src/HOL/Real/PReal.thy
 author nipkow Mon Aug 16 14:22:27 2004 +0200 (2004-08-16) changeset 15131 c69542757a4d parent 15055 aed573241bea child 15140 322485b816ac permissions -rw-r--r--
```     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 theory PReal
```
```    11 import Rational
```
```    12 begin
```
```    13
```
```    14 text{*Could be generalized and moved to @{text Ring_and_Field}*}
```
```    15 lemma add_eq_exists: "\<exists>x. a+x = (b::rat)"
```
```    16 by (rule_tac x="b-a" in exI, simp)
```
```    17
```
```    18 text{*As a special case, the sum of two positives is positive.  One of the
```
```    19 premises could be weakened to the relation @{text "\<le>"}.*}
```
```    20 lemma pos_add_strict: "[|0<a; b<c|] ==> b < a + (c::'a::ordered_semidom)"
```
```    21 by (insert add_strict_mono [of 0 a b c], simp)
```
```    22
```
```    23 lemma interval_empty_iff:
```
```    24      "({y::'a::ordered_field. x < y & y < z} = {}) = (~(x < z))"
```
```    25 by (blast dest: dense intro: order_less_trans)
```
```    26
```
```    27
```
```    28 constdefs
```
```    29   cut :: "rat set => bool"
```
```    30     "cut A == {} \<subset> A &
```
```    31               A < {r. 0 < r} &
```
```    32               (\<forall>y \<in> A. ((\<forall>z. 0<z & z < y --> z \<in> A) & (\<exists>u \<in> A. y < u)))"
```
```    33
```
```    34
```
```    35 lemma cut_of_rat:
```
```    36   assumes q: "0 < q" shows "cut {r::rat. 0 < r & r < q}"
```
```    37 proof -
```
```    38   let ?A = "{r::rat. 0 < r & r < q}"
```
```    39   from q have pos: "?A < {r. 0 < r}" by force
```
```    40   have nonempty: "{} \<subset> ?A"
```
```    41   proof
```
```    42     show "{} \<subseteq> ?A" by simp
```
```    43     show "{} \<noteq> ?A"
```
```    44       by (force simp only: q eq_commute [of "{}"] interval_empty_iff)
```
```    45   qed
```
```    46   show ?thesis
```
```    47     by (simp add: cut_def pos nonempty,
```
```    48         blast dest: dense intro: order_less_trans)
```
```    49 qed
```
```    50
```
```    51
```
```    52 typedef preal = "{A. cut A}"
```
```    53   by (blast intro: cut_of_rat [OF zero_less_one])
```
```    54
```
```    55 instance preal :: "{ord, plus, minus, times, inverse}" ..
```
```    56
```
```    57 constdefs
```
```    58   preal_of_rat :: "rat => preal"
```
```    59      "preal_of_rat q == Abs_preal({x::rat. 0 < x & x < q})"
```
```    60
```
```    61   psup       :: "preal set => preal"
```
```    62     "psup(P)   == Abs_preal(\<Union>X \<in> P. Rep_preal(X))"
```
```    63
```
```    64   add_set :: "[rat set,rat set] => rat set"
```
```    65     "add_set A B == {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x + y}"
```
```    66
```
```    67   diff_set :: "[rat set,rat set] => rat set"
```
```    68     "diff_set A B == {w. \<exists>x. 0 < w & 0 < x & x \<notin> B & x + w \<in> A}"
```
```    69
```
```    70   mult_set :: "[rat set,rat set] => rat set"
```
```    71     "mult_set A B == {w. \<exists>x \<in> A. \<exists>y \<in> B. w = x * y}"
```
```    72
```
```    73   inverse_set :: "rat set => rat set"
```
```    74     "inverse_set A == {x. \<exists>y. 0 < x & x < y & inverse y \<notin> A}"
```
```    75
```
```    76
```
```    77 defs (overloaded)
```
```    78
```
```    79   preal_less_def:
```
```    80     "R < (S::preal) == Rep_preal R < Rep_preal S"
```
```    81
```
```    82   preal_le_def:
```
```    83     "R \<le> (S::preal) == Rep_preal R \<subseteq> Rep_preal S"
```
```    84
```
```    85   preal_add_def:
```
```    86     "R + S == Abs_preal (add_set (Rep_preal R) (Rep_preal S))"
```
