src/HOL/Real/PReal.thy
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```     1 (*  Title       : PReal.thy
```
```     2     ID          : \$Id\$
```
```     3     Author      : Jacques D. Fleuriot
```
```     4     Copyright   : 1998  University of Cambridge
```
```     5     Description : The positive reals as Dedekind sections of positive
```
```     6          rationals. Fundamentals of Abstract Analysis [Gleason- p. 121]
```
```     7                   provides some of the definitions.
```
```     8 *)
```
```     9
```
```    10 header {* Positive real numbers *}
```
```    11
```
```    12 theory PReal
```
```    13 imports Rational
```
```    14 begin
```
```    15
```
```    16 text{*Could be generalized and moved to @{text Ring_and_Field}*}
```
```    17 lemma add_eq_exists: "\<exists>x. a+x = (b::rat)"
```
```    18 by (rule_tac x="b-a" in exI, simp)
```
```    19
```
```    20 definition
```
```    21   cut :: "rat set => bool" where
```
```    22   "cut A = ({} \<subset> A &
```
```    23             A < {r. 0 < r} &
```
```    24             (\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u))))"
```
```    25
```
```    26 lemma cut_of_rat:
```
```    27   assumes q: "0 < q" shows "cut {r::rat. 0 < r & r < q}" (is "cut ?A")
```
```    28 proof -
```
```    29   from q have pos: "?A < {r. 0 < r}" by force
```
```    30   have nonempty: "{} \<subset> ?A"
```
```    31   proof
```
```    32     show "{} \<subseteq> ?A" by simp
```
```    33     show "{} \<noteq> ?A"
```
```    34       by (force simp only: q eq_commute [of "{}"] interval_empty_iff)
```
```    35   qed
```
```    36   show ?thesis
```
```    37     by (simp add: cut_def pos nonempty,
```
```    38         blast dest: dense intro: order_less_trans)
```
```    39 qed
```
```    40
```
```    41
```
```    42 typedef preal = "{A. cut A}"
```
```    43   by (blast intro: cut_of_rat [OF zero_less_one])
```
```    44
```
```    45 instance preal :: "{ord, plus, minus, times, inverse, one}" ..
```
```    46
```
```    47 definition
```
```    48   preal_of_rat :: "rat => preal" where
```
```    49   "preal_of_rat q = Abs_preal {x::rat. 0 < x & x < q}"
```
```    50
```
```    51 definition
```
```    52   psup :: "preal set => preal" where
```
```    53   "psup P = Abs_preal (\<Union>X \<in> P. Rep_preal X)"
```
```    54
```
```    55 definition
```
```    56   add_set :: "[rat set,rat set] => rat set" where
```
```    57   "add_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"
```
```    58
```
```    59 definition
```
```    60   diff_set :: "[rat set,rat set] => rat set" where
```
```    61   "diff_set A B = {w. \<exists>x. 0 < w & 0 < x & x \<notin> B & x + w \<in> A}"
```
```    62
```
```    63 definition
```
```    64   mult_set :: "[rat set,rat set] => rat set" where
```
```    65   "mult_set A B = {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}"
```
```    66
```
```    67 definition
```
```    68   inverse_set :: "rat set => rat set" where
```
```    69   "inverse_set A = {x. \<exists>y. 0 < x & x < y & inverse y \<notin> A}"
```
```    70
```
```    71
```
```    72 defs (overloaded)
```
```    73
```
```    74   preal_less_def:
```
```    75     "R < S == Rep_preal R < Rep_preal S"
```
```    76
```
```    77   preal_le_def:
```
```    78     "R \<le> S == Rep_preal R \<subseteq> Rep_preal S"
```
```    79
```
```    80   preal_add_def:
```
```    81     "R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))"
```
```    82
```
```    83   preal_diff_def:
```
```    84     "R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))"
```
```    85
```
```    86   preal_mult_def:
```
```    87     "R * S == Abs_preal (mult_set (Rep_preal R) (Rep_preal S))"
```
```    88
```
```    89   preal_inverse_def:
```
```    90     "inverse R == Abs_preal (inverse_set (Rep_preal R))"
```
```    91
```
```    92   preal_one_def:
```
```    93     "1 == preal_of_rat 1"
```
```    94
```
```    95
```
```    96 text{*Reduces equality on abstractions to equality on representatives*}
```
```    97 declare Abs_preal_inject [simp]
```
```    98 declare Abs_preal_inverse [simp]
```
```    99
```
```   100 lemma rat_mem_preal: "0 < q ==> {r::rat. 0 < r & r < q} \<in> preal"
```
```   101 by (simp add: preal_def cut_of_rat)
```
```   102
```
```   103 lemma preal_nonempty: "A \<in> preal ==> \<exists>x\<in>A. 0 < x"
```
```   104 by (unfold preal_def cut_def, blast)
```
```   105
```
```   106 lemma preal_Ex_mem: "A \<in> preal \<Longrightarrow> \<exists>x. x \<in> A"
```
```   107 by (drule preal_nonempty, fast)
```
```   108
```
```   109 lemma preal_imp_psubset_positives: "A \<in> preal ==> A < {r. 0 < r}"
```
```   110 by (force simp add: preal_def cut_def)
```
```   111
```
```   112 lemma preal_exists_bound: "A \<in> preal ==> \<exists>x. 0 < x & x \<notin> A"
```
```   113 by (drule preal_imp_psubset_positives, auto)
```
```   114
```
```   115 lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"
```
```   116 by (unfold preal_def cut_def, blast)
```
```   117
```
```   118 lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A"
```
```   119 by (unfold preal_def cut_def, blast)
```
```   120
```
```   121 text{*Relaxing the final premise*}
```
```   122 lemma preal_downwards_closed':
```
```   123      "[| A \<in> preal; y \<in> A; 0 < z; z \<le> y |] ==> z \<in> A"
```
```   124 apply (simp add: order_le_less)
```
```   125 apply (blast intro: preal_downwards_closed)
```
```   126 done
```
```   127
```
```   128 text{*A positive fraction not in a positive real is an upper bound.
```
```   129  Gleason p. 122 - Remark (1)*}
```
```   130
```
```   131 lemma not_in_preal_ub:
```
```   132   assumes A: "A \<in> preal"
```
```   133     and notx: "x \<notin> A"
```
```   134     and y: "y \<in> A"
```
```   135     and pos: "0 < x"
```
```   136   shows "y < x"
```
```   137 proof (cases rule: linorder_cases)
```
```   138   assume "x<y"
```
```   139   with notx show ?thesis
```
```   140     by (simp add:  preal_downwards_closed [OF A y] pos)
```
```   141 next
```
```   142   assume "x=y"
```
```   143   with notx and y show ?thesis by simp
```
```   144 next
```
```   145   assume "y<x"
```
```   146   thus ?thesis .
