: ``` paulson@14365 ` 89` ``` assumes "x + y1 = x1 + y" ``` paulson@14365 ` 90` ``` and "x + y2 = x2 + y" ``` paulson@14365 ` 91` ``` shows "x1 + y2 = x2 + (y1::preal)" ``` paulson@14365 ` 92` ```proof - ``` paulson@14365 ` 93` ``` have "(x1 + y2) + x = (x + y2) + x1" by (simp add: preal_add_ac) ``` paulson@14365 ` 94` ``` also have "... = (x2 + y) + x1" by (simp add: prems) ``` paulson@14365 ` 95` ``` also have "... = x2 + (x1 + y)" by (simp add: preal_add_ac) ``` paulson@14365 ` 96` ``` also have "... = x2 + (x + y1)" by (simp add: prems) ``` paulson@14365 ` 97` ``` also have "... = (x2 + y1) + x" by (simp add: preal_add_ac) ``` paulson@14365 ` 98` ``` finally have "(x1 + y2) + x = (x2 + y1) + x" . ``` paulson@14365 ` 99` ``` thus ?thesis by (simp add: preal_add_right_cancel_iff) ``` paulson@14365 ` 100` ```qed ``` paulson@14365 ` 101` paulson@14269 ` 102` paulson@14484 ` 103` ```lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \ realrel) = (x1 + y2 = x2 + y1)" ``` paulson@14484 ` 104` ```by (simp add: realrel_def) ``` paulson@14269 ` 105` paulson@14269 ` 106` ```lemma equiv_realrel: "equiv UNIV realrel" ``` paulson@14365 ` 107` ```apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def) ``` paulson@14365 ` 108` ```apply (blast dest: preal_trans_lemma) ``` paulson@14269 ` 109` ```done ``` paulson@14269 ` 110` paulson@14497 ` 111` ```text{*Reduces equality of equivalence classes to the @{term realrel} relation: ``` paulson@14497 ` 112` ``` @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \ realrel)"} *} ``` paulson@14269 ` 113` ```lemmas equiv_realrel_iff = ``` paulson@14269 ` 114` ``` eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I] ``` paulson@14269 ` 115` paulson@14269 ` 116` ```declare equiv_realrel_iff [simp] ``` paulson@14269 ` 117` paulson@14497 ` 118` paulson@14484 ` 119` ```lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real" ``` paulson@14484 ` 120` ```by (simp add: Real_def realrel_def quotient_def, blast) ``` paulson@14269 ` 121` paulson@14365 ` 122` paulson@14484 ` 123` ```lemma inj_on_Abs_Real: "inj_on Abs_Real Real" ``` paulson@14269 ` 124` ```apply (rule inj_on_inverseI) ``` paulson@14484 ` 125` ```apply (erule Abs_Real_inverse) ``` paulson@14269 ` 126` ```done ``` paulson@14269 ` 127` paulson@14484 ` 128` ```declare inj_on_Abs_Real [THEN inj_on_iff, simp] ``` paulson@14484 ` 129` ```declare Abs_Real_inverse [simp] ``` paulson@14269 ` 130` paulson@14269 ` 131` paulson@14484 ` 132` ```text{*Case analysis on the representation of a real number as an equivalence ``` paulson@14484 ` 133` ``` class of pairs of positive reals.*} ``` paulson@14484 ` 134` ```lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: ``` paulson@14484 ` 135` ``` "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P" ``` paulson@14484 ` 136` ```apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE]) ``` paulson@14484 ` 137` ```apply (drule arg_cong [where f=Abs_Real]) ``` paulson@14484 ` 138` ```apply (auto simp add: Rep_Real_inverse) ``` paulson@14269 ` 139` ```done ``` paulson@14269 ` 140` paulson@14269 ` 141` paulson@14329 ` 142` ```subsection{*Congruence property for addition*} ``` paulson@14269 ` 143` paulson@14269 ` 144` ```lemma real_add_congruent2_lemma: ``` paulson@14269 ` 145` ``` "[|a + ba = aa + b; ab + bc = ac + bb|] ``` paulson@14269 ` 146` ``` ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))" ``` paulson@14269 ` 147` ```apply (simp add: preal_add_assoc) ``` paulson@14269 ` 148` ```apply (rule preal_add_left_commute [of ab, THEN ssubst]) ``` paulson@14269 ` 149` ```apply (simp add: preal_add_assoc [symmetric]) ``` paulson@14269 ` 150` ```apply (simp add: preal_add_ac) ``` paulson@14269 ` 151` ```done ``` paulson@14269 ` 152` paulson@14269 ` 153` ```lemma real_add: ``` paulson@14497 ` 154` ``` "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) = ``` paulson@14497 ` 155` ``` Abs_Real (realrel``{(x+u, y+v)})" ``` paulson@14497 ` 156` ```proof - ``` paulson@15169 ` 157` ``` have "(\z w. (\(x,y). (\(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z) ``` paulson@15169 ` 158` ``` respects2 realrel" ``` paulson@14497 ` 159` ``` by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) ``` paulson@14497 ` 160` ``` thus ?thesis ``` paulson@14497 ` 161` ``` by (simp add: real_add_def UN_UN_split_split_eq ``` paulson@14658 ` 162` ``` UN_equiv_class2 [OF equiv_realrel equiv_realrel]) ``` paulson@14497 ` 163` ```qed ``` paulson@14269 ` 164` paulson@14269 ` 165` ```lemma real_add_commute: "(z::real) + w = w + z" ``` paulson@14497 ` 166` ```by (cases z, cases w, simp add: real_add preal_add_ac) ``` paulson@14269 ` 167` paulson@14269 ` 168` ```lemma real_add_assoc: "((z1::real) + z2) + z3 = z1 + (z2 + z3)" ``` paulson@14497 ` 169` ```by (cases z1, cases z2, cases z3, simp add: real_add preal_add_assoc) ``` paulson@14269 ` 170` paulson@14269 ` 171` ```lemma real_add_zero_left: "(0::real) + z = z" ``` paulson@14497 ` 172` ```by (cases z, simp add: real_add real_zero_def preal_add_ac) ``` paulson@14269 ` 173` obua@14738 ` 174` ```instance real :: comm_monoid_add ``` paulson@14269 ` 175` ``` by (intro_classes, ``` paulson@14269 ` 176` ``` (assumption | ``` paulson@14269 ` 177` ``` rule real_add_commute real_add_assoc real_add_zero_left)+) ``` paulson@14269 ` 178` paulson@14269 ` 179` paulson@14334 ` 180` ```subsection{*Additive Inverse on real*} ``` paulson@14334 ` 181` paulson@14484 ` 182` ```lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})" ``` paulson@14484 ` 183` ```proof - ``` paulson@15169 ` 184` ``` have "(\(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel" ``` paulson@14484 ` 185` ``` by (simp add: congruent_def preal_add_commute) ``` paulson@14484 ` 186` ``` thus ?thesis ``` paulson@14484 ` 187` ``` by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel]) ``` paulson@14484 ` 188` ```qed ``` paulson@14334 ` 189` paulson@14334 ` 190` ```lemma real_add_minus_left: "(-z) + z = (0::real)" ``` paulson@14497 ` 191` ```by (cases z, simp add: real_minus real_add real_zero_def preal_add_commute) ``` paulson@14269 ` 192` paulson@14269 ` 193` paulson@14329 ` 194` ```subsection{*Congruence property for multiplication*} ``` paulson@14269 ` 195` paulson@14329 ` 196` ```lemma real_mult_congruent2_lemma: ``` paulson@14329 ` 197` ``` "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> ``` paulson@14484 ` 198` ``` x * x1 + y * y1 + (x * y2 + y * x2) = ``` paulson@14484 ` 199` ``` x * x2 + y * y2 + (x * y1 + y * x1)" ``` paulson@14484 ` 200` ```apply (simp add: preal_add_left_commute preal_add_assoc [symmetric]) ``` paulson@14269 ` 201` ```apply (simp add: preal_add_assoc preal_add_mult_distrib2 [symmetric]) ``` paulson@14269 ` 202` ```apply (simp add: preal_add_commute) ``` paulson@14269 ` 203` ```done ``` paulson@14269 ` 204` paulson@14269 ` 205` ```lemma real_mult_congruent2: ``` paulson@15169 ` 206` ``` "(%p1 p2. ``` paulson@14484 ` 207` ``` (%(x1,y1). (%(x2,y2). ``` paulson@15169 ` 208` ``` { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1) ``` paulson@15169 ` 209` ``` respects2 realrel" ``` paulson@14658 ` 210` ```apply (rule congruent2_commuteI [OF equiv_realrel], clarify) ``` paulson@14269 ` 211` ```apply (simp add: preal_mult_commute preal_add_commute) ``` paulson@14269 ` 212` ```apply (auto simp add: real_mult_congruent2_lemma) ``` paulson@14269 ` 213` ```done ``` paulson@14269 ` 214` paulson@14269 ` 215` ```lemma real_mult: ``` paulson@14484 ` 216` ``` "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) = ``` paulson@14484 ` 217` ``` Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})" ``` paulson@14484 ` 218` ```by (simp add: real_mult_def UN_UN_split_split_eq ``` paulson@14658 ` 219` ``` UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2]) ``` paulson@14269 ` 220` paulson@14269 ` 221` ```lemma real_mult_commute: "(z::real) * w = w * z" ``` paulson@14497 ` 222` ```by (cases z, cases w, simp add: real_mult preal_add_ac preal_mult_ac) ``` paulson@14269 ` 223` paulson@14269 ` 224` ```lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)" ``` paulson@14484 ` 225` ```apply (cases z1, cases z2, cases z3) ``` paulson@14484 ` 226` ```apply (simp add: real_mult preal_add_mult_distrib2 preal_add_ac preal_mult_ac) ``` paulson@14269 ` 227` ```done ``` paulson@14269 ` 228` paulson@14269 ` 229` ```lemma real_mult_1: "(1::real) * z = z" ``` paulson@14484 ` 230` ```apply (cases z) ``` paulson@14484 ` 231` ```apply (simp add: real_mult real_one_def preal_add_mult_distrib2 ``` paulson@14484 ` 232` ``` preal_mult_1_right preal_mult_ac preal_add_ac) ``` paulson@14269 ` 233` ```done ``` paulson@14269 ` 234` paulson@14269 ` 235` ```lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)" ``` paulson@14484 ` 236` ```apply (cases z1, cases z2, cases w) ``` paulson@14484 ` 237` ```apply (simp add: real_add real_mult preal_add_mult_distrib2 ``` paulson@14484 ` 238` ``` preal_add_ac preal_mult_ac) ``` paulson@14269 ` 239` ```done ``` paulson@14269 ` 240` paulson@14329 ` 241` ```text{*one and zero are distinct*} ``` paulson@14365 ` 242` ```lemma real_zero_not_eq_one: "0 \ (1::real)" ``` paulson@14484 ` 243` ```proof - ``` paulson@14484 ` 244` ``` have "preal_of_rat 1 < preal_of_rat 1 + preal_of_rat 1" ``` paulson@14484 ` 245` ``` by (simp add: preal_self_less_add_left) ``` paulson@14484 ` 246` ``` thus ?thesis ``` paulson@14484 ` 247` ``` by (simp add: real_zero_def real_one_def preal_add_right_cancel_iff) ``` paulson@14484 ` 248` ```qed ``` paulson@14269 ` 249` paulson@14329 ` 250` ```subsection{*existence of inverse*} ``` paulson@14365 ` 251` paulson@14484 ` 252` ```lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0" ``` paulson@14497 ` 253` ```by (simp add: real_zero_def preal_add_commute) ``` paulson@14269 ` 254` paulson@14365 ` 255` ```text{*Instead of using an existential quantifier and constructing the inverse ``` paulson@14365 ` 256` ```within the proof, we could define the inverse explicitly.*} ``` paulson@14365 ` 257` paulson@14365 ` 258` ```lemma real_mult_inverse_left_ex: "x \ 0 ==> \y. y*x = (1::real)" ``` paulson@14484 ` 259` ```apply (simp add: real_zero_def real_one_def, cases x) ``` paulson@14269 ` 260` ```apply (cut_tac x = xa and y = y in linorder_less_linear) ``` paulson@14365 ` 261` ```apply (auto dest!: less_add_left_Ex simp add: real_zero_iff) ``` paulson@14334 ` 262` ```apply (rule_tac ``` paulson@14484 ` 263` ``` x = "Abs_Real (realrel `` { (preal_of_rat 1, ``` paulson@14365 ` 264` ``` inverse (D) + preal_of_rat 1)}) " ``` paulson@14334 ` 265` ``` in exI) ``` paulson@14334 ` 266` ```apply (rule_tac [2] ``` paulson@14484 ` 267` ``` x = "Abs_Real (realrel `` { (inverse (D) + preal_of_rat 1, ``` paulson@14365 ` 268` ``` preal_of_rat 1)})" ``` paulson@14334 ` 269` ``` in exI) ``` paulson@14365 ` 270` ```apply (auto simp add: real_mult preal_mult_1_right ``` paulson@14329 ` 271` ``` preal_add_mult_distrib2 preal_add_mult_distrib preal_mult_1 ``` paulson@14365 ` 272` ``` preal_mult_inverse_right preal_add_ac preal_mult_ac) ``` paulson@14269 ` 273` ```done ``` paulson@14269 ` 274` paulson@14365 ` 275` ```lemma real_mult_inverse_left: "x \ 0 ==> inverse(x)*x = (1::real)" ``` paulson@14484 ` 276` ```apply (simp add: real_inverse_def) ``` paulson@14365 ` 277` ```apply (frule real_mult_inverse_left_ex, safe) ``` paulson@14269 ` 278` ```apply (rule someI2, auto) ``` paulson@14269 ` 279` ```done ``` paulson@14334 ` 280` paulson@14341 ` 281` paulson@14341 ` 282` ```subsection{*The Real Numbers form a Field*} ``` paulson@14341 ` 283` paulson@14334 ` 284` ```instance real :: field ``` paulson@14334 ` 285` ```proof ``` paulson@14334 ` 286` ``` fix x y z :: real ``` paulson@14334 ` 287` ``` show "- x + x = 0" by (rule real_add_minus_left) ``` paulson@14334 ` 288` ``` show "x - y = x + (-y)" by (simp add: real_diff_def) ``` paulson@14334 ` 289` ``` show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc) ``` paulson@14334 ` 290` ``` show "x * y = y * x" by (rule real_mult_commute) ``` paulson@14334 ` 291` ``` show "1 * x = x" by (rule real_mult_1) ``` paulson@14334 ` 292` ``` show "(x + y) * z = x * z + y * z" by (simp add: real_add_mult_distrib) ``` paulson@14334 ` 293` ``` show "0 \ (1::real)" by (rule real_zero_not_eq_one) ``` paulson@14365 ` 294` ``` show "x \ 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left) ``` paulson@14430 ` 295` ``` show "x / y = x * inverse y" by (simp add: real_divide_def) ``` paulson@14334 ` 296` ```qed ``` paulson@14334 ` 297` paulson@14334 ` 298` paulson@14341 ` 299` ```text{*Inverse of zero! Useful to simplify certain equations*} ``` paulson@14269 ` 300` paulson@14334 ` 301` ```lemma INVERSE_ZERO: "inverse 0 = (0::real)" ``` paulson@14484 ` 302` ```by (simp add: real_inverse_def) ``` paulson@14334 ` 303` paulson@14334 ` 304` ```instance real :: division_by_zero ``` paulson@14334 ` 305` ```proof ``` paulson@14334 ` 306` ``` show "inverse 0 = (0::real)" by (rule INVERSE_ZERO) ``` paulson@14334 ` 307` ```qed ``` paulson@14334 ` 308` paulson@14334 ` 309` paulson@14334 ` 310` ```(*Pull negations out*) ``` paulson@14334 ` 311` ```declare minus_mult_right [symmetric, simp] ``` paulson@14334 ` 312` ``` minus_mult_left [symmetric, simp] ``` paulson@14334 ` 313` paulson@14334 ` 314` ```lemma real_mult_1_right: "z * (1::real) = z" ``` obua@14738 ` 315` ``` by (rule OrderedGroup.mult_1_right) ``` paulson@14269 ` 316` paulson@14269 ` 317` paulson@14365 ` 318` ```subsection{*The @{text "\"} Ordering*} ``` paulson@14269 ` 319` paulson@14365 ` 320` ```lemma real_le_refl: "w \ (w::real)" ``` paulson@14484 ` 321` ```by (cases w, force simp add: real_le_def) ``` paulson@14269 ` 322` paulson@14378 ` 323` ```text{*The arithmetic decision procedure is not set up for type preal. ``` paulson@14378 ` 324` ``` This lemma is currently unused, but it could simplify the proofs of the ``` paulson@14378 ` 325` ``` following two lemmas.