```    87
```
```    88   preal_diff_def:
```
```    89     "R - S == Abs_preal (diff_set (Rep_preal R) (Rep_preal S))"
```
```    90
```
```    91   preal_mult_def:
```
```    92     "R * S == Abs_preal(mult_set (Rep_preal R) (Rep_preal S))"
```
```    93
```
```    94   preal_inverse_def:
```
```    95     "inverse R == Abs_preal(inverse_set (Rep_preal R))"
```
```    96
```
```    97
```
```    98 lemma inj_on_Abs_preal: "inj_on Abs_preal preal"
```
```    99 apply (rule inj_on_inverseI)
```
```   100 apply (erule Abs_preal_inverse)
```
```   101 done
```
```   102
```
```   103 declare inj_on_Abs_preal [THEN inj_on_iff, simp]
```
```   104
```
```   105 lemma inj_Rep_preal: "inj(Rep_preal)"
```
```   106 apply (rule inj_on_inverseI)
```
```   107 apply (rule Rep_preal_inverse)
```
```   108 done
```
```   109
```
```   110 lemma preal_nonempty: "A \<in> preal ==> \<exists>x\<in>A. 0 < x"
```
```   111 by (unfold preal_def cut_def, blast)
```
```   112
```
```   113 lemma preal_imp_psubset_positives: "A \<in> preal ==> A < {r. 0 < r}"
```
```   114 by (force simp add: preal_def cut_def)
```
```   115
```
```   116 lemma preal_exists_bound: "A \<in> preal ==> \<exists>x. 0 < x & x \<notin> A"
```
```   117 by (drule preal_imp_psubset_positives, auto)
```
```   118
```
```   119 lemma preal_exists_greater: "[| A \<in> preal; y \<in> A |] ==> \<exists>u \<in> A. y < u"
```
```   120 by (unfold preal_def cut_def, blast)
```
```   121
```
```   122 lemma mem_Rep_preal_Ex: "\<exists>x. x \<in> Rep_preal X"
```
```   123 apply (insert Rep_preal [of X])
```
```   124 apply (unfold preal_def cut_def, blast)
```
```   125 done
```
```   126
```
```   127 declare Abs_preal_inverse [simp]
```
```   128
```
```   129 lemma preal_downwards_closed: "[| A \<in> preal; y \<in> A; 0 < z; z < y |] ==> z \<in> A"
```
```   130 by (unfold preal_def cut_def, blast)
```
```   131
```
```   132 text{*Relaxing the final premise*}
```
```   133 lemma preal_downwards_closed':
```
```   134      "[| A \<in> preal; y \<in> A; 0 < z; z \<le> y |] ==> z \<in> A"
```
```   135 apply (simp add: order_le_less)
```
```   136 apply (blast intro: preal_downwards_closed)
```
```   137 done
```
```   138
```
```   139 lemma Rep_preal_exists_bound: "\<exists>x. 0 < x & x \<notin> Rep_preal X"
```
```   140 apply (cut_tac x = X in Rep_preal)
```
```   141 apply (drule preal_imp_psubset_positives)
```
```   142 apply (auto simp add: psubset_def)
```
```   143 done
```
```   144
```
```   145
```
```   146 subsection{*@{term preal_of_prat}: the Injection from prat to preal*}
```
```   147
```
```   148 lemma rat_less_set_mem_preal: "0 < y ==> {u::rat. 0 < u & u < y} \<in> preal"
```
```   149 apply (auto simp add: preal_def cut_def intro: order_less_trans)
```
```   150 apply (force simp only: eq_commute [of "{}"] interval_empty_iff)
```
```   151 apply (blast dest: dense intro: order_less_trans)
```
```   152 done
```
```   153
```
```   154 lemma rat_subset_imp_le:
```
```   155      "[|{u::rat. 0 < u & u < x} \<subseteq> {u. 0 < u & u < y}; 0<x|] ==> x \<le> y"
```
```   156 apply (simp add: linorder_not_less [symmetric])
```
```   157 apply (blast dest: dense intro: order_less_trans)
```
```   158 done
```
```   159
```
```   160 lemma rat_set_eq_imp_eq:
```
```   161      "[|{u::rat. 0 < u & u < x} = {u. 0 < u & u < y};
```
```   162         0 < x; 0 < y|] ==> x = y"
```
```   163 by (blast intro: rat_subset_imp_le order_antisym)
```
```   164
```
```   165
```
```   166
```
```   167 subsection{*Theorems for Ordering*}
```
```   168
```
```   169 text{*A positive fraction not in a positive real is an upper bound.