```
```   147 qed
```
```   148
```
```   149 text {* preal lemmas instantiated to @{term "Rep_preal X"} *}
```
```   150
```
```   151 lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"
```
```   152 by (rule preal_Ex_mem [OF Rep_preal])
```
```   153
```
```   154 lemma Rep_preal_exists_bound: "\<exists>x>0. x \<notin> Rep_preal X"
```
```   155 by (rule preal_exists_bound [OF Rep_preal])
```
```   156
```
```   157 lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal]
```
```   158
```
```   159
```
```   160
```
```   161 subsection{*@{term preal_of_prat}: the Injection from prat to preal*}
```
```   162
```
```   163 lemma rat_less_set_mem_preal: "0 < y ==> {u::rat. 0 < u & u < y} \<in> preal"
```
```   164 by (simp add: preal_def cut_of_rat)
```
```   165
```
```   166 lemma rat_subset_imp_le:
```
```   167      "[|{u::rat. 0 < u & u < x} \<subseteq> {u. 0 < u & u < y}; 0<x|] ==> x \<le> y"
```
```   168 apply (simp add: linorder_not_less [symmetric])
```
```   169 apply (blast dest: dense intro: order_less_trans)
```
```   170 done
```
```   171
```
```   172 lemma rat_set_eq_imp_eq:
```
```   173      "[|{u::rat. 0 < u & u < x} = {u. 0 < u & u < y};
```
```   174         0 < x; 0 < y|] ==> x = y"
```
```   175 by (blast intro: rat_subset_imp_le order_antisym)
```
```   176
```
```   177
```
```   178
```
```   179 subsection{*Properties of Ordering*}
```
```   180
```
```   181 lemma preal_le_refl: "w \<le> (w::preal)"
```
```   182 by (simp add: preal_le_def)
```
```   183
```
```   184 lemma preal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::preal)"
```
```   185 by (force simp add: preal_le_def)
```
```   186
```
```   187 lemma preal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::preal)"
```
```   188 apply (simp add: preal_le_def)
```
```   189 apply (rule Rep_preal_inject [THEN iffD1], blast)
```
```   190 done
```
```   191
```
```   192 (* Axiom 'order_less_le' of class 'order': *)
```
```   193 lemma preal_less_le: "((w::preal) < z) = (w \<le> z & w \<noteq> z)"
```
```   194 by (simp add: preal_le_def preal_less_def Rep_preal_inject psubset_def)
```
```   195
```
```   196 instance preal :: order
```
```   197   by intro_classes
```
```   198     (assumption |
```
```   199       rule preal_le_refl preal_le_trans preal_le_anti_sym preal_less_le)+
```
```   200
```
```   201 lemma preal_imp_pos: "[|A \<in> preal; r \<in> A|] ==> 0 < r"
```
```   202 by (insert preal_imp_psubset_positives, blast)
```
```   203
```
```   204 lemma preal_le_linear: "x <= y | y <= (x::preal)"
```
```   205 apply (auto simp add: preal_le_def)
```
```   206 apply (rule ccontr)
```
```   207 apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal]
```
```   208              elim: order_less_asym)
```
```   209 done
```
```   210
```
```   211 instance preal :: linorder
```
```   212   by intro_classes (rule preal_le_linear)
```
```   213
```
```   214 instance preal :: distrib_lattice
```
```   215   "inf \<equiv> min"
```
```   216   "sup \<equiv> max"
```
```   217   by intro_classes
```
```   218     (auto simp add: inf_preal_def sup_preal_def min_max.sup_inf_distrib1)
```
```   219
```
```   220
```
```   221 subsection{*Properties of Addition*}
```
```   222
```
```   223 lemma preal_add_commute: "(x::preal) + y = y + x"
```
```   224 apply (unfold preal_add_def add_set_def)
```
```   225 apply (rule_tac f = Abs_preal in arg_cong)
```
```   226 apply (force simp add: add_commute)
```
```   227 done
```
```   228
```
```   229 text{*Lemmas for proving that addition of two positive reals gives
```
```   230  a positive real*}
```
```   231
```
```   232 lemma empty_psubset_nonempty: "a \<in> A ==> {} \<subset> A"
```
```   233 by blast
```
```   234
```
```   235 text{*Part 1 of Dedekind sections definition*}
```
```   236 lemma add_set_not_empty:
```
```   237      "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> add_set A B"
```
```   238 apply (drule preal_nonempty)+
```
```   239 apply (auto simp add: add_set_def)
```
```   240 done
```
```   241
```
```   242 text{*Part 2 of Dedekind sections definition.  A structured version of
```
```   243 this proof is @{text preal_not_mem_mult_set_Ex} below.*}
```
```   244 lemma preal_not_mem_add_set_Ex:
```
```   245      "[|A \<in> preal; B \<in> preal|] ==> \<exists>q>0. q \<notin> add_set A B"
```
```   246 apply (insert preal_exists_bound [of A] preal_exists_bound [of B], auto)
```
```   247 apply (rule_tac x = "x+xa" in exI)
```
```   248 apply (simp add: add_set_def, clarify)
```
```   249 apply (drule (3) not_in_preal_ub)+
```
```   250 apply (force dest: add_strict_mono)
```
```   251 done
```
```   252
```
```   253 lemma add_set_not_rat_set:
```
```   254    assumes A: "A \<in> preal"
```
```   255        and B: "B \<in> preal"
```
```   256      shows "add_set A B < {r. 0 < r}"
```
```   257 proof
```
```   258   from preal_imp_pos [OF A] preal_imp_pos [OF B]
```
```   259   show "add_set A B \<subseteq> {r. 0 < r}" by (force simp add: add_set_def)
```
```   260 next
```
```   261   show "add_set A B \<noteq> {r. 0 < r}"
```
```   262     by (insert preal_not_mem_add_set_Ex [OF A B], blast)
```
```   263 qed
```
```   264
```
```   265 text{*Part 3 of Dedekind sections definition*}
```
```   266 lemma add_set_lemma3:
```
```   267      "[|A \<in> preal; B \<in> preal; u \<in> add_set A B; 0 < z; z < u|]
```
```   268       ==> z \<in> add_set A B"
```
```   269 proof (unfold add_set_def, clarify)
```
```   270   fix x::rat and y::rat
```
```   271   assume A: "A \<in> preal"
```
```   272     and B: "B \<in> preal"
```
```   273     and [simp]: "0 < z"
```
```   274     and zless: "z < x + y"
```
```   275     and x:  "x \<in> A"
```
```   276     and y:  "y \<in> B"
```
```   277   have xpos [simp]: "0<x" by (rule preal_imp_pos [OF A x])
```
```   278   have ypos [simp]: "0<y" by (rule preal_imp_pos [OF B y])
```
```   279   have xypos [simp]: "0 < x+y" by (simp add: pos_add_strict)
```
```   280   let ?f = "z/(x+y)"
```
```   281   have fless: "?f < 1" by (simp add: zless pos_divide_less_eq)
```
```   282   show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'"
```
```   283   proof (intro bexI)
```
```   284     show "z = x*?f + y*?f"
```
```   285       by (simp add: left_distrib [symmetric] divide_inverse mult_ac
```
```   286           order_less_imp_not_eq2)
```
```   287   next
```
```   288     show "y * ?f \<in> B"
```
```   289     proof (rule preal_downwards_closed [OF B y])
```
```   290       show "0 < y * ?f"
```
```   291         by (simp add: divide_inverse zero_less_mult_iff)
```
```   292     next
```
```   293       show "y * ?f < y"
```
```   294         by (insert mult_strict_left_mono [OF fless ypos], simp)
```
```   295     qed
```
```   296   next
```
```   297     show "x * ?f \<in> A"
```
```   298     proof (rule preal_downwards_closed [OF A x])
```
```   299       show "0 < x * ?f"
```
```   300 	by (simp add: divide_inverse zero_less_mult_iff)
```
```   301     next
```
```   302       show "x * ?f < x"
```
```   303 	by (insert mult_strict_left_mono [OF fless xpos], simp)
```
```   304     qed
```
```   305   qed
```
```   306 qed
```
```   307
```
```   308 text{*Part 4 of Dedekind sections definition*}
```
```   309 lemma add_set_lemma4:
```
```   310      "[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u"
```
```   311 apply (auto simp add: add_set_def)
```
```   312 apply (frule preal_exists_greater [of A], auto)
```
```   313 apply (rule_tac x="u + y" in exI)
```
```   314 apply (auto intro: add_strict_left_mono)
```
```   315 done
```
```   316
```
```   317 lemma mem_add_set:
```
```   318      "[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal"
```
```   319 apply (simp (no_asm_simp) add: preal_def cut_def)
```
```   320 apply (blast intro!: add_set_not_empty add_set_not_rat_set
```
```   321                      add_set_lemma3 add_set_lemma4)
```
```   322 done
```
```   323
```
```   324 lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
```
```   325 apply (simp add: preal_add_def mem_add_set Rep_preal)