*} ``` paulson@14378 ` 326` ```lemma preal_eq_le_imp_le: ``` paulson@14378 ` 327` ``` assumes eq: "a+b = c+d" and le: "c \ a" ``` paulson@14378 ` 328` ``` shows "b \ (d::preal)" ``` paulson@14378 ` 329` ```proof - ``` paulson@14378 ` 330` ``` have "c+d \ a+d" by (simp add: prems preal_cancels) ``` paulson@14378 ` 331` ``` hence "a+b \ a+d" by (simp add: prems) ``` paulson@14378 ` 332` ``` thus "b \ d" by (simp add: preal_cancels) ``` paulson@14378 ` 333` ```qed ``` paulson@14378 ` 334` paulson@14378 ` 335` ```lemma real_le_lemma: ``` paulson@14378 ` 336` ``` assumes l: "u1 + v2 \ u2 + v1" ``` paulson@14378 ` 337` ``` and "x1 + v1 = u1 + y1" ``` paulson@14378 ` 338` ``` and "x2 + v2 = u2 + y2" ``` paulson@14378 ` 339` ``` shows "x1 + y2 \ x2 + (y1::preal)" ``` paulson@14365 ` 340` ```proof - ``` paulson@14378 ` 341` ``` have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems) ``` paulson@14378 ` 342` ``` hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: preal_add_ac) ``` paulson@14378 ` 343` ``` also have "... \ (x2+y1) + (u2+v1)" ``` paulson@14365 ` 344` ``` by (simp add: prems preal_add_le_cancel_left) ``` paulson@14378 ` 345` ``` finally show ?thesis by (simp add: preal_add_le_cancel_right) ``` paulson@14378 ` 346` ```qed ``` paulson@14378 ` 347` paulson@14378 ` 348` ```lemma real_le: ``` paulson@14484 ` 349` ``` "(Abs_Real(realrel``{(x1,y1)}) \ Abs_Real(realrel``{(x2,y2)})) = ``` paulson@14484 ` 350` ``` (x1 + y2 \ x2 + y1)" ``` paulson@14378 ` 351` ```apply (simp add: real_le_def) ``` paulson@14387 ` 352` ```apply (auto intro: real_le_lemma) ``` paulson@14378 ` 353` ```done ``` paulson@14378 ` 354` paulson@14378 ` 355` ```lemma real_le_anti_sym: "[| z \ w; w \ z |] ==> z = (w::real)" ``` nipkow@15542 ` 356` ```by (cases z, cases w, simp add: real_le) ``` paulson@14378 ` 357` paulson@14378 ` 358` ```lemma real_trans_lemma: ``` paulson@14378 ` 359` ``` assumes "x + v \ u + y" ``` paulson@14378 ` 360` ``` and "u + v' \ u' + v" ``` paulson@14378 ` 361` ``` and "x2 + v2 = u2 + y2" ``` paulson@14378 ` 362` ``` shows "x + v' \ u' + (y::preal)" ``` paulson@14378 ` 363` ```proof - ``` paulson@14378 ` 364` ``` have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: preal_add_ac) ``` paulson@14378 ` 365` ``` also have "... \ (u+y) + (u+v')" ``` paulson@14378 ` 366` ``` by (simp add: preal_add_le_cancel_right prems) ``` paulson@14378 ` 367` ``` also have "... \ (u+y) + (u'+v)" ``` paulson@14378 ` 368` ``` by (simp add: preal_add_le_cancel_left prems) ``` paulson@14378 ` 369` ``` also have "... = (u'+y) + (u+v)" by (simp add: preal_add_ac) ``` paulson@14378 ` 370` ``` finally show ?thesis by (simp add: preal_add_le_cancel_right) ``` nipkow@15542 ` 371` ```qed ``` paulson@14269 ` 372` paulson@14365 ` 373` ```lemma real_le_trans: "[| i \ j; j \ k |] ==> i \ (k::real)" ``` paulson@14484 ` 374` ```apply (cases i, cases j, cases k) ``` paulson@14484 ` 375` ```apply (simp add: real_le) ``` paulson@14378 ` 376` ```apply (blast intro: real_trans_lemma) ``` paulson@14334 ` 377` ```done ``` paulson@14334 ` 378` paulson@14334 ` 379` ```(* Axiom 'order_less_le' of class 'order': *) ``` paulson@14334 ` 380` ```lemma real_less_le: "((w::real) < z) = (w \ z & w \ z)" ``` paulson@14365 ` 381` ```by (simp add: real_less_def) ``` paulson@14365 ` 382` paulson@14365 ` 383` ```instance real :: order ``` paulson@14365 ` 384` ```proof qed ``` paulson@14365 ` 385` ``` (assumption | ``` paulson@14365 ` 386` ``` rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+ ``` paulson@14365 ` 387` paulson@14378 ` 388` ```(* Axiom 'linorder_linear' of class 'linorder': *) ``` paulson@14378 ` 389` ```lemma real_le_linear: "(z::real) \ w | w \ z" ``` paulson@14484 ` 390` ```apply (cases z, cases w) ``` paulson@14378 ` 391` ```apply (auto simp add: real_le real_zero_def preal_add_ac preal_cancels) ``` paulson@14334 ` 392` ```done ``` paulson@14334 ` 393` paulson@14334 ` 394` paulson@14334 ` 395` ```instance real :: linorder ``` paulson@14334 ` 396` ``` by (intro_classes, rule real_le_linear) ``` paulson@14334 ` 397` paulson@14334 ` 398` paulson@14378 ` 399` ```lemma real_le_eq_diff: "(x \ y) = (x-y \ (0::real))" ``` paulson@14484 ` 400` ```apply (cases x, cases y) ``` paulson@14378 ` 401` ```apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus ``` paulson@14378 ` 402` ``` preal_add_ac) ``` paulson@14378 ` 403` ```apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) ``` nipkow@15542 ` 404` ```done ``` paulson@14378 ` 405` paulson@14484 ` 406` ```lemma real_add_left_mono: ``` paulson@14484 ` 407` ``` assumes le: "x \ y" shows "z + x \ z + (y::real)" ``` paulson@14484 ` 408` ```proof - ``` paulson@14484 ` 409` ``` have "z + x - (z + y) = (z + -z) + (x - y)" ``` paulson@14484 ` 410` ``` by (simp add: diff_minus add_ac) ``` paulson@14484 ` 411` ``` with le show ?thesis ``` obua@14754 ` 412` ``` by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus) ``` paulson@14484 ` 413` ```qed ``` paulson@14334 ` 414` paulson@14365 ` 415` ```lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)" ``` paulson@14365 ` 416` ```by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) ``` paulson@14365 ` 417` paulson@14365 ` 418` ```lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))" ``` paulson@14365 ` 419` ```by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus) ``` paulson@14334 ` 420` paulson@14334 ` 421` ```lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y" ``` paulson@14484 ` 422` ```apply (cases x, cases y) ``` paulson@14378 ` 423` ```apply (simp add: linorder_not_le [where 'a = real, symmetric] ``` paulson@14378 ` 424` ``` linorder_not_le [where 'a = preal] ``` paulson@14378 ` 425` ``` real_zero_def real_le real_mult) ``` paulson@14365 ` 426` ``` --{*Reduce to the (simpler) @{text "\"} relation *} ``` wenzelm@16973 ` 427` ```apply (auto dest!: less_add_left_Ex ``` paulson@14365 ` 428` ``` simp add: preal_add_ac preal_mult_ac ``` wenzelm@16973 ` 429` ``` preal_add_mult_distrib2 preal_cancels preal_self_less_add_left) ``` paulson@14334 ` 430` ```done ``` paulson@14334 ` 431` paulson@14334 ` 432` ```lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y" ``` paulson@14334 ` 433` ```apply (rule real_sum_gt_zero_less) ``` paulson@14334 ` 434` ```apply (drule real_less_sum_gt_zero [of x y]) ``` paulson@14334 ` 435` ```apply (drule real_mult_order, assumption) ``` paulson@14334 ` 436` ```apply (simp add: right_distrib) ``` paulson@14334 ` 437` ```done ``` paulson@14334 ` 438` paulson@14365 ` 439` ```text{*lemma for proving @{term "0<(1::real)"}*} ``` paulson@14365 ` 440` ```lemma real_zero_le_one: "0 \ (1::real)" ``` paulson@14387 ` 441` ```by (simp add: real_zero_def real_one_def real_le ``` paulson@14378 ` 442` ``` preal_self_less_add_left order_less_imp_le) ``` paulson@14334 ` 443` paulson@14378 ` 444` paulson@14334 ` 445` ```subsection{*The Reals Form an Ordered Field*} ``` paulson@14334 ` 446` paulson@14334 ` 447` ```instance real :: ordered_field ``` paulson@14334 ` 448` ```proof ``` paulson@14334 ` 449` ``` fix x y z :: real ``` paulson@14334 ` 450` ``` show "x \ y ==> z + x \ z + y" by (rule real_add_left_mono) ``` paulson@14334 ` 451` ``` show "x < y ==> 0 < z ==> z * x < z * y" by (simp add: real_mult_less_mono2) ``` paulson@14334 ` 452` ``` show "\x\ = (if x < 0 then -x else x)" ``` paulson@14334 ` 453` ``` by (auto dest: order_le_less_trans simp add: real_abs_def linorder_not_le) ``` paulson@14334 ` 454` ```qed ``` paulson@14334 ` 455` paulson@14365 ` 456` paulson@14365 ` 457` paulson@14365 ` 458` ```text{*The function @{term real_of_preal} requires many proofs, but it seems ``` paulson@14365 ` 459` ```to be essential for proving completeness of the reals from that of the ``` paulson@14365 ` 460` ```positive reals.*} ``` paulson@14365 ` 461` paulson@14365 ` 462` ```lemma real_of_preal_add: ``` paulson@14365 ` 463` ``` "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y" ``` paulson@14365 ` 464` ```by (simp add: real_of_preal_def real_add preal_add_mult_distrib preal_mult_1 ``` paulson@14365 ` 465` ``` preal_add_ac) ``` paulson@14365 ` 466` paulson@14365 ` 467` ```lemma real_of_preal_mult: ``` paulson@14365 ` 468` ``` "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y" ``` paulson@14365 ` 469` ```by (simp add: real_of_preal_def real_mult preal_add_mult_distrib2 ``` paulson@14365 ` 470` ``` preal_mult_1 preal_mult_1_right preal_add_ac preal_mult_ac) ``` paulson@14365 ` 471` paulson@14365 ` 472` paulson@14365 ` 473` ```text{*Gleason prop 9-4.4 p 127*} ``` paulson@14365 ` 474` ```lemma real_of_preal_trichotomy: ``` paulson@14365 ` 475` ``` "\m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)" ``` paulson@14484 ` 476` ```apply (simp add: real_of_preal_def real_zero_def, cases x) ``` paulson@14365 ` 477` ```apply (auto simp add: real_minus preal_add_ac) ``` paulson@14365 ` 478` ```apply (cut_tac x = x and y = y in linorder_less_linear) ``` paulson@14365 ` 479` ```apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc [symmetric]) ``` paulson@14365 ` 480` ```done ``` paulson@14365 ` 481` paulson@14365 ` 482` ```lemma real_of_preal_leD: ``` paulson@14365 ` 483` ``` "real_of_preal m1 \ real_of_preal m2 ==> m1 \ m2" ``` paulson@14484 ` 484` ```by (simp add: real_of_preal_def real_le preal_cancels) ``` paulson@14365 ` 485` paulson@14365 ` 486` ```lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2" ``` paulson@14365 ` 487` ```by (auto simp add: real_of_preal_leD linorder_not_le [symmetric]) ``` paulson@14365 ` 488` paulson@14365 ` 489` ```lemma real_of_preal_lessD: ``` paulson@14365 ` 490` ``` "real_of_preal m1 < real_of_preal m2 ==> m1 < m2" ``` paulson@14484 ` 491` ```by (simp add: real_of_preal_def real_le linorder_not_le [symmetric] ``` paulson@14484 ` 492` ``` preal_cancels) ``` paulson@14484 ` 493` paulson@14365 ` 494` paulson@14365 ` 495` ```lemma real_of_preal_less_iff [simp]: ``` paulson@14365 ` 496` ``` "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)" ``` paulson@14365 ` 497` ```by (blast intro: real_of_preal_lessI real_of_preal_lessD) ``` paulson@14365 ` 498` paulson@14365 ` 499` ```lemma real_of_preal_le_iff: ``` paulson@14365 ` 500` ``` "(real_of_preal m1 \ real_of_preal m2) = (m1 \ m2)" ``` paulson@14365 ` 501` ```by (simp add: linorder_not_less [symmetric]) ``` paulson@14365 ` 502` paulson@14365 ` 503` ```lemma real_of_preal_zero_less: "0 < real_of_preal m" ``` paulson@14365 ` 504` ```apply (auto simp add: real_zero_def real_of_preal_def real_less_def real_le_def ``` paulson@14365 ` 505` ``` preal_add_ac preal_cancels) ``` paulson@14365 ` 506` ```apply (simp_all add: preal_add_assoc [symmetric] preal_cancels) ``` paulson@14365 ` 507` ```apply (blast intro: preal_self_less_add_left order_less_imp_le) ``` paulson@14365 ` 508` ```apply (insert preal_not_eq_self [of "preal_of_rat 1" m]) ``` paulson@14365 ` 509` ```apply (simp add: preal_add_ac) ``` paulson@14365 ` 510` ```done ``` paulson@14365 ` 511` paulson@14365 ` 512` ```lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0" ``` paulson@14365 ` 513` ```by (simp add: real_of_preal_zero_less) ``` paulson@14365 ` 514` paulson@14365 ` 515` ```lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m" ``` paulson@14484 ` 516` ```proof - ``` paulson@14484 ` 517` ``` from real_of_preal_minus_less_zero ``` paulson@14484 ` 518` ``` show ?thesis by (blast dest: order_less_trans) ``` paulson@14484 ` 519` ```qed ``` paulson@14365 ` 520` paulson@14365 ` 521` paulson@14365 ` 522` ```subsection{*Theorems About the Ordering*} ``` paulson@14365 ` 523` paulson@14365 ` 524` ```text{*obsolete but used a lot*} ``` paulson@14365 ` 525` paulson@14365 ` 526` ```lemma real_not_refl2: "x < y ==> x \ (y::real)" ``` paulson@14365 ` 527` ```by blast ``` paulson@14365 ` 528` paulson@14365 ` 529` ```lemma real_le_imp_less_or_eq: "!!