```
```   170  Gleason p. 122 - Remark (1)*}
```
```   171
```
```   172 lemma not_in_preal_ub:
```
```   173      assumes A: "A \<in> preal"
```
```   174          and notx: "x \<notin> A"
```
```   175          and y: "y \<in> A"
```
```   176          and pos: "0 < x"
```
```   177         shows "y < x"
```
```   178 proof (cases rule: linorder_cases)
```
```   179   assume "x<y"
```
```   180   with notx show ?thesis
```
```   181     by (simp add:  preal_downwards_closed [OF A y] pos)
```
```   182 next
```
```   183   assume "x=y"
```
```   184   with notx and y show ?thesis by simp
```
```   185 next
```
```   186   assume "y<x"
```
```   187   thus ?thesis by assumption
```
```   188 qed
```
```   189
```
```   190 lemmas not_in_Rep_preal_ub = not_in_preal_ub [OF Rep_preal]
```
```   191
```
```   192
```
```   193 subsection{*The @{text "\<le>"} Ordering*}
```
```   194
```
```   195 lemma preal_le_refl: "w \<le> (w::preal)"
```
```   196 by (simp add: preal_le_def)
```
```   197
```
```   198 lemma preal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::preal)"
```
```   199 by (force simp add: preal_le_def)
```
```   200
```
```   201 lemma preal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::preal)"
```
```   202 apply (simp add: preal_le_def)
```
```   203 apply (rule Rep_preal_inject [THEN iffD1], blast)
```
```   204 done
```
```   205
```
```   206 (* Axiom 'order_less_le' of class 'order': *)
```
```   207 lemma preal_less_le: "((w::preal) < z) = (w \<le> z & w \<noteq> z)"
```
```   208 by (simp add: preal_le_def preal_less_def Rep_preal_inject psubset_def)
```
```   209
```
```   210 instance preal :: order
```
```   211   by intro_classes
```
```   212     (assumption |
```
```   213       rule preal_le_refl preal_le_trans preal_le_anti_sym preal_less_le)+
```
```   214
```
```   215 lemma preal_imp_pos: "[|A \<in> preal; r \<in> A|] ==> 0 < r"
```
```   216 by (insert preal_imp_psubset_positives, blast)
```
```   217
```
```   218 lemma preal_le_linear: "x <= y | y <= (x::preal)"
```
```   219 apply (auto simp add: preal_le_def)
```
```   220 apply (rule ccontr)
```
```   221 apply (blast dest: not_in_Rep_preal_ub intro: preal_imp_pos [OF Rep_preal]
```
```   222              elim: order_less_asym)
```
```   223 done
```
```   224
```
```   225 instance preal :: linorder
```
```   226   by intro_classes (rule preal_le_linear)
```
```   227
```
```   228
```
```   229
```
```   230 subsection{*Properties of Addition*}
```
```   231
```
```   232 lemma preal_add_commute: "(x::preal) + y = y + x"
```
```   233 apply (unfold preal_add_def add_set_def)
```
```   234 apply (rule_tac f = Abs_preal in arg_cong)
```
```   235 apply (force simp add: add_commute)
```
```   236 done
```
```   237
```
```   238 text{*Lemmas for proving that addition of two positive reals gives
```
```   239  a positive real*}
```
```   240
```
```   241 lemma empty_psubset_nonempty: "a \<in> A ==> {} \<subset> A"
```
```   242 by blast
```
```   243
```
```   244 text{*Part 1 of Dedekind sections definition*}
```
```   245 lemma add_set_not_empty:
```
```   246      "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> add_set A B"
```
```   247 apply (insert preal_nonempty [of A] preal_nonempty [of B])
```
```   248 apply (auto simp add: add_set_def)
```
```   249 done
```
```   250
```
```   251 text{*Part 2 of Dedekind sections definition.  A structured version of
```
```   252 this proof is @{text preal_not_mem_mult_set_Ex} below.*}
```
```   253 lemma preal_not_mem_add_set_Ex:
```
```   254      "[|A \<in> preal; B \<in> preal|] ==> \<exists>q. 0 < q & q \<notin> add_set A B"
```
```   255 apply (insert preal_exists_bound [of A] preal_exists_bound [of B], auto)
```
```   256 apply (rule_tac x = "x+xa" in exI)
```
```   257 apply (simp add: add_set_def, clarify)
```
```   258 apply (drule not_in_preal_ub, assumption+)+
```
```   259 apply (force dest: add_strict_mono)
```
```   260 done
```
```   261
```
```   262 lemma add_set_not_rat_set:
```
```   263    assumes A: "A \<in> preal"