```
```   326 apply (force simp add: add_set_def add_ac)
```
```   327 done
```
```   328
```
```   329 instance preal :: ab_semigroup_add
```
```   330 proof
```
```   331   fix a b c :: preal
```
```   332   show "(a + b) + c = a + (b + c)" by (rule preal_add_assoc)
```
```   333   show "a + b = b + a" by (rule preal_add_commute)
```
```   334 qed
```
```   335
```
```   336 lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)"
```
```   337 by (rule add_left_commute)
```
```   338
```
```   339 text{* Positive Real addition is an AC operator *}
```
```   340 lemmas preal_add_ac = preal_add_assoc preal_add_commute preal_add_left_commute
```
```   341
```
```   342
```
```   343 subsection{*Properties of Multiplication*}
```
```   344
```
```   345 text{*Proofs essentially same as for addition*}
```
```   346
```
```   347 lemma preal_mult_commute: "(x::preal) * y = y * x"
```
```   348 apply (unfold preal_mult_def mult_set_def)
```
```   349 apply (rule_tac f = Abs_preal in arg_cong)
```
```   350 apply (force simp add: mult_commute)
```
```   351 done
```
```   352
```
```   353 text{*Multiplication of two positive reals gives a positive real.*}
```
```   354
```
```   355 text{*Lemmas for proving positive reals multiplication set in @{typ preal}*}
```
```   356
```
```   357 text{*Part 1 of Dedekind sections definition*}
```
```   358 lemma mult_set_not_empty:
```
```   359      "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B"
```
```   360 apply (insert preal_nonempty [of A] preal_nonempty [of B])
```
```   361 apply (auto simp add: mult_set_def)
```
```   362 done
```
```   363
```
```   364 text{*Part 2 of Dedekind sections definition*}
```
```   365 lemma preal_not_mem_mult_set_Ex:
```
```   366    assumes A: "A \<in> preal"
```
```   367        and B: "B \<in> preal"
```
```   368      shows "\<exists>q. 0 < q & q \<notin> mult_set A B"
```
```   369 proof -
```
```   370   from preal_exists_bound [OF A]
```
```   371   obtain x where [simp]: "0 < x" "x \<notin> A" by blast
```
```   372   from preal_exists_bound [OF B]
```
```   373   obtain y where [simp]: "0 < y" "y \<notin> B" by blast
```
```   374   show ?thesis
```
```   375   proof (intro exI conjI)
```
```   376     show "0 < x*y" by (simp add: mult_pos_pos)
```
```   377     show "x * y \<notin> mult_set A B"
```
```   378     proof -
```
```   379       { fix u::rat and v::rat
```
```   380 	      assume "u \<in> A" and "v \<in> B" and "x*y = u*v"
```
```   381 	      moreover
```
```   382 	      with prems have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
```
```   383 	      moreover
```
```   384 	      with prems have "0\<le>v"
```
```   385 	        by (blast intro: preal_imp_pos [OF B]  order_less_imp_le prems)
```
```   386 	      moreover
```
```   387         from calculation
```
```   388 	      have "u*v < x*y" by (blast intro: mult_strict_mono prems)
```
```   389 	      ultimately have False by force }
```
```   390       thus ?thesis by (auto simp add: mult_set_def)
```
```   391     qed
```
```   392   qed
```
```   393 qed
```
```   394
```
```   395 lemma mult_set_not_rat_set:
```
```   396   assumes A: "A \<in> preal"
```
```   397     and B: "B \<in> preal"
```
```   398   shows "mult_set A B < {r. 0 < r}"
```
```   399 proof
```
```   400   show "mult_set A B \<subseteq> {r. 0 < r}"
```
```   401     by (force simp add: mult_set_def
```
```   402       intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos_pos)
```
```   403   show "mult_set A B \<noteq> {r. 0 < r}"
```
```   404     using preal_not_mem_mult_set_Ex [OF A B] by blast
```
```   405 qed
```
```   406
```
```   407
```
```   408
```
```   409 text{*Part 3 of Dedekind sections definition*}
```
```   410 lemma mult_set_lemma3:
```
```   411      "[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|]
```
```   412       ==> z \<in> mult_set A B"
```
```   413 proof (unfold mult_set_def, clarify)
```
```   414   fix x::rat and y::rat
```
```   415   assume A: "A \<in> preal"
```
```   416     and B: "B \<in> preal"
```
```   417     and [simp]: "0 < z"
```
```   418     and zless: "z < x * y"
```
```   419     and x:  "x \<in> A"
```
```   420     and y:  "y \<in> B"
```
```   421   have [simp]: "0<y" by (rule preal_imp_pos [OF B y])
```
```   422   show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'"
```
```   423   proof
```
```   424     show "\<exists>y'\<in>B. z = (z/y) * y'"
```
```   425     proof
```
```   426       show "z = (z/y)*y"
```
```   427 	by (simp add: divide_inverse mult_commute [of y] mult_assoc
```
```   428 		      order_less_imp_not_eq2)
```
```   429       show "y \<in> B" by fact
```
```   430     qed
```
```   431   next
```
```   432     show "z/y \<in> A"
```
```   433     proof (rule preal_downwards_closed [OF A x])
```
```   434       show "0 < z/y"
```
```   435 	by (simp add: zero_less_divide_iff)
```
```   436       show "z/y < x" by (simp add: pos_divide_less_eq zless)
```
```   437     qed
```
```   438   qed
```
```   439 qed
```
```   440
```
```   441 text{*Part 4 of Dedekind sections definition*}
```
```   442 lemma mult_set_lemma4:
```
```   443      "[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u"
```
```   444 apply (auto simp add: mult_set_def)
```
```   445 apply (frule preal_exists_greater [of A], auto)
```
```   446 apply (rule_tac x="u * y" in exI)
```
```   447 apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B]
```
```   448                    mult_strict_right_mono)
```
```   449 done
```
```   450
```
```   451
```
```   452 lemma mem_mult_set:
```
```   453      "[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal"
```
```   454 apply (simp (no_asm_simp) add: preal_def cut_def)
```
```   455 apply (blast intro!: mult_set_not_empty mult_set_not_rat_set
```
```   456                      mult_set_lemma3 mult_set_lemma4)
```
```   457 done
```
```   458
```
```   459 lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
```
```   460 apply (simp add: preal_mult_def mem_mult_set Rep_preal)
```
```   461 apply (force simp add: mult_set_def mult_ac)
```
```   462 done
```
```   463
```
```   464 instance preal :: ab_semigroup_mult
```
```   465 proof
```
```   466   fix a b c :: preal
```
```   467   show "(a * b) * c = a * (b * c)" by (rule preal_mult_assoc)
```
```   468   show "a * b = b * a" by (rule preal_mult_commute)
```
```   469 qed
```
```   470
```
```   471 lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)"
```
```   472 by (rule mult_left_commute)
```
```   473
```
```   474
```
```   475 text{* Positive Real multiplication is an AC operator *}
```
```   476 lemmas preal_mult_ac =
```
```   477        preal_mult_assoc preal_mult_commute preal_mult_left_commute
```
```   478
```
```   479
```
```   480 text{* Positive real 1 is the multiplicative identity element *}
```
```   481
```
```   482 lemma preal_mult_1: "(1::preal) * z = z"
```
```   483 unfolding preal_one_def
```
```   484 proof (induct z)
```
```   485   fix A :: "rat set"
```
```   486   assume A: "A \<in> preal"
```
```   487   have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A")
```
```   488   proof
```
```   489     show "?lhs \<subseteq> A"
```
```   490     proof clarify
```
```   491       fix x::rat and u::rat and v::rat
```
```   492       assume upos: "0<u" and "u<1" and v: "v \<in> A"
```
```   493       have vpos: "0<v" by (rule preal_imp_pos [OF A v])
```
```   494       hence "u*v < 1*v" by (simp only: mult_strict_right_mono prems)
```
```   495       thus "u * v \<in> A"
```
```   496         by (force intro: preal_downwards_closed [OF A v] mult_pos_pos
```
```   497           upos vpos)
```
```   498     qed
```
```   499   next
```
```   500     show "A \<subseteq> ?lhs"
```
```   501     proof clarify
```
```   502       fix x::rat
```
```   503       assume x: "x \<in> A"
```
```   504       have xpos: "0<x" by (rule preal_imp_pos [OF A x])
```
```   505       from preal_exists_greater [OF A x]
```
```   506       obtain v where v: "v \<in> A" and xlessv: "x < v" ..