(x::real). x \ y ==> x < y | x = y" ``` paulson@14365 ` 530` ```by (simp add: order_le_less) ``` paulson@14365 ` 531` paulson@14365 ` 532` ```lemma real_gt_zero_preal_Ex: "(0 < x) = (\y. x = real_of_preal y)" ``` paulson@14365 ` 533` ```apply (auto simp add: real_of_preal_zero_less) ``` paulson@14365 ` 534` ```apply (cut_tac x = x in real_of_preal_trichotomy) ``` paulson@14365 ` 535` ```apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE]) ``` paulson@14365 ` 536` ```done ``` paulson@14365 ` 537` paulson@14365 ` 538` ```lemma real_gt_preal_preal_Ex: ``` paulson@14365 ` 539` ``` "real_of_preal z < x ==> \y. x = real_of_preal y" ``` paulson@14365 ` 540` ```by (blast dest!: real_of_preal_zero_less [THEN order_less_trans] ``` paulson@14365 ` 541` ``` intro: real_gt_zero_preal_Ex [THEN iffD1]) ``` paulson@14365 ` 542` paulson@14365 ` 543` ```lemma real_ge_preal_preal_Ex: ``` paulson@14365 ` 544` ``` "real_of_preal z \ x ==> \y. x = real_of_preal y" ``` paulson@14365 ` 545` ```by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex) ``` paulson@14365 ` 546` paulson@14365 ` 547` ```lemma real_less_all_preal: "y \ 0 ==> \x. y < real_of_preal x" ``` paulson@14365 ` 548` ```by (auto elim: order_le_imp_less_or_eq [THEN disjE] ``` paulson@14365 ` 549` ``` intro: real_of_preal_zero_less [THEN [2] order_less_trans] ``` paulson@14365 ` 550` ``` simp add: real_of_preal_zero_less) ``` paulson@14365 ` 551` paulson@14365 ` 552` ```lemma real_less_all_real2: "~ 0 < y ==> \x. y < real_of_preal x" ``` paulson@14365 ` 553` ```by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1]) ``` paulson@14365 ` 554` paulson@14334 ` 555` ```lemma real_add_less_le_mono: "[| w'z |] ==> w' + z' < w + (z::real)" ``` obua@14738 ` 556` ``` by (rule OrderedGroup.add_less_le_mono) ``` paulson@14334 ` 557` paulson@14334 ` 558` ```lemma real_add_le_less_mono: ``` paulson@14334 ` 559` ``` "!!z z'::real. [| w'\w; z' w' + z' < w + z" ``` obua@14738 ` 560` ``` by (rule OrderedGroup.add_le_less_mono) ``` paulson@14334 ` 561` paulson@14334 ` 562` ```lemma real_le_square [simp]: "(0::real) \ x*x" ``` paulson@14334 ` 563` ``` by (rule Ring_and_Field.zero_le_square) ``` paulson@14334 ` 564` paulson@14334 ` 565` paulson@14334 ` 566` ```subsection{*More Lemmas*} ``` paulson@14334 ` 567` paulson@14334 ` 568` ```lemma real_mult_left_cancel: "(c::real) \ 0 ==> (c*a=c*b) = (a=b)" ``` paulson@14334 ` 569` ```by auto ``` paulson@14334 ` 570` paulson@14334 ` 571` ```lemma real_mult_right_cancel: "(c::real) \ 0 ==> (a*c=b*c) = (a=b)" ``` paulson@14334 ` 572` ```by auto ``` paulson@14334 ` 573` paulson@14334 ` 574` ```text{*The precondition could be weakened to @{term "0\x"}*} ``` paulson@14334 ` 575` ```lemma real_mult_less_mono: ``` paulson@14334 ` 576` ``` "[| u u*x < v* y" ``` paulson@14334 ` 577` ``` by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) ``` paulson@14334 ` 578` paulson@14334 ` 579` ```lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" ``` paulson@14334 ` 580` ``` by (force elim: order_less_asym ``` paulson@14334 ` 581` ``` simp add: Ring_and_Field.mult_less_cancel_right) ``` paulson@14334 ` 582` paulson@14334 ` 583` ```lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \ y*z) = (x\y)" ``` paulson@14365 ` 584` ```apply (simp add: mult_le_cancel_right) ``` paulson@14365 ` 585` ```apply (blast intro: elim: order_less_asym) ``` paulson@14365 ` 586` ```done ``` paulson@14334 ` 587` paulson@14334 ` 588` ```lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \ z*y) = (x\y)" ``` nipkow@15923 ` 589` ```by(simp add:mult_commute) ``` paulson@14334 ` 590` paulson@14334 ` 591` ```text{*Only two uses?*} ``` paulson@14334 ` 592` ```lemma real_mult_less_mono': ``` paulson@14334 ` 593` ``` "[| x < y; r1 < r2; (0::real) \ r1; 0 \ x|] ==> r1 * x < r2 * y" ``` paulson@14334 ` 594` ``` by (rule Ring_and_Field.mult_strict_mono') ``` paulson@14334 ` 595` paulson@14334 ` 596` ```text{*FIXME: delete or at least combine the next two lemmas*} ``` paulson@14334 ` 597` ```lemma real_sum_squares_cancel: "x * x + y * y = 0 ==> x = (0::real)" ``` obua@14738 ` 598` ```apply (drule OrderedGroup.equals_zero_I [THEN sym]) ``` paulson@14334 ` 599` ```apply (cut_tac x = y in real_le_square) ``` paulson@14476 ` 600` ```apply (auto, drule order_antisym, auto) ``` paulson@14334 ` 601` ```done ``` paulson@14334 ` 602` paulson@14334 ` 603` ```lemma real_sum_squares_cancel2: "x * x + y * y = 0 ==> y = (0::real)" ``` paulson@14334 ` 604` ```apply (rule_tac y = x in real_sum_squares_cancel) ``` paulson@14476 ` 605` ```apply (simp add: add_commute) ``` paulson@14334 ` 606` ```done ``` paulson@14334 ` 607` paulson@14334 ` 608` ```lemma real_add_order: "[| 0 < x; 0 < y |] ==> (0::real) < x + y" ``` paulson@14365 ` 609` ```by (drule add_strict_mono [of concl: 0 0], assumption, simp) ``` paulson@14334 ` 610` paulson@14334 ` 611` ```lemma real_le_add_order: "[| 0 \ x; 0 \ y |] ==> (0::real) \ x + y" ``` paulson@14334 ` 612` ```apply (drule order_le_imp_less_or_eq)+ ``` paulson@14334 ` 613` ```apply (auto intro: real_add_order order_less_imp_le) ``` paulson@14334 ` 614` ```done ``` paulson@14334 ` 615` paulson@14365 ` 616` ```lemma real_inverse_unique: "x*y = (1::real) ==> y = inverse x" ``` paulson@14365 ` 617` ```apply (case_tac "x \ 0") ``` paulson@14365 ` 618` ```apply (rule_tac c1 = x in real_mult_left_cancel [THEN iffD1], auto) ``` paulson@14365 ` 619` ```done ``` paulson@14334 ` 620` paulson@14365 ` 621` ```lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x" ``` paulson@14365 ` 622` ```by (auto dest: less_imp_inverse_less) ``` paulson@14334 ` 623` paulson@14365 ` 624` ```lemma real_mult_self_sum_ge_zero: "(0::real) \ x*x + y*y" ``` paulson@14365 ` 625` ```proof - ``` paulson@14365 ` 626` ``` have "0 + 0 \ x*x + y*y" by (blast intro: add_mono zero_le_square) ``` paulson@14365 ` 627` ``` thus ?thesis by simp ``` paulson@14365 ` 628` ```qed ``` paulson@14365 ` 629` paulson@14334 ` 630` paulson@14365 ` 631` ```subsection{*Embedding the Integers into the Reals*} ``` paulson@14365 ` 632` paulson@14378 ` 633` ```defs (overloaded) ``` paulson@14378 ` 634` ``` real_of_nat_def: "real z == of_nat z" ``` paulson@14378 ` 635` ``` real_of_int_def: "real z == of_int z" ``` paulson@14365 ` 636` avigad@16819 ` 637` ```lemma real_eq_of_nat: "real = of_nat" ``` avigad@16819 ` 638` ``` apply (rule ext) ``` avigad@16819 ` 639` ``` apply (unfold real_of_nat_def) ``` avigad@16819 ` 640` ``` apply (rule refl) ``` avigad@16819 ` 641` ``` done ``` avigad@16819 ` 642` avigad@16819 ` 643` ```lemma real_eq_of_int: "real = of_int" ``` avigad@16819 ` 644` ``` apply (rule ext) ``` avigad@16819 ` 645` ``` apply (unfold real_of_int_def) ``` avigad@16819 ` 646` ``` apply (rule refl) ``` avigad@16819 ` 647` ``` done ``` avigad@16819 ` 648` paulson@14365 ` 649` ```lemma real_of_int_zero [simp]: "real (0::int) = 0" ``` paulson@14378 ` 650` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 651` paulson@14365 ` 652` ```lemma real_of_one [simp]: "real (1::int) = (1::real)" ``` paulson@14378 ` 653` ```by (simp add: real_of_int_def) ``` paulson@14334 ` 654` avigad@16819 ` 655` ```lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y" ``` paulson@14378 ` 656` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 657` avigad@16819 ` 