```
```   264        and B: "B \<in> preal"
```
```   265      shows "add_set A B < {r. 0 < r}"
```
```   266 proof
```
```   267   from preal_imp_pos [OF A] preal_imp_pos [OF B]
```
```   268   show "add_set A B \<subseteq> {r. 0 < r}" by (force simp add: add_set_def)
```
```   269 next
```
```   270   show "add_set A B \<noteq> {r. 0 < r}"
```
```   271     by (insert preal_not_mem_add_set_Ex [OF A B], blast)
```
```   272 qed
```
```   273
```
```   274 text{*Part 3 of Dedekind sections definition*}
```
```   275 lemma add_set_lemma3:
```
```   276      "[|A \<in> preal; B \<in> preal; u \<in> add_set A B; 0 < z; z < u|]
```
```   277       ==> z \<in> add_set A B"
```
```   278 proof (unfold add_set_def, clarify)
```
```   279   fix x::rat and y::rat
```
```   280   assume A: "A \<in> preal"
```
```   281      and B: "B \<in> preal"
```
```   282      and [simp]: "0 < z"
```
```   283      and zless: "z < x + y"
```
```   284      and x:  "x \<in> A"
```
```   285      and y:  "y \<in> B"
```
```   286   have xpos [simp]: "0<x" by (rule preal_imp_pos [OF A x])
```
```   287   have ypos [simp]: "0<y" by (rule preal_imp_pos [OF B y])
```
```   288   have xypos [simp]: "0 < x+y" by (simp add: pos_add_strict)
```
```   289   let ?f = "z/(x+y)"
```
```   290   have fless: "?f < 1" by (simp add: zless pos_divide_less_eq)
```
```   291   show "\<exists>x' \<in> A. \<exists>y'\<in>B. z = x' + y'"
```
```   292   proof
```
```   293     show "\<exists>y' \<in> B. z = x*?f + y'"
```
```   294     proof
```
```   295       show "z = x*?f + y*?f"
```
```   296 	by (simp add: left_distrib [symmetric] divide_inverse mult_ac
```
```   297 		      order_less_imp_not_eq2)
```
```   298     next
```
```   299       show "y * ?f \<in> B"
```
```   300       proof (rule preal_downwards_closed [OF B y])
```
```   301         show "0 < y * ?f"
```
```   302           by (simp add: divide_inverse zero_less_mult_iff)
```
```   303       next
```
```   304         show "y * ?f < y"
```
```   305           by (insert mult_strict_left_mono [OF fless ypos], simp)
```
```   306       qed
```
```   307     qed
```
```   308   next
```
```   309     show "x * ?f \<in> A"
```
```   310     proof (rule preal_downwards_closed [OF A x])
```
```   311       show "0 < x * ?f"
```
```   312 	by (simp add: divide_inverse zero_less_mult_iff)
```
```   313     next
```
```   314       show "x * ?f < x"
```
```   315 	by (insert mult_strict_left_mono [OF fless xpos], simp)
```
```   316     qed
```
```   317   qed
```
```   318 qed
```
```   319
```
```   320 text{*Part 4 of Dedekind sections definition*}
```
```   321 lemma add_set_lemma4:
```
```   322      "[|A \<in> preal; B \<in> preal; y \<in> add_set A B|] ==> \<exists>u \<in> add_set A B. y < u"
```
```   323 apply (auto simp add: add_set_def)
```
```   324 apply (frule preal_exists_greater [of A], auto)
```
```   325 apply (rule_tac x="u + y" in exI)
```
```   326 apply (auto intro: add_strict_left_mono)
```
```   327 done
```
```   328
```
```   329 lemma mem_add_set:
```
```   330      "[|A \<in> preal; B \<in> preal|] ==> add_set A B \<in> preal"
```
```   331 apply (simp (no_asm_simp) add: preal_def cut_def)
```
```   332 apply (blast intro!: add_set_not_empty add_set_not_rat_set
```
```   333                      add_set_lemma3 add_set_lemma4)
```
```   334 done
```
```   335
```
```   336 lemma preal_add_assoc: "((x::preal) + y) + z = x + (y + z)"
```
```   337 apply (simp add: preal_add_def mem_add_set Rep_preal)
```
```   338 apply (force simp add: add_set_def add_ac)
```
```   339 done
```
```   340
```
```   341 lemma preal_add_left_commute: "x + (y + z) = y + ((x + z)::preal)"
```
```   342   apply (rule mk_left_commute [of "op +"])
```
```   343   apply (rule preal_add_assoc)
```
```   344   apply (rule preal_add_commute)
```
```   345   done
```
```   346
```
```   347 text{* Positive Real addition is an AC operator *}
```
```   348 lemmas preal_add_ac = preal_add_assoc preal_add_commute preal_add_left_commute
```
```   349
```