```
```   507       have vpos: "0<v" by (rule preal_imp_pos [OF A v])
```
```   508       show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)"
```
```   509       proof (intro exI conjI)
```
```   510         show "0 < x/v"
```
```   511           by (simp add: zero_less_divide_iff xpos vpos)
```
```   512 	show "x / v < 1"
```
```   513           by (simp add: pos_divide_less_eq vpos xlessv)
```
```   514         show "\<exists>v'\<in>A. x = (x / v) * v'"
```
```   515         proof
```
```   516           show "x = (x/v)*v"
```
```   517 	    by (simp add: divide_inverse mult_assoc vpos
```
```   518                           order_less_imp_not_eq2)
```
```   519           show "v \<in> A" by fact
```
```   520         qed
```
```   521       qed
```
```   522     qed
```
```   523   qed
```
```   524   thus "preal_of_rat 1 * Abs_preal A = Abs_preal A"
```
```   525     by (simp add: preal_of_rat_def preal_mult_def mult_set_def
```
```   526                   rat_mem_preal A)
```
```   527 qed
```
```   528
```
```   529 instance preal :: comm_monoid_mult
```
```   530 by intro_classes (rule preal_mult_1)
```
```   531
```
```   532 lemma preal_mult_1_right: "z * (1::preal) = z"
```
```   533 by (rule mult_1_right)
```
```   534
```
```   535
```
```   536 subsection{*Distribution of Multiplication across Addition*}
```
```   537
```
```   538 lemma mem_Rep_preal_add_iff:
```
```   539       "(z \<in> Rep_preal(R+S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x + y)"
```
```   540 apply (simp add: preal_add_def mem_add_set Rep_preal)
```
```   541 apply (simp add: add_set_def)
```
```   542 done
```
```   543
```
```   544 lemma mem_Rep_preal_mult_iff:
```
```   545       "(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)"
```
```   546 apply (simp add: preal_mult_def mem_mult_set Rep_preal)
```
```   547 apply (simp add: mult_set_def)
```
```   548 done
```
```   549
```
```   550 lemma distrib_subset1:
```
```   551      "Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)"
```
```   552 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
```
```   553 apply (force simp add: right_distrib)
```
```   554 done
```
```   555
```
```   556 lemma preal_add_mult_distrib_mean:
```
```   557   assumes a: "a \<in> Rep_preal w"
```
```   558     and b: "b \<in> Rep_preal w"
```
```   559     and d: "d \<in> Rep_preal x"
```
```   560     and e: "e \<in> Rep_preal y"
```
```   561   shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)"
```
```   562 proof
```
```   563   let ?c = "(a*d + b*e)/(d+e)"
```
```   564   have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e"
```
```   565     by (blast intro: preal_imp_pos [OF Rep_preal] a b d e pos_add_strict)+
```
```   566   have cpos: "0 < ?c"
```
```   567     by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
```
```   568   show "a * d + b * e = ?c * (d + e)"
```
```   569     by (simp add: divide_inverse mult_assoc order_less_imp_not_eq2)
```
```   570   show "?c \<in> Rep_preal w"
```
```   571   proof (cases rule: linorder_le_cases)
```
```   572     assume "a \<le> b"
```
```   573     hence "?c \<le> b"
```
```   574       by (simp add: pos_divide_le_eq right_distrib mult_right_mono
```
```   575                     order_less_imp_le)
```
```   576     thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])
```
```   577   next
```
```   578     assume "b \<le> a"
```
```   579     hence "?c \<le> a"
```
```   580       by (simp add: pos_divide_le_eq right_distrib mult_right_mono
```
```   581                     order_less_imp_le)
```
```   582     thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos])
```
```   583   qed
```
```   584 qed
```
```   585
```
```   586 lemma distrib_subset2:
```
```   587      "Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))"
```
```   588 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
```
```   589 apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto)
```
```   590 done
```
```   591
```
```   592 lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"
```
```   593 apply (rule Rep_preal_inject [THEN iffD1])
```
```   594 apply (rule equalityI [OF distrib_subset1 distrib_subset2])
```
```   595 done
```
```   596
```
```   597 lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)"
```
```   598 by (simp add: preal_mult_commute preal_add_mult_distrib2)
```
```   599
```
```   600 instance preal :: comm_semiring
```
```   601 by intro_classes (rule preal_add_mult_distrib)
```
```   602
```
```   603
```
```   604 subsection{*Existence of Inverse, a Positive Real*}
```
```   605
```
```   606 lemma mem_inv_set_ex:
```
```   607   assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A"
```
```   608 proof -
```
```   609   from preal_exists_bound [OF A]
```
```   610   obtain x where [simp]: "0<x" "x \<notin> A" by blast
```
```   611   show ?thesis
```
```   612   proof (intro exI conjI)
```
```   613     show "0 < inverse (x+1)"
```
```   614       by (simp add: order_less_trans [OF _ less_add_one])
```
```   615     show "inverse(x+1) < inverse x"
```
```   616       by (simp add: less_imp_inverse_less less_add_one)
```
```   617     show "inverse (inverse x) \<notin> A"
```
```   618       by (simp add: order_less_imp_not_eq2)
```
```   619   qed
```
```   620 qed
```
```   621
```
```   622 text{*Part 1 of Dedekind sections definition*}
```
```   623 lemma inverse_set_not_empty:
```
```   624      "A \<in> preal ==> {} \<subset> inverse_set A"
```
```   625 apply (insert mem_inv_set_ex [of A])
```
```   626 apply (auto simp add: inverse_set_def)
```
```   627 done
```
```   628
```
```   629 text{*Part 2 of Dedekind sections definition*}
```
```   630
```
```   631 lemma preal_not_mem_inverse_set_Ex:
```
```   632    assumes A: "A \<in> preal"  shows "\<exists>q. 0 < q & q \<notin> inverse_set A"
```
```   633 proof -
```
```   634   from preal_nonempty [OF A]
```
```   635   obtain x where x: "x \<in> A" and  xpos [simp]: "0<x" ..
```
```   636   show ?thesis
```
```   637   proof (intro exI conjI)
```
```   638     show "0 < inverse x" by simp
```
```   639     show "inverse x \<notin> inverse_set A"
```
```   640     proof -
```
```   641       { fix y::rat
```
```   642 	assume ygt: "inverse x < y"
```
```   643 	have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt])
```
```   644 	have iyless: "inverse y < x"
```
```   645 	  by (simp add: inverse_less_imp_less [of x] ygt)
```
```   646 	have "inverse y \<in> A"
```
```   647 	  by (simp add: preal_downwards_closed [OF A x] iyless)}
```
```   648      thus ?thesis by (auto simp add: inverse_set_def)
```
```   649     qed
```
```   650   qed
```
```   651 qed
```
```   652
```
```   653 lemma inverse_set_not_rat_set:
```
```   654    assumes A: "A \<in> preal"  shows "inverse_set A < {r. 0 < r}"
```
```   655 proof
```
```   656   show "inverse_set A \<subseteq> {r. 0 < r}"  by (force simp add: inverse_set_def)
```
```   657 next
```
```   658   show "inverse_set A \<noteq> {r. 0 < r}"
```
```   659     by (insert preal_not_mem_inverse_set_Ex [OF A], blast)
```
```   660 qed
```
```   661
```
```   662 text{*Part 3 of Dedekind sections definition*}
```
```   663 lemma inverse_set_lemma3:
```
```   664      "[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|]
```
```   665       ==> z \<in> inverse_set A"
```
```   666 apply (auto simp add: inverse_set_def)
```
```   667 apply (auto intro: order_less_trans)
```
```   668 done
```
```   669
```
```   670 text{*Part 4 of Dedekind sections definition*}
```
```   671 lemma inverse_set_lemma4:
```
```   672      "[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u"
```
```   673 apply (auto simp add: inverse_set_def)
```
```   674 apply (drule dense [of y])
```
```   675 apply (blast intro: order_less_trans)
```
```   676 done
```
```   677
```
```   678
```
```   679 lemma mem_inverse_set:
```
```   680      "A \<in> preal ==> inverse_set A \<in> preal"
```
```   681 apply (simp (no_asm_simp) add: preal_def cut_def)
```
```   682 apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set
```
```   683                      inverse_set_lemma3 inverse_set_lemma4)
```
```   684 done
```
```   685
```
```   686
```
```   687 subsection{*Gleason's Lemma 9-3.4, page 122*}
```
```   688
```
```   689 lemma Gleason9_34_exists:
```
```   690   assumes A: "A \<in> preal"
```
```   691     and "\<forall>x\<in>A. x + u \<in> A"
```
```   692     and "0 \<le> z"
```
```   693   shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A"
```
```   694 proof (cases z rule: int_cases)
```
```   695   case (nonneg n)
```
```   696   show ?thesis
```
```   697   proof (simp add: prems, induct n)
```
```   698     case 0
```
```   699       from preal_nonempty [OF A]
```
```   700       show ?case  by force
```
```   701     case (Suc k)
```
```   702       from this obtain b where "b \<in> A" "b + of_nat k * u \<in> A" ..