658` ```lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)" ``` paulson@14378 ` 659` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 660` avigad@16819 ` 661` ```lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y" ``` avigad@16819 ` 662` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 663` avigad@16819 ` 664` ```lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y" ``` paulson@14378 ` 665` ```by (simp add: real_of_int_def) ``` paulson@14334 ` 666` avigad@16819 ` 667` ```lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" ``` avigad@16819 ` 668` ``` apply (subst real_eq_of_int)+ ``` avigad@16819 ` 669` ``` apply (rule of_int_setsum) ``` avigad@16819 ` 670` ```done ``` avigad@16819 ` 671` avigad@16819 ` 672` ```lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = ``` avigad@16819 ` 673` ``` (PROD x:A. real(f x))" ``` avigad@16819 ` 674` ``` apply (subst real_eq_of_int)+ ``` avigad@16819 ` 675` ``` apply (rule of_int_setprod) ``` avigad@16819 ` 676` ```done ``` paulson@14365 ` 677` paulson@14365 ` 678` ```lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))" ``` paulson@14378 ` 679` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 680` paulson@14365 ` 681` ```lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)" ``` paulson@14378 ` 682` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 683` paulson@14365 ` 684` ```lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)" ``` paulson@14378 ` 685` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 686` paulson@14365 ` 687` ```lemma real_of_int_le_iff [simp]: "(real (x::int) \ real y) = (x \ y)" ``` paulson@14378 ` 688` ```by (simp add: real_of_int_def) ``` paulson@14365 ` 689` avigad@16819 ` 690` ```lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)" ``` avigad@16819 ` 691` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 692` avigad@16819 ` 693` ```lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)" ``` avigad@16819 ` 694` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 695` avigad@16819 ` 696` ```lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)" ``` avigad@16819 ` 697` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 698` avigad@16819 ` 699` ```lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)" ``` avigad@16819 ` 700` ```by (simp add: real_of_int_def) ``` avigad@16819 ` 701` avigad@16888 ` 702` ```lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" ``` avigad@16888 ` 703` ```by (auto simp add: abs_if) ``` avigad@16888 ` 704` avigad@16819 ` 705` ```lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" ``` avigad@16819 ` 706` ``` apply (subgoal_tac "real n + 1 = real (n + 1)") ``` avigad@16819 ` 707` ``` apply (simp del: real_of_int_add) ``` avigad@16819 ` 708` ``` apply auto ``` avigad@16819 ` 709` ```done ``` avigad@16819 ` 710` avigad@16819 ` 711` ```lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" ``` avigad@16819 ` 712` ``` apply (subgoal_tac "real m + 1 = real (m + 1)") ``` avigad@16819 ` 713` ``` apply (simp del: real_of_int_add) ``` avigad@16819 ` 714` ``` apply simp ``` avigad@16819 ` 715` ```done ``` avigad@16819 ` 716` avigad@16819 ` 717` ```lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = ``` avigad@16819 ` 718` ``` real (x div d) + (real (x mod d)) / (real d)" ``` avigad@16819 ` 719` ```proof - ``` avigad@16819 ` 720` ``` assume "d ~= 0" ``` avigad@16819 ` 721` ``` have "x = (x div d) * d + x mod d" ``` avigad@16819 ` 722` ``` by auto ``` avigad@16819 ` 723` ``` then have "real x = real (x div d) * real d + real(x mod d)" ``` avigad@16819 ` 724` ``` by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) ``` avigad@16819 ` 725` ``` then have "real x / real d = ... / real d" ``` avigad@16819 ` 726` ``` by simp ``` avigad@16819 ` 727` ``` then show ?thesis ``` avigad@16819 ` 728` ``` by (auto simp add: add_divide_distrib ring_eq_simps prems) ``` avigad@16819 ` 729` ```qed ``` avigad@16819 ` 730` avigad@16819 ` 731` ```lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> ``` avigad@16819 ` 732` ``` real(n div d) = real n / real d" ``` avigad@16819 ` 733` ``` apply (frule real_of_int_div_aux [of d n]) ``` avigad@16819 ` 734` ``` apply simp ``` avigad@16819 ` 735` ``` apply (simp add: zdvd_iff_zmod_eq_0) ``` avigad@16819 ` 736` ```done ``` avigad@16819 ` 737` avigad@16819 ` 738` ```lemma real_of_int_div2: ``` avigad@16819 ` 739` ``` "0 <= real (n::int) / real (x) - real (n div x)" ``` avigad@16819 ` 740` ``` apply (case_tac "x = 0") ``` avigad@16819 ` 741` ``` apply simp ``` avigad@16819 ` 742` ``` apply (case_tac "0 < x") ``` avigad@16819 ` 743` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 744` ``` apply (subst real_of_int_div_aux) ``` avigad@16819 ` 745` ``` apply simp ``` avigad@16819 ` 746` ``` apply simp ``` avigad@16819 ` 747` ``` apply (subst zero_le_divide_iff) ``` avigad@16819 ` 748` ``` apply auto ``` avigad@16819 ` 749` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 750` ``` apply (subst real_of_int_div_aux) ``` avigad@16819 ` 751` ``` apply simp ``` avigad@16819 ` 752` ``` apply simp ``` avigad@16819 ` 753` ``` apply (subst zero_le_divide_iff) ``` avigad@16819 ` 754` ``` apply auto ``` avigad@16819 ` 755` ```done ``` avigad@16819 ` 756` avigad@16819 ` 757` ```lemma real_of_int_div3: ``` avigad@16819 ` 758` ``` "real (n::int) / real (x) - real (n div x) <= 1" ``` avigad@16819 ` 759` ``` apply(case_tac "x = 0") ``` avigad@16819 ` 760` ``` apply simp ``` avigad@16819 ` 761` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 762` ``` apply (subst real_of_int_div_aux) ``` avigad@16819 ` 763` ``` apply assumption ``` avigad@16819 ` 764` ``` apply simp ``` avigad@16819 ` 765` ``` apply (subst divide_le_eq) ``` avigad@16819 ` 766` ``` apply clarsimp ``` avigad@16819 ` 767` ``` apply (rule conjI) ``` avigad@16819 ` 768` ``` apply (rule impI) ``` avigad@16819 ` 769` ``` apply (rule order_less_imp_le) ``` avigad@16819 ` 770` ``` apply simp ``` avigad@16819 ` 771` ``` apply (rule impI) ``` avigad@16819 ` 772` ``` apply (rule order_less_imp_le) ``` avigad@16819 ` 773` ``` apply simp ``` avigad@16819 ` 774` ```done ``` avigad@16819 ` 775` avigad@16819 ` 776` ```lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" ``` avigad@16819 ` 777` ``` by (insert real_of_int_div2 [of n x], simp) ``` paulson@14365 ` 778` paulson@14365 ` 779` ```subsection{*Embedding the Naturals into the Reals*} ``` paulson@14365 ` 780` paulson@14334 ` 781` ```lemma real_of_nat_zero [simp]: "real (0::nat) = 0" ``` paulson@14365 ` 782` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 783` paulson@14334 ` 