```   350
```
```   351 subsection{*Properties of Multiplication*}
```
```   352
```
```   353 text{*Proofs essentially same as for addition*}
```
```   354
```
```   355 lemma preal_mult_commute: "(x::preal) * y = y * x"
```
```   356 apply (unfold preal_mult_def mult_set_def)
```
```   357 apply (rule_tac f = Abs_preal in arg_cong)
```
```   358 apply (force simp add: mult_commute)
```
```   359 done
```
```   360
```
```   361 text{*Multiplication of two positive reals gives a positive real.*}
```
```   362
```
```   363 text{*Lemmas for proving positive reals multiplication set in @{typ preal}*}
```
```   364
```
```   365 text{*Part 1 of Dedekind sections definition*}
```
```   366 lemma mult_set_not_empty:
```
```   367      "[|A \<in> preal; B \<in> preal|] ==> {} \<subset> mult_set A B"
```
```   368 apply (insert preal_nonempty [of A] preal_nonempty [of B])
```
```   369 apply (auto simp add: mult_set_def)
```
```   370 done
```
```   371
```
```   372 text{*Part 2 of Dedekind sections definition*}
```
```   373 lemma preal_not_mem_mult_set_Ex:
```
```   374    assumes A: "A \<in> preal"
```
```   375        and B: "B \<in> preal"
```
```   376      shows "\<exists>q. 0 < q & q \<notin> mult_set A B"
```
```   377 proof -
```
```   378   from preal_exists_bound [OF A]
```
```   379   obtain x where [simp]: "0 < x" "x \<notin> A" by blast
```
```   380   from preal_exists_bound [OF B]
```
```   381   obtain y where [simp]: "0 < y" "y \<notin> B" by blast
```
```   382   show ?thesis
```
```   383   proof (intro exI conjI)
```
```   384     show "0 < x*y" by (simp add: mult_pos)
```
```   385     show "x * y \<notin> mult_set A B"
```
```   386     proof -
```
```   387       { fix u::rat and v::rat
```
```   388 	      assume "u \<in> A" and "v \<in> B" and "x*y = u*v"
```
```   389 	      moreover
```
```   390 	      with prems have "u<x" and "v<y" by (blast dest: not_in_preal_ub)+
```
```   391 	      moreover
```
```   392 	      with prems have "0\<le>v"
```
```   393 	        by (blast intro: preal_imp_pos [OF B]  order_less_imp_le prems)
```
```   394 	      moreover
```
```   395         from calculation
```
```   396 	      have "u*v < x*y" by (blast intro: mult_strict_mono prems)
```
```   397 	      ultimately have False by force }
```
```   398       thus ?thesis by (auto simp add: mult_set_def)
```
```   399     qed
```
```   400   qed
```
```   401 qed
```
```   402
```
```   403 lemma mult_set_not_rat_set:
```
```   404    assumes A: "A \<in> preal"
```
```   405        and B: "B \<in> preal"
```
```   406      shows "mult_set A B < {r. 0 < r}"
```
```   407 proof
```
```   408   show "mult_set A B \<subseteq> {r. 0 < r}"
```
```   409     by (force simp add: mult_set_def
```
```   410               intro: preal_imp_pos [OF A] preal_imp_pos [OF B] mult_pos)
```
```   411 next
```
```   412   show "mult_set A B \<noteq> {r. 0 < r}"
```
```   413     by (insert preal_not_mem_mult_set_Ex [OF A B], blast)
```
```   414 qed
```
```   415
```
```   416
```
```   417
```
```   418 text{*Part 3 of Dedekind sections definition*}
```
```   419 lemma mult_set_lemma3:
```
```   420      "[|A \<in> preal; B \<in> preal; u \<in> mult_set A B; 0 < z; z < u|]
```
```   421       ==> z \<in> mult_set A B"
```
```   422 proof (unfold mult_set_def, clarify)
```
```   423   fix x::rat and y::rat
```
```   424   assume A: "A \<in> preal"
```
```   425      and B: "B \<in> preal"
```
```   426      and [simp]: "0 < z"
```
```   427      and zless: "z < x * y"
```
```   428      and x:  "x \<in> A"
```
```   429      and y:  "y \<in> B"
```
```   430   have [simp]: "0<y" by (rule preal_imp_pos [OF B y])
```
```   431   show "\<exists>x' \<in> A. \<exists>y' \<in> B. z = x' * y'"
```
```   432   proof
```
```   433     show "\<exists>y'\<in>B. z = (z/y) * y'"
```
```   434     proof
```
```   435       show "z = (z/y)*y"
```
```   436 	by (simp add: divide_inverse mult_commute [of y] mult_assoc
```
```   437 		      order_less_imp_not_eq2)
```
```   438       show "y \<in> B" .