```
```   703       hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems)
```
```   704       thus ?case by (force simp add: left_distrib add_ac prems)
```
```   705   qed
```
```   706 next
```
```   707   case (neg n)
```
```   708   with prems show ?thesis by simp
```
```   709 qed
```
```   710
```
```   711 lemma Gleason9_34_contra:
```
```   712   assumes A: "A \<in> preal"
```
```   713     shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False"
```
```   714 proof (induct u, induct y)
```
```   715   fix a::int and b::int
```
```   716   fix c::int and d::int
```
```   717   assume bpos [simp]: "0 < b"
```
```   718     and dpos [simp]: "0 < d"
```
```   719     and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A"
```
```   720     and upos: "0 < Fract c d"
```
```   721     and ypos: "0 < Fract a b"
```
```   722     and notin: "Fract a b \<notin> A"
```
```   723   have cpos [simp]: "0 < c"
```
```   724     by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos)
```
```   725   have apos [simp]: "0 < a"
```
```   726     by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos)
```
```   727   let ?k = "a*d"
```
```   728   have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)"
```
```   729   proof -
```
```   730     have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"
```
```   731       by (simp add: mult_rat le_rat order_less_imp_not_eq2 mult_ac)
```
```   732     moreover
```
```   733     have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"
```
```   734       by (rule mult_mono,
```
```   735           simp_all add: int_one_le_iff_zero_less zero_less_mult_iff
```
```   736                         order_less_imp_le)
```
```   737     ultimately
```
```   738     show ?thesis by simp
```
```   739   qed
```
```   740   have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff)
```
```   741   from Gleason9_34_exists [OF A closed k]
```
```   742   obtain z where z: "z \<in> A"
```
```   743              and mem: "z + of_int ?k * Fract c d \<in> A" ..
```
```   744   have less: "z + of_int ?k * Fract c d < Fract a b"
```
```   745     by (rule not_in_preal_ub [OF A notin mem ypos])
```
```   746   have "0<z" by (rule preal_imp_pos [OF A z])
```
```   747   with frle and less show False by (simp add: Fract_of_int_eq)
```
```   748 qed
```
```   749
```
```   750
```
```   751 lemma Gleason9_34:
```
```   752   assumes A: "A \<in> preal"
```
```   753     and upos: "0 < u"
```
```   754   shows "\<exists>r \<in> A. r + u \<notin> A"
```
```   755 proof (rule ccontr, simp)
```
```   756   assume closed: "\<forall>r\<in>A. r + u \<in> A"
```
```   757   from preal_exists_bound [OF A]
```
```   758   obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast
```
```   759   show False
```
```   760     by (rule Gleason9_34_contra [OF A closed upos ypos y])
```
```   761 qed
```
```   762
```
```   763
```
```   764
```
```   765 subsection{*Gleason's Lemma 9-3.6*}
```
```   766
```
```   767 lemma lemma_gleason9_36:
```
```   768   assumes A: "A \<in> preal"
```
```   769     and x: "1 < x"
```
```   770   shows "\<exists>r \<in> A. r*x \<notin> A"
```
```   771 proof -
```
```   772   from preal_nonempty [OF A]
```
```   773   obtain y where y: "y \<in> A" and  ypos: "0<y" ..
```
```   774   show ?thesis
```
```   775   proof (rule classical)
```
```   776     assume "~(\<exists>r\<in>A. r * x \<notin> A)"
```
```   777     with y have ymem: "y * x \<in> A" by blast
```
```   778     from ypos mult_strict_left_mono [OF x]
```
```   779     have yless: "y < y*x" by simp
```
```   780     let ?d = "y*x - y"
```
```   781     from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto
```
```   782     from Gleason9_34 [OF A dpos]
```
```   783     obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" ..
```
```   784     have rpos: "0<r" by (rule preal_imp_pos [OF A r])
```
```   785     with dpos have rdpos: "0 < r + ?d" by arith
```
```   786     have "~ (r + ?d \<le> y + ?d)"
```
```   787     proof
```
```   788       assume le: "r + ?d \<le> y + ?d"
```
```   789       from ymem have yd: "y + ?d \<in> A" by (simp add: eq)
```
```   790       have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le])
```
```   791       with notin show False by simp
```
```   792     qed
```
```   793     hence "y < r" by simp
```
```   794     with ypos have  dless: "?d < (r * ?d)/y"
```
```   795       by (simp add: pos_less_divide_eq mult_commute [of ?d]
```
```   796                     mult_strict_right_mono dpos)
```
```   797     have "r + ?d < r*x"
```
```   798     proof -
```
```   799       have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
```
```   800       also with ypos have "... = (r/y) * (y + ?d)"
```
```   801 	by (simp only: right_distrib divide_inverse mult_ac, simp)
```
```   802       also have "... = r*x" using ypos
```
```   803 	by (simp add: times_divide_eq_left)
```
```   804       finally show "r + ?d < r*x" .
```
```   805     qed
```
```   806     with r notin rdpos
```
```   807     show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest:  preal_downwards_closed [OF A])
```
```   808   qed
```
```   809 qed
```
```   810
```
```   811 subsection{*Existence of Inverse: Part 2*}
```
```   812
```
```   813 lemma mem_Rep_preal_inverse_iff:
```
```   814       "(z \<in> Rep_preal(inverse R)) =
```
```   815        (0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))"
```
```   816 apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
```
```   817 apply (simp add: inverse_set_def)
```
```   818 done
```
```   819
```
```   820 lemma Rep_preal_of_rat:
```
```   821      "0 < q ==> Rep_preal (preal_of_rat q) = {x. 0 < x \<and> x < q}"
```
```   822 by (simp add: preal_of_rat_def rat_mem_preal)
```
```   823
```
```   824 lemma subset_inverse_mult_lemma:
```
```   825   assumes xpos: "0 < x" and xless: "x < 1"
```
```   826   shows "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R &
```
```   827     u \<in> Rep_preal R & x = r * u"
```
```   828 proof -
```
```   829   from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)
```
```   830   from lemma_gleason9_36 [OF Rep_preal this]
```
```   831   obtain r where r: "r \<in> Rep_preal R"
```
```   832              and notin: "r * (inverse x) \<notin> Rep_preal R" ..
```
```   833   have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
```
```   834   from preal_exists_greater [OF Rep_preal r]
```
```   835   obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" ..
```
```   836   have upos: "0<u" by (rule preal_imp_pos [OF Rep_preal u])
```
```   837   show ?thesis
```
```   838   proof (intro exI conjI)
```
```   839     show "0 < x/u" using xpos upos
```
```   840       by (simp add: zero_less_divide_iff)
```
```   841     show "x/u < x/r" using xpos upos rpos
```
```   842       by (simp add: divide_inverse mult_less_cancel_left rless)
```
```   843     show "inverse (x / r) \<notin> Rep_preal R" using notin
```
```   844       by (simp add: divide_inverse mult_commute)
```
```   845     show "u \<in> Rep_preal R" by (rule u)
```
```   846     show "x = x / u * u" using upos
```
```   847       by (simp add: divide_inverse mult_commute)
```
```   848   qed
```
```   849 qed
```
```   850
```
```   851 lemma subset_inverse_mult:
```
```   852      "Rep_preal(preal_of_rat 1) \<subseteq> Rep_preal(inverse R * R)"
```
```   853 apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff
```
```   854                       mem_Rep_preal_mult_iff)
```
```   855 apply (blast dest: subset_inverse_mult_lemma)
```
```   856 done
```
```   857
```
```   858 lemma inverse_mult_subset_lemma:
```
```   859   assumes rpos: "0 < r"
```
```   860     and rless: "r < y"
```
```   861     and notin: "inverse y \<notin> Rep_preal R"
```
```   862     and q: "q \<in> Rep_preal R"
```
```   863   shows "r*q < 1"
```
```   864 proof -
```
```   865   have "q < inverse y" using rpos rless
```
```   866     by (simp add: not_in_preal_ub [OF Rep_preal notin] q)
```
```   867   hence "r * q < r/y" using rpos
```
```   868     by (simp add: divide_inverse mult_less_cancel_left)
```
```   869   also have "... \<le> 1" using rpos rless
```
```   870     by (simp add: pos_divide_le_eq)
```
```   871   finally show ?thesis .