784` ```lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" ``` paulson@14365 ` 785` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 786` paulson@14365 ` 787` ```lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n" ``` paulson@14378 ` 788` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 789` paulson@14334 ` 790` ```(*Not for addsimps: often the LHS is used to represent a positive natural*) ``` paulson@14334 ` 791` ```lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" ``` paulson@14378 ` 792` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 793` paulson@14334 ` 794` ```lemma real_of_nat_less_iff [iff]: ``` paulson@14334 ` 795` ``` "(real (n::nat) < real m) = (n < m)" ``` paulson@14365 ` 796` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 797` paulson@14334 ` 798` ```lemma real_of_nat_le_iff [iff]: "(real (n::nat) \ real m) = (n \ m)" ``` paulson@14378 ` 799` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 800` paulson@14334 ` 801` ```lemma real_of_nat_ge_zero [iff]: "0 \ real (n::nat)" ``` paulson@14378 ` 802` ```by (simp add: real_of_nat_def zero_le_imp_of_nat) ``` paulson@14334 ` 803` paulson@14365 ` 804` ```lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)" ``` paulson@14378 ` 805` ```by (simp add: real_of_nat_def del: of_nat_Suc) ``` paulson@14365 ` 806` paulson@14334 ` 807` ```lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" ``` paulson@14378 ` 808` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 809` avigad@16819 ` 810` ```lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = ``` avigad@16819 ` 811` ``` (SUM x:A. real(f x))" ``` avigad@16819 ` 812` ``` apply (subst real_eq_of_nat)+ ``` avigad@16819 ` 813` ``` apply (rule of_nat_setsum) ``` avigad@16819 ` 814` ```done ``` avigad@16819 ` 815` avigad@16819 ` 816` ```lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = ``` avigad@16819 ` 817` ``` (PROD x:A. real(f x))" ``` avigad@16819 ` 818` ``` apply (subst real_eq_of_nat)+ ``` avigad@16819 ` 819` ``` apply (rule of_nat_setprod) ``` avigad@16819 ` 820` ```done ``` avigad@16819 ` 821` avigad@16819 ` 822` ```lemma real_of_card: "real (card A) = setsum (%x.1) A" ``` avigad@16819 ` 823` ``` apply (subst card_eq_setsum) ``` avigad@16819 ` 824` ``` apply (subst real_of_nat_setsum) ``` avigad@16819 ` 825` ``` apply simp ``` avigad@16819 ` 826` ```done ``` avigad@16819 ` 827` paulson@14334 ` 828` ```lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)" ``` paulson@14378 ` 829` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 830` paulson@14387 ` 831` ```lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)" ``` paulson@14378 ` 832` ```by (simp add: real_of_nat_def) ``` paulson@14334 ` 833` paulson@14365 ` 834` ```lemma real_of_nat_diff: "n \ m ==> real (m - n) = real (m::nat) - real n" ``` paulson@14378 ` 835` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 836` paulson@14365 ` 837` ```lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)" ``` paulson@14378 ` 838` ```by (simp add: add: real_of_nat_def) ``` paulson@14365 ` 839` paulson@14365 ` 840` ```lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \ 0) = (n = 0)" ``` paulson@14378 ` 841` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 842` paulson@14365 ` 843` ```lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0" ``` paulson@14378 ` 844` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 845` paulson@14365 ` 846` ```lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \ real (n::nat)) = (0 \ n)" ``` paulson@14378 ` 847` ```by (simp add: add: real_of_nat_def) ``` paulson@14334 ` 848` avigad@16819 ` 849` ```lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)" ``` avigad@16819 ` 850` ``` apply (subgoal_tac "real n + 1 = real (Suc n)") ``` avigad@16819 ` 851` ``` apply simp ``` avigad@16819 ` 852` ``` apply (auto simp add: real_of_nat_Suc) ``` avigad@16819 ` 853` ```done ``` avigad@16819 ` 854` avigad@16819 ` 855` ```lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)" ``` avigad@16819 ` 856` ``` apply (subgoal_tac "real m + 1 = real (Suc m)") ``` avigad@16819 ` 857` ``` apply (simp add: less_Suc_eq_le) ``` avigad@16819 ` 858` ``` apply (simp add: real_of_nat_Suc) ``` avigad@16819 ` 859` ```done ``` avigad@16819 ` 860` avigad@16819 ` 861` ```lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = ``` avigad@16819 ` 862` ``` real (x div d) + (real (x mod d)) / (real d)" ``` avigad@16819 ` 863` ```proof - ``` avigad@16819 ` 864` ``` assume "0 < d" ``` avigad@16819 ` 865` ``` have "x = (x div d) * d + x mod d" ``` avigad@16819 ` 866` ``` by auto ``` avigad@16819 ` 867` ``` then have "real x = real (x div d) * real d + real(x mod d)" ``` avigad@16819 ` 868` ``` by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym]) ``` avigad@16819 ` 869` ``` then have "real x / real d = \ / real d" ``` avigad@16819 ` 870` ``` by simp ``` avigad@16819 ` 871` ``` then show ?thesis ``` avigad@16819 ` 872` ``` by (auto simp add: add_divide_distrib ring_eq_simps prems) ``` avigad@16819 ` 873` ```qed ``` avigad@16819 ` 874` avigad@16819 ` 875` ```lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==> ``` avigad@16819 ` 876` ``` real(n div d) = real n / real d" ``` avigad@16819 ` 877` ``` apply (frule real_of_nat_div_aux [of d n]) ``` avigad@16819 ` 878` ``` apply simp ``` avigad@16819 ` 879` ``` apply (subst dvd_eq_mod_eq_0 [THEN sym]) ``` avigad@16819 ` 880` ``` apply assumption ``` avigad@16819 ` 881` ```done ``` avigad@16819 ` 882` avigad@16819 ` 883` ```lemma real_of_nat_div2: ``` avigad@16819 ` 884` ``` "0 <= real (n::nat) / real (x) - real (n div x)" ``` avigad@16819 ` 885` ``` apply(case_tac "x = 0") ``` avigad@16819 ` 886` ``` apply simp ``` avigad@16819 ` 887` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 888` ``` apply (subst real_of_nat_div_aux) ``` avigad@16819 ` 889` ``` apply assumption ``` avigad@16819 ` 890` ``` apply simp ``` avigad@16819 ` 891` ``` apply (subst zero_le_divide_iff) ``` avigad@16819 ` 892` ``` apply simp ``` avigad@16819 ` 893` ```done ``` avigad@16819 ` 894` avigad@16819 ` 895` ```lemma real_of_nat_div3: ``` avigad@16819 ` 896` ``` "real (n::nat) / real (x) - real (n div x) <= 1" ``` avigad@16819 ` 897` ``` apply(case_tac "x = 0") ``` avigad@16819 ` 898` ``` apply simp ``` avigad@16819 ` 899` ``` apply (simp add: compare_rls) ``` avigad@16819 ` 900` ``` apply (subst real_of_nat_div_aux) ``` avigad@16819 ` 901` ``` apply assumption ``` avigad@16819 ` 902` ``` apply simp ``` avigad@16819 ` 903` ```done ``` avigad@16819 ` 904` avigad@16819 ` 905` ```lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" ``` avigad@16819 ` 