```
```   439     qed
```
```   440   next
```
```   441     show "z/y \<in> A"
```
```   442     proof (rule preal_downwards_closed [OF A x])
```
```   443       show "0 < z/y"
```
```   444 	by (simp add: zero_less_divide_iff)
```
```   445       show "z/y < x" by (simp add: pos_divide_less_eq zless)
```
```   446     qed
```
```   447   qed
```
```   448 qed
```
```   449
```
```   450 text{*Part 4 of Dedekind sections definition*}
```
```   451 lemma mult_set_lemma4:
```
```   452      "[|A \<in> preal; B \<in> preal; y \<in> mult_set A B|] ==> \<exists>u \<in> mult_set A B. y < u"
```
```   453 apply (auto simp add: mult_set_def)
```
```   454 apply (frule preal_exists_greater [of A], auto)
```
```   455 apply (rule_tac x="u * y" in exI)
```
```   456 apply (auto intro: preal_imp_pos [of A] preal_imp_pos [of B]
```
```   457                    mult_strict_right_mono)
```
```   458 done
```
```   459
```
```   460
```
```   461 lemma mem_mult_set:
```
```   462      "[|A \<in> preal; B \<in> preal|] ==> mult_set A B \<in> preal"
```
```   463 apply (simp (no_asm_simp) add: preal_def cut_def)
```
```   464 apply (blast intro!: mult_set_not_empty mult_set_not_rat_set
```
```   465                      mult_set_lemma3 mult_set_lemma4)
```
```   466 done
```
```   467
```
```   468 lemma preal_mult_assoc: "((x::preal) * y) * z = x * (y * z)"
```
```   469 apply (simp add: preal_mult_def mem_mult_set Rep_preal)
```
```   470 apply (force simp add: mult_set_def mult_ac)
```
```   471 done
```
```   472
```
```   473 lemma preal_mult_left_commute: "x * (y * z) = y * ((x * z)::preal)"
```
```   474   apply (rule mk_left_commute [of "op *"])
```
```   475   apply (rule preal_mult_assoc)
```
```   476   apply (rule preal_mult_commute)
```
```   477   done
```
```   478
```
```   479
```
```   480 text{* Positive Real multiplication is an AC operator *}
```
```   481 lemmas preal_mult_ac =
```
```   482        preal_mult_assoc preal_mult_commute preal_mult_left_commute
```
```   483
```
```   484
```
```   485 text{* Positive real 1 is the multiplicative identity element *}
```
```   486
```
```   487 lemma rat_mem_preal: "0 < q ==> {r::rat. 0 < r & r < q} \<in> preal"
```
```   488 by (simp add: preal_def cut_of_rat)
```
```   489
```
```   490 lemma preal_mult_1: "(preal_of_rat 1) * z = z"
```
```   491 proof (induct z)
```
```   492   fix A :: "rat set"
```
```   493   assume A: "A \<in> preal"
```
```   494   have "{w. \<exists>u. 0 < u \<and> u < 1 & (\<exists>v \<in> A. w = u * v)} = A" (is "?lhs = A")
```
```   495   proof
```
```   496     show "?lhs \<subseteq> A"
```
```   497     proof clarify
```
```   498       fix x::rat and u::rat and v::rat
```
```   499       assume upos: "0<u" and "u<1" and v: "v \<in> A"
```
```   500       have vpos: "0<v" by (rule preal_imp_pos [OF A v])
```
```   501       hence "u*v < 1*v" by (simp only: mult_strict_right_mono prems)
```
```   502       thus "u * v \<in> A"
```
```   503         by (force intro: preal_downwards_closed [OF A v] mult_pos upos vpos)
```
```   504     qed
```
```   505   next
```
```   506     show "A \<subseteq> ?lhs"
```
```   507     proof clarify
```
```   508       fix x::rat
```
```   509       assume x: "x \<in> A"
```
```   510       have xpos: "0<x" by (rule preal_imp_pos [OF A x])
```
```   511       from preal_exists_greater [OF A x]
```
```   512       obtain v where v: "v \<in> A" and xlessv: "x < v" ..