```
```   872 qed
```
```   873
```
```   874 lemma inverse_mult_subset:
```
```   875      "Rep_preal(inverse R * R) \<subseteq> Rep_preal(preal_of_rat 1)"
```
```   876 apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff
```
```   877                       mem_Rep_preal_mult_iff)
```
```   878 apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal])
```
```   879 apply (blast intro: inverse_mult_subset_lemma)
```
```   880 done
```
```   881
```
```   882 lemma preal_mult_inverse: "inverse R * R = (1::preal)"
```
```   883 unfolding preal_one_def
```
```   884 apply (rule Rep_preal_inject [THEN iffD1])
```
```   885 apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult])
```
```   886 done
```
```   887
```
```   888 lemma preal_mult_inverse_right: "R * inverse R = (1::preal)"
```
```   889 apply (rule preal_mult_commute [THEN subst])
```
```   890 apply (rule preal_mult_inverse)
```
```   891 done
```
```   892
```
```   893
```
```   894 text{*Theorems needing @{text Gleason9_34}*}
```
```   895
```
```   896 lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)"
```
```   897 proof
```
```   898   fix r
```
```   899   assume r: "r \<in> Rep_preal R"
```
```   900   have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
```
```   901   from mem_Rep_preal_Ex
```
```   902   obtain y where y: "y \<in> Rep_preal S" ..
```
```   903   have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
```
```   904   have ry: "r+y \<in> Rep_preal(R + S)" using r y
```
```   905     by (auto simp add: mem_Rep_preal_add_iff)
```
```   906   show "r \<in> Rep_preal(R + S)" using r ypos rpos
```
```   907     by (simp add:  preal_downwards_closed [OF Rep_preal ry])
```
```   908 qed
```
```   909
```
```   910 lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)"
```
```   911 proof -
```
```   912   from mem_Rep_preal_Ex
```
```   913   obtain y where y: "y \<in> Rep_preal S" ..
```
```   914   have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
```
```   915   from  Gleason9_34 [OF Rep_preal ypos]
```
```   916   obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" ..
```
```   917   have "r + y \<in> Rep_preal (R + S)" using r y
```
```   918     by (auto simp add: mem_Rep_preal_add_iff)
```
```   919   thus ?thesis using notin by blast
```
```   920 qed
```
```   921
```
```   922 lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)"
```
```   923 by (insert Rep_preal_sum_not_subset, blast)
```
```   924
```
```   925 text{*at last, Gleason prop. 9-3.5(iii) page 123*}
```
```   926 lemma preal_self_less_add_left: "(R::preal) < R + S"
```
```   927 apply (unfold preal_less_def psubset_def)
```
```   928 apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
```
```   929 done
```
```   930
```
```   931 lemma preal_self_less_add_right: "(R::preal) < S + R"
```
```   932 by (simp add: preal_add_commute preal_self_less_add_left)
```
```   933
```
```   934 lemma preal_not_eq_self: "x \<noteq> x + (y::preal)"
```
```   935 by (insert preal_self_less_add_left [of x y], auto)
```
```   936
```
```   937
```
```   938 subsection{*Subtraction for Positive Reals*}
```
```   939
```
```   940 text{*Gleason prop. 9-3.5(iv), page 123: proving @{prop "A < B ==> \<exists>D. A + D =
```
```   941 B"}. We define the claimed @{term D} and show that it is a positive real*}
```
```   942
```
```   943 text{*Part 1 of Dedekind sections definition*}
```
```   944 lemma diff_set_not_empty:
```
```   945      "R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)"
```
```   946 apply (auto simp add: preal_less_def diff_set_def elim!: equalityE)
```
```   947 apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater])
```
```   948 apply (drule preal_imp_pos [OF Rep_preal], clarify)
```
```   949 apply (cut_tac a=x and b=u in add_eq_exists, force)
```
```   950 done
```
```   951
```
```   952 text{*Part 2 of Dedekind sections definition*}
```
```   953 lemma diff_set_nonempty:
```
```   954      "\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)"
```
```   955 apply (cut_tac X = S in Rep_preal_exists_bound)
```
```   956 apply (erule exE)
```
```   957 apply (rule_tac x = x in exI, auto)
```
```   958 apply (simp add: diff_set_def)
```
```   959 apply (auto dest: Rep_preal [THEN preal_downwards_closed])
```
```   960 done
```
```   961
```
```   962 lemma diff_set_not_rat_set:
```
```   963   "diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs")
```
```   964 proof
```
```   965   show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def)
```
```   966   show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast
```
```   967 qed
```
```   968
```
```   969 text{*Part 3 of Dedekind sections definition*}
```
```   970 lemma diff_set_lemma3:
```
```   971      "[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|]
```
```   972       ==> z \<in> diff_set (Rep_preal S) (Rep_preal R)"
```
```   973 apply (auto simp add: diff_set_def)
```
```   974 apply (rule_tac x=x in exI)
```
```   975 apply (drule Rep_preal [THEN preal_downwards_closed], auto)
```
```   976 done
```
```   977
```
```   978 text{*Part 4 of Dedekind sections definition*}
```
```   979 lemma diff_set_lemma4:
```
```   980      "[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|]
```
```   981       ==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u"
```
```   982 apply (auto simp add: diff_set_def)
```
```   983 apply (drule Rep_preal [THEN preal_exists_greater], clarify)
```
```   984 apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify)
```
```   985 apply (rule_tac x="y+xa" in exI)
```
```   986 apply (auto simp add: add_ac)
```
```   987 done
```
```   988
```
```   989 lemma mem_diff_set:
```
```   990      "R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal"
```
```   991 apply (unfold preal_def cut_def)
```
```   992 apply (blast intro!: diff_set_not_empty diff_set_not_rat_set
```
```   993                      diff_set_lemma3 diff_set_lemma4)
```
```   994 done
```
```   995
```
```   996 lemma mem_Rep_preal_diff_iff:
```
```   997       "R < S ==>
```
```   998        (z \<in> Rep_preal(S-R)) =
```
```   999        (\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)"
```
```  1000 apply (simp add: preal_diff_def mem_diff_set Rep_preal)
```
```  1001 apply (force simp add: diff_set_def)
```
```  1002 done
```
```  1003
```
```  1004
```
```  1005 text{*proving that @{term "R + D \<le> S"}*}
```
```  1006
```
```  1007 lemma less_add_left_lemma:
```
```  1008   assumes Rless: "R < S"
```
```  1009     and a: "a \<in> Rep_preal R"
```
```  1010     and cb: "c + b \<in> Rep_preal S"
```
```  1011     and "c \<notin> Rep_preal R"
```
```  1012     and "0 < b"
```
```  1013     and "0 < c"
```
```  1014   shows "a + b \<in> Rep_preal S"
```
```  1015 proof -
```
```  1016   have "0<a" by (rule preal_imp_pos [OF Rep_preal a])
```
```  1017   moreover
```
```  1018   have "a < c" using prems
```
```  1019     by (blast intro: not_in_Rep_preal_ub )
```
```  1020   ultimately show ?thesis using prems
```
```  1021     by (simp add: preal_downwards_closed [OF Rep_preal cb])
```
```  1022 qed
```
```  1023
```
```  1024 lemma less_add_left_le1:
```
```  1025        "R < (S::preal) ==> R + (S-R) \<le> S"
```
```  1026 apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff
```
```  1027                       mem_Rep_preal_diff_iff)
```
```  1028 apply (blast intro: less_add_left_lemma)
```
```  1029 done
```
```  1030
```
```  1031 subsection{*proving that @{term "S \<le> R + D"} --- trickier*}
```
```  1032
```
```  1033 lemma lemma_sum_mem_Rep_preal_ex:
```
```  1034      "x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S"
```
```  1035 apply (drule Rep_preal [THEN preal_exists_greater], clarify)
```
```  1036 apply (cut_tac a=x and b=u in add_eq_exists, auto)
```
```  1037 done
```
```  1038
```
```  1039 lemma less_add_left_lemma2:
```
```  1040   assumes Rless: "R < S"
```
```  1041     and x:     "x \<in> Rep_preal S"
```
```  1042     and xnot: "x \<notin>  Rep_preal R"
```
```  1043   shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R &
```
```  1044                      z + v \<in> Rep_preal S & x = u + v"
```
```  1045 proof -
```
```  1046   have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x])
```
```  1047   from lemma_sum_mem_Rep_preal_ex [OF x]
```
```  1048   obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast
```
```  1049   from  Gleason9_34 [OF Rep_preal epos]
```
```  1050   obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" ..