906` ``` by (insert real_of_nat_div2 [of n x], simp) ``` avigad@16819 ` 907` paulson@14365 ` 908` ```lemma real_of_int_real_of_nat: "real (int n) = real n" ``` paulson@14378 ` 909` ```by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat) ``` paulson@14378 ` 910` paulson@14426 ` 911` ```lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n" ``` paulson@14426 ` 912` ```by (simp add: real_of_int_def real_of_nat_def) ``` paulson@14334 ` 913` avigad@16819 ` 914` ```lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x" ``` avigad@16819 ` 915` ``` apply (subgoal_tac "real(int(nat x)) = real(nat x)") ``` avigad@16819 ` 916` ``` apply force ``` avigad@16819 ` 917` ``` apply (simp only: real_of_int_real_of_nat) ``` avigad@16819 ` 918` ```done ``` paulson@14387 ` 919` paulson@14387 ` 920` ```subsection{*Numerals and Arithmetic*} ``` paulson@14387 ` 921` paulson@14387 ` 922` ```instance real :: number .. ``` paulson@14387 ` 923` paulson@15013 ` 924` ```defs (overloaded) ``` paulson@15013 ` 925` ``` real_number_of_def: "(number_of w :: real) == of_int (Rep_Bin w)" ``` paulson@15013 ` 926` ``` --{*the type constraint is essential!*} ``` paulson@14387 ` 927` paulson@14387 ` 928` ```instance real :: number_ring ``` paulson@15013 ` 929` ```by (intro_classes, simp add: real_number_of_def) ``` paulson@14387 ` 930` paulson@14387 ` 931` ```text{*Collapse applications of @{term real} to @{term number_of}*} ``` paulson@14387 ` 932` ```lemma real_number_of [simp]: "real (number_of v :: int) = number_of v" ``` paulson@14387 ` 933` ```by (simp add: real_of_int_def of_int_number_of_eq) ``` paulson@14387 ` 934` paulson@14387 ` 935` ```lemma real_of_nat_number_of [simp]: ``` paulson@14387 ` 936` ``` "real (number_of v :: nat) = ``` paulson@14387 ` 937` ``` (if neg (number_of v :: int) then 0 ``` paulson@14387 ` 938` ``` else (number_of v :: real))" ``` paulson@14387 ` 939` ```by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of) ``` paulson@14387 ` 940` ``` ``` paulson@14387 ` 941` paulson@14387 ` 942` ```use "real_arith.ML" ``` paulson@14387 ` 943` paulson@14387 ` 944` ```setup real_arith_setup ``` paulson@14387 ` 945` kleing@19023 ` 946` kleing@19023 ` 947` ```lemma real_diff_mult_distrib: ``` kleing@19023 ` 948` ``` fixes a::real ``` kleing@19023 ` 949` ``` shows "a * (b - c) = a * b - a * c" ``` kleing@19023 ` 950` ```proof - ``` kleing@19023 ` 951` ``` have "a * (b - c) = a * (b + -c)" by simp ``` kleing@19023 ` 952` ``` also have "\ = (b + -c) * a" by simp ``` kleing@19023 ` 953` ``` also have "\ = b*a + (-c)*a" by (rule real_add_mult_distrib) ``` kleing@19023 ` 954` ``` also have "\ = a*b - a*c" by simp ``` kleing@19023 ` 955` ``` finally show ?thesis . ``` kleing@19023 ` 956` ```qed ``` kleing@19023 ` 957` kleing@19023 ` 958` paulson@14387 ` 959` ```subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*} ``` paulson@14387 ` 960` paulson@14387 ` 961` ```text{*Needed in this non-standard form by Hyperreal/Transcendental*} ``` paulson@14387 ` 962` ```lemma real_0_le_divide_iff: ``` paulson@14387 ` 963` ``` "((0::real) \ x/y) = ((x \ 0 | 0 \ y) & (0 \ x | y \ 0))" ``` paulson@14387 ` 964` ```by (simp add: real_divide_def zero_le_mult_iff, auto) ``` paulson@14387 ` 965` paulson@14387 ` 966` ```lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" ``` paulson@14387 ` 967` ```by arith ``` paulson@14387 ` 968` paulson@15085 ` 969` ```lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)" ``` paulson@14387 ` 970` ```by auto ``` paulson@14387 ` 971` paulson@15085 ` 972` ```lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)" ``` paulson@14387 ` 973` ```by auto ``` paulson@14387 ` 974` paulson@15085 ` 975` ```lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)" ``` paulson@14387 ` 976` ```by auto ``` paulson@14387 ` 977` paulson@15085 ` 978` ```lemma real_add_le_0_iff: "(x+y \ (0::real)) = (y \ -x)" ``` paulson@14387 ` 979` ```by auto ``` paulson@14387 ` 980` paulson@15085 ` 981` ```lemma real_0_le_add_iff: "((0::real) \ x+y) = (-x \ y)" ``` paulson@14387 ` 982` ```by auto ``` paulson@14387 ` 983` paulson@14387 ` 984` paulson@14387 ` 985` ```(* ``` paulson@14387 ` 986` ```FIXME: we should have this, as for type int, but many proofs would break. ``` paulson@14387 ` 987` ```It replaces x+-y by x-y. ``` paulson@15086 ` 988` ```declare real_diff_def [symmetric, simp] ``` paulson@14387 ` 989` ```*) ``` paulson@14387 ` 990` paulson@14387 ` 991` paulson@14387 ` 992` ```subsubsection{*Density of the Reals*} ``` paulson@14387 ` 993` paulson@14387 ` 994` ```lemma real_lbound_gt_zero: ``` paulson@14387 ` 995` ``` "[| (0::real) < d1; 0 < d2 |] ==> \e. 0 < e & e < d1 & e < d2" ``` paulson@14387 ` 996` ```apply (rule_tac x = " (min d1 d2) /2" in exI) ``` paulson@14387 ` 997` ```apply (simp add: min_def) ``` paulson@14387 ` 998` ```done ``` paulson@14387 ` 999` paulson@14387 ` 1000` paulson@14387 ` 1001` ```text{*Similar results are proved in @{text Ring_and_Field}*} ``` paulson@14387 ` 1002` ```lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)" ``` paulson@14387 ` 1003` ``` by auto ``` paulson@14387 ` 1004` paulson@14387 ` 1005` ```lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y" ``` paulson@14387 ` 1006` ``` by auto ``` paulson@14387 ` 1007` paulson@14387 ` 1008` paulson@14387 ` 1009` ```subsection{*Absolute Value Function for the Reals*} ``` paulson@14387 ` 1010` paulson@14387 ` 1011` ```lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))" ``` paulson@15003 ` 1012` ```by (simp add: abs_if) ``` paulson@14387 ` 1013` paulson@14387 ` 1014` ```lemma abs_interval_iff: "(abs x < r) = (-r < x & x < (r::real))" ``` paulson@14387 ` 1015` ```by (force simp add: Ring_and_Field.abs_less_iff) ``` paulson@14387 ` 1016` paulson@14387 ` 1017` ```lemma abs_le_interval_iff: "(abs x \ r) = (-r\x & x\(r::real))" ``` obua@14738 ` 1018` ```by (force simp add: OrderedGroup.abs_le_iff) ``` paulson@14387 ` 1019` paulson@14387 ` 1020` ```lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)" ``` paulson@15003 ` 1021` ```by (simp add: abs_if) ``` paulson@14387 ` 1022` paulson@14387 ` 1023` ```lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)" ``` paulson@15229 ` 1024` ```by (simp add: real_of_nat_ge_zero) ``` paulson@14387 ` 1025` paulson@14387 ` 1026` ```lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x" ``` webertj@20217 ` 1027` ```by simp ``` paulson@14387 ` 1028` ``` ``` paulson@14387 ` 1029` ```lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \ abs(x + -l) + abs(y + -m)" ``` webertj@20217 ` 1030` ```by simp ``` paulson@14387 ` 1031` paulson@5588 ` 1032` ```end ```