```
```   513       have vpos: "0<v" by (rule preal_imp_pos [OF A v])
```
```   514       show "\<exists>u. 0 < u \<and> u < 1 \<and> (\<exists>v\<in>A. x = u * v)"
```
```   515       proof (intro exI conjI)
```
```   516         show "0 < x/v"
```
```   517           by (simp add: zero_less_divide_iff xpos vpos)
```
```   518 	show "x / v < 1"
```
```   519           by (simp add: pos_divide_less_eq vpos xlessv)
```
```   520         show "\<exists>v'\<in>A. x = (x / v) * v'"
```
```   521         proof
```
```   522           show "x = (x/v)*v"
```
```   523 	    by (simp add: divide_inverse mult_assoc vpos
```
```   524                           order_less_imp_not_eq2)
```
```   525           show "v \<in> A" .
```
```   526         qed
```
```   527       qed
```
```   528     qed
```
```   529   qed
```
```   530   thus "preal_of_rat 1 * Abs_preal A = Abs_preal A"
```
```   531     by (simp add: preal_of_rat_def preal_mult_def mult_set_def
```
```   532                   rat_mem_preal A)
```
```   533 qed
```
```   534
```
```   535
```
```   536 lemma preal_mult_1_right: "z * (preal_of_rat 1) = z"
```
```   537 apply (rule preal_mult_commute [THEN subst])
```
```   538 apply (rule preal_mult_1)
```
```   539 done
```
```   540
```
```   541
```
```   542 subsection{*Distribution of Multiplication across Addition*}
```
```   543
```
```   544 lemma mem_Rep_preal_add_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_add_def mem_add_set Rep_preal)
```
```   547 apply (simp add: add_set_def)
```
```   548 done
```
```   549
```
```   550 lemma mem_Rep_preal_mult_iff:
```
```   551       "(z \<in> Rep_preal(R*S)) = (\<exists>x \<in> Rep_preal R. \<exists>y \<in> Rep_preal S. z = x * y)"
```
```   552 apply (simp add: preal_mult_def mem_mult_set Rep_preal)
```
```   553 apply (simp add: mult_set_def)
```
```   554 done
```
```   555
```
```   556 lemma distrib_subset1:
```
```   557      "Rep_preal (w * (x + y)) \<subseteq> Rep_preal (w * x + w * y)"
```
```   558 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
```
```   559 apply (force simp add: right_distrib)
```
```   560 done
```
```   561
```
```   562 lemma linorder_le_cases [case_names le ge]:
```
```   563     "((x::'a::linorder) <= y ==> P) ==> (y <= x ==> P) ==> P"
```
```   564   apply (insert linorder_linear, blast)
```
```   565   done
```
```   566
```
```   567 lemma preal_add_mult_distrib_mean:
```
```   568   assumes a: "a \<in> Rep_preal w"
```
```   569       and b: "b \<in> Rep_preal w"
```
```   570       and d: "d \<in> Rep_preal x"
```
```   571       and e: "e \<in> Rep_preal y"
```
```   572      shows "\<exists>c \<in> Rep_preal w. a * d + b * e = c * (d + e)"
```
```   573 proof
```
```   574   let ?c = "(a*d + b*e)/(d+e)"
```
```   575   have [simp]: "0<a" "0<b" "0<d" "0<e" "0<d+e"
```
```   576     by (blast intro: preal_imp_pos [OF Rep_preal] a b d e pos_add_strict)+
```
```   577   have cpos: "0 < ?c"
```
```   578     by (simp add: zero_less_divide_iff zero_less_mult_iff pos_add_strict)
```
```   579   show "a * d + b * e = ?c * (d + e)"
```
```   580     by (simp add: divide_inverse mult_assoc order_less_imp_not_eq2)
```
```   581   show "?c \<in> Rep_preal w"
```
```   582     proof (cases rule: linorder_le_cases)
```
```   583       assume "a \<le> b"