```
```  1051   with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)
```
```  1052   from add_eq_exists [of r x]
```
```  1053   obtain y where eq: "x = r+y" by auto
```
```  1054   show ?thesis
```
```  1055   proof (intro exI conjI)
```
```  1056     show "r \<in> Rep_preal R" by (rule r)
```
```  1057     show "r + e \<notin> Rep_preal R" by (rule notin)
```
```  1058     show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac)
```
```  1059     show "x = r + y" by (simp add: eq)
```
```  1060     show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r]
```
```  1061       by simp
```
```  1062     show "0 < y" using rless eq by arith
```
```  1063   qed
```
```  1064 qed
```
```  1065
```
```  1066 lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)"
```
```  1067 apply (auto simp add: preal_le_def)
```
```  1068 apply (case_tac "x \<in> Rep_preal R")
```
```  1069 apply (cut_tac Rep_preal_self_subset [of R], force)
```
```  1070 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff)
```
```  1071 apply (blast dest: less_add_left_lemma2)
```
```  1072 done
```
```  1073
```
```  1074 lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"
```
```  1075 by (blast intro: preal_le_anti_sym [OF less_add_left_le1 less_add_left_le2])
```
```  1076
```
```  1077 lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S"
```
```  1078 by (fast dest: less_add_left)
```
```  1079
```
```  1080 lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"
```
```  1081 apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc)
```
```  1082 apply (rule_tac y1 = D in preal_add_commute [THEN subst])
```
```  1083 apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric])
```
```  1084 done
```
```  1085
```
```  1086 lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S"
```
```  1087 by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of T])
```
```  1088
```
```  1089 lemma preal_add_right_less_cancel: "R + T < S + T ==> R < (S::preal)"
```
```  1090 apply (insert linorder_less_linear [of R S], auto)
```
```  1091 apply (drule_tac R = S and T = T in preal_add_less2_mono1)
```
```  1092 apply (blast dest: order_less_trans)
```
```  1093 done
```
```  1094
```
```  1095 lemma preal_add_left_less_cancel: "T + R < T + S ==> R <  (S::preal)"
```
```  1096 by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of T])
```
```  1097
```
```  1098 lemma preal_add_less_cancel_right: "((R::preal) + T < S + T) = (R < S)"
```
```  1099 by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel)
```
```  1100
```
```  1101 lemma preal_add_less_cancel_left: "(T + (R::preal) < T + S) = (R < S)"
```
```  1102 by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)
```
```  1103
```
```  1104 lemma preal_add_le_cancel_right: "((R::preal) + T \<le> S + T) = (R \<le> S)"
```
```  1105 by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_right)
```
```  1106
```
```  1107 lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)"
```
```  1108 by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left)
```
```  1109
```
```  1110 lemma preal_add_less_mono:
```
```  1111      "[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)"
```
```  1112 apply (auto dest!: less_add_left_Ex simp add: preal_add_ac)
```
```  1113 apply (rule preal_add_assoc [THEN subst])
```
```  1114 apply (rule preal_self_less_add_right)
```
```  1115 done
```
```  1116
```
```  1117 lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S"
```
```  1118 apply (insert linorder_less_linear [of R S], safe)
```
```  1119 apply (drule_tac [!] T = T in preal_add_less2_mono1, auto)
```
```  1120 done
```
```  1121
```
```  1122 lemma preal_add_left_cancel: "C + A = C + B ==> A = (B::preal)"
```
```  1123 by (auto intro: preal_add_right_cancel simp add: preal_add_commute)
```
```  1124
```
```  1125 lemma preal_add_left_cancel_iff: "(C + A = C + B) = ((A::preal) = B)"
```
```  1126 by (fast intro: preal_add_left_cancel)
```
```  1127
```
```  1128 lemma preal_add_right_cancel_iff: "(A + C = B + C) = ((A::preal) = B)"
```
```  1129 by (fast intro: preal_add_right_cancel)
```
```  1130
```
```  1131 lemmas preal_cancels =
```
```  1132     preal_add_less_cancel_right preal_add_less_cancel_left
```
```  1133     preal_add_le_cancel_right preal_add_le_cancel_left
```
```  1134     preal_add_left_cancel_iff preal_add_right_cancel_iff
```
```  1135
```
```  1136 instance preal :: ordered_cancel_ab_semigroup_add
```
```  1137 proof
```
```  1138   fix a b c :: preal
```
```  1139   show "a + b = a + c \<Longrightarrow> b = c" by (rule preal_add_left_cancel)
```
```  1140   show "a \<le> b \<Longrightarrow> c + a \<le> c + b" by (simp only: preal_add_le_cancel_left)
```
```  1141 qed
```
```  1142
```
```  1143
```
```  1144 subsection{*Completeness of type @{typ preal}*}
```
```  1145
```
```  1146 text{*Prove that supremum is a cut*}
```
```  1147
```
```  1148 text{*Part 1 of Dedekind sections definition*}
```
```  1149
```
```  1150 lemma preal_sup_set_not_empty:
```
```  1151      "P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))"
```
```  1152 apply auto
```
```  1153 apply (cut_tac X = x in mem_Rep_preal_Ex, auto)
```
```  1154 done
```
```  1155
```
```  1156
```
```  1157 text{*Part 2 of Dedekind sections definition*}
```
```  1158
```
```  1159 lemma preal_sup_not_exists:
```
```  1160      "\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))"
```
```  1161 apply (cut_tac X = Y in Rep_preal_exists_bound)
```
```  1162 apply (auto simp add: preal_le_def)
```
```  1163 done
```
```  1164
```
```  1165 lemma preal_sup_set_not_rat_set:
```
```  1166      "\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}"
```
```  1167 apply (drule preal_sup_not_exists)
```
```  1168 apply (blast intro: preal_imp_pos [OF Rep_preal])
```
```  1169 done
```
```  1170
```
```  1171 text{*Part 3 of Dedekind sections definition*}
```
```  1172 lemma preal_sup_set_lemma3:
```
```  1173      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|]
```
```  1174       ==> z \<in> (\<Union>X \<in> P. Rep_preal(X))"
```
```  1175 by (auto elim: Rep_preal [THEN preal_downwards_closed])
```
```  1176
```
```  1177 text{*Part 4 of Dedekind sections definition*}
```
```  1178 lemma preal_sup_set_lemma4:
```
```  1179      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |]
```
```  1180           ==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u"
```
```  1181 by (blast dest: Rep_preal [THEN preal_exists_greater])
```
```  1182
```
```  1183 lemma preal_sup:
```
```  1184      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal"
```
```  1185 apply (unfold preal_def cut_def)
```
```  1186 apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set
```
```  1187                      preal_sup_set_lemma3 preal_sup_set_lemma4)
```
```  1188 done
```
```  1189
```
```  1190 lemma preal_psup_le:
```
```  1191      "[| \<forall>X \<in> P. X \<le> Y;  x \<in> P |] ==> x \<le> psup P"
```
```  1192 apply (simp (no_asm_simp) add: preal_le_def)
```
```  1193 apply (subgoal_tac "P \<noteq> {}")
```
```  1194 apply (auto simp add: psup_def preal_sup)
```
```  1195 done
```
```  1196
```
```  1197 lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"
```
```  1198 apply (simp (no_asm_simp) add: preal_le_def)
```
```  1199 apply (simp add: psup_def preal_sup)
```
```  1200 apply (auto simp add: preal_le_def)
```
```  1201 done
```
```  1202
```
```  1203 text{*Supremum property*}
```
```  1204 lemma preal_complete:
```
```  1205      "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)"
```
```  1206 apply (simp add: preal_less_def psup_def preal_sup)
```
```  1207 apply (auto simp add: preal_le_def)
```
```  1208 apply (rename_tac U)
```
```  1209 apply (cut_tac x = U and y = Z in linorder_less_linear)
```
```  1210 apply (auto simp add: preal_less_def)
```
```  1211 done
```
```  1212
```
```  1213
```
```  1214 subsection{*The Embedding from @{typ rat} into @{typ preal}*}
```
```  1215
```
```  1216 lemma preal_of_rat_add_lemma1:
```
```  1217      "[|x < y + z; 0 < x; 0 < y|] ==> x * y * inverse (y + z) < (y::rat)"
```
```  1218 apply (frule_tac c = "y * inverse (y + z) " in mult_strict_right_mono)
```
```  1219 apply (simp add: zero_less_mult_iff)
```
```  1220 apply (simp add: mult_ac)
```
```  1221 done
```
```  1222
```
```  1223 lemma preal_of_rat_add_lemma2:
```
```  1224   assumes "u < x + y"
```
```  1225     and "0 < x"
```
```  1226     and "0 < y"
```
```  1227     and "0 < u"
```
```  1228   shows "\<exists>v w::rat. w < y & 0 < v & v < x & 0 < w & u = v + w"
```
```  1229 proof (intro exI conjI)
```
```  1230   show "u * x * inverse(x+y) < x" using prems
```
```  1231     by (simp add: preal_of_rat_add_lemma1)
```
```  1232   show "u * y * inverse(x+y) < y" using prems
```
```  1233     by (simp add: preal_of_rat_add_lemma1 add_commute [of x])
```
```  1234   show "0 < u * x * inverse (x + y)" using prems
```
```  1235     by (simp add: zero_less_mult_iff)
```
```  1236   show "0 < u * y * inverse (x + y)" using prems
```
```  1237     by (simp add: zero_less_mult_iff)
```
```  1238   show "u = u * x * inverse (x + y) + u * y * inverse (x + y)" using prems
```
```  1239     by (simp add: left_distrib [symmetric] right_distrib [symmetric] mult_ac)
```
```  1240 qed
```
```  1241
```
```  1242 lemma preal_of_rat_add:
```
```  1243      "[| 0 < x; 0 < y|]
```
```  1244       ==> preal_of_rat ((x::rat) + y) = preal_of_rat x + preal_of_rat y"
```
```  1245 apply (unfold preal_of_rat_def preal_add_def)
```
```  1246 apply (simp add: rat_mem_preal)
```
```  1247 apply (rule_tac f = Abs_preal in arg_cong)
```
```  1248 apply (auto simp add: add_set_def)
```
```  1249 apply (blast dest: preal_of_rat_add_lemma2)
```
```  1250 done
```
```  1251
```
```  1252 lemma preal_of_rat_mult_lemma1:
```
```  1253      "[|x < y; 0 < x; 0 < z|] ==> x * z * inverse y < (z::rat)"
```
```  1254 apply (frule_tac c = "z * inverse y" in mult_strict_right_mono)
```
```  1255 apply (simp add: zero_less_mult_iff)
```
```  1256 apply (subgoal_tac "y * (z * inverse y) = z * (y * inverse y)")
```
```  1257 apply (simp_all add: mult_ac)
```
```  1258 done
```
```  1259
```
```  1260 lemma preal_of_rat_mult_lemma2:
```
```  1261   assumes xless: "x < y * z"
```
```  1262     and xpos: "0 < x"
```
```  1263     and ypos: "0 < y"
```
```  1264   shows "x * z * inverse y * inverse z < (z::rat)"
```
```  1265 proof -
```
```  1266   have "0 < y * z" using prems by simp
```
```  1267   hence zpos:  "0 < z" using prems by (simp add: zero_less_mult_iff)
```
```  1268   have "x * z * inverse y * inverse z = x * inverse y * (z * inverse z)"
```
```  1269     by (simp add: mult_ac)
```
```  1270   also have "... = x/y" using zpos
```
```  1271     by (simp add: divide_inverse)
```
```  1272   also from xless have "... < z"
```
```  1273     by (simp add: pos_divide_less_eq [OF ypos] mult_commute)
```
```  1274   finally show ?thesis .
```
```  1275 qed
```
```  1276
```
```  1277 lemma preal_of_rat_mult_lemma3:
```
```  1278   assumes uless: "u < x * y"
```
```  1279     and "0 < x"
```
```  1280     and "0 < y"
```
```  1281     and "0 < u"
```
```  1282   shows "\<exists>v w::rat. v < x & w < y & 0 < v & 0 < w & u = v * w"
```
```  1283 proof -
```
```  1284   from dense [OF uless]
```
```  1285   obtain r where "u < r" "r < x * y" by blast
```
```  1286   thus ?thesis
```
```  1287   proof (intro exI conjI)
```
```  1288   show "u * x * inverse r < x" using prems
```
```  1289     by (simp add: preal_of_rat_mult_lemma1)
```
```  1290   show "r * y * inverse x * inverse y < y" using prems
```
```  1291     by (simp add: preal_of_rat_mult_lemma2)
```
```  1292   show "0 < u * x * inverse r" using prems
```
```  1293     by (simp add: zero_less_mult_iff)
```
```  1294   show "0 < r * y * inverse x * inverse y" using prems
```
```  1295     by (simp add: zero_less_mult_iff)
```
```  1296   have "u * x * inverse r * (r * y * inverse x * inverse y) =
```
```  1297         u * (r * inverse r) * (x * inverse x) * (y * inverse y)"
```
```  1298     by (simp only: mult_ac)
```
```  1299   thus "u = u * x * inverse r * (r * y * inverse x * inverse y)" using prems
```
```  1300     by simp
```
```  1301   qed
```
```  1302 qed
```
```  1303
```
```  1304 lemma preal_of_rat_mult:
```
```  1305      "[| 0 < x; 0 < y|]
```
```  1306       ==> preal_of_rat ((x::rat) * y) = preal_of_rat x * preal_of_rat y"
```
```  1307 apply (unfold preal_of_rat_def preal_mult_def)
```
```  1308 apply (simp add: rat_mem_preal)
```
```  1309 apply (rule_tac f = Abs_preal in arg_cong)
```
```  1310 apply (auto simp add: zero_less_mult_iff mult_strict_mono mult_set_def)
```
```  1311 apply (blast dest: preal_of_rat_mult_lemma3)
```
```  1312 done
```
```  1313
```
```  1314 lemma preal_of_rat_less_iff:
```
```  1315       "[| 0 < x; 0 < y|] ==> (preal_of_rat x < preal_of_rat y) = (x < y)"
```
```  1316 by (force simp add: preal_of_rat_def preal_less_def rat_mem_preal)
```
```  1317
```
```  1318 lemma preal_of_rat_le_iff:
```
```  1319       "[| 0 < x; 0 < y|] ==> (preal_of_rat x \<le> preal_of_rat y) = (x \<le> y)"
```
```  1320 by (simp add: preal_of_rat_less_iff linorder_not_less [symmetric])
```
```  1321
```
```  1322 lemma preal_of_rat_eq_iff:
```
```  1323       "[| 0 < x; 0 < y|] ==> (preal_of_rat x = preal_of_rat y) = (x = y)"
```
```  1324 by (simp add: preal_of_rat_le_iff order_eq_iff)
```
```  1325
```
```  1326 end
```