```
```   584       hence "?c \<le> b"
```
```   585 	by (simp add: pos_divide_le_eq right_distrib mult_right_mono
```
```   586                       order_less_imp_le)
```
```   587       thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal b cpos])
```
```   588     next
```
```   589       assume "b \<le> a"
```
```   590       hence "?c \<le> a"
```
```   591 	by (simp add: pos_divide_le_eq right_distrib mult_right_mono
```
```   592                       order_less_imp_le)
```
```   593       thus ?thesis by (rule preal_downwards_closed' [OF Rep_preal a cpos])
```
```   594     qed
```
```   595   qed
```
```   596
```
```   597 lemma distrib_subset2:
```
```   598      "Rep_preal (w * x + w * y) \<subseteq> Rep_preal (w * (x + y))"
```
```   599 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_mult_iff)
```
```   600 apply (drule_tac w=w and x=x and y=y in preal_add_mult_distrib_mean, auto)
```
```   601 done
```
```   602
```
```   603 lemma preal_add_mult_distrib2: "(w * ((x::preal) + y)) = (w * x) + (w * y)"
```
```   604 apply (rule inj_Rep_preal [THEN injD])
```
```   605 apply (rule equalityI [OF distrib_subset1 distrib_subset2])
```
```   606 done
```
```   607
```
```   608 lemma preal_add_mult_distrib: "(((x::preal) + y) * w) = (x * w) + (y * w)"
```
```   609 by (simp add: preal_mult_commute preal_add_mult_distrib2)
```
```   610
```
```   611
```
```   612 subsection{*Existence of Inverse, a Positive Real*}
```
```   613
```
```   614 lemma mem_inv_set_ex:
```
```   615   assumes A: "A \<in> preal" shows "\<exists>x y. 0 < x & x < y & inverse y \<notin> A"
```
```   616 proof -
```
```   617   from preal_exists_bound [OF A]
```
```   618   obtain x where [simp]: "0<x" "x \<notin> A" by blast
```
```   619   show ?thesis
```
```   620   proof (intro exI conjI)
```
```   621     show "0 < inverse (x+1)"
```
```   622       by (simp add: order_less_trans [OF _ less_add_one])
```
```   623     show "inverse(x+1) < inverse x"
```
```   624       by (simp add: less_imp_inverse_less less_add_one)
```
```   625     show "inverse (inverse x) \<notin> A"
```
```   626       by (simp add: order_less_imp_not_eq2)
```
```   627   qed
```
```   628 qed
```
```   629
```
```   630 text{*Part 1 of Dedekind sections definition*}
```
```   631 lemma inverse_set_not_empty:
```
```   632      "A \<in> preal ==> {} \<subset> inverse_set A"
```
```   633 apply (insert mem_inv_set_ex [of A])
```
```   634 apply (auto simp add: inverse_set_def)
```
```   635 done
```
```   636
```
```   637 text{*Part 2 of Dedekind sections definition*}
```
```   638
```
```   639 lemma preal_not_mem_inverse_set_Ex:
```
```   640    assumes A: "A \<in> preal"  shows "\<exists>q. 0 < q & q \<notin> inverse_set A"
```
```   641 proof -
```
```   642   from preal_nonempty [OF A]
```
```   643   obtain x where x: "x \<in> A" and  xpos [simp]: "0<x" ..
```
```   644   show ?thesis
```
```   645   proof (intro exI conjI)
```
```   646     show "0 < inverse x" by simp
```
```   647     show "inverse x \<notin> inverse_set A"
```
```   648     proof -
```
```   649       { fix y::rat
```
```   650 	assume ygt: "inverse x < y"
```
```   651 	have [simp]: "0 < y" by (simp add: order_less_trans [OF _ ygt])
```
```   652 	have iyless: "inverse y < x"
```
```   653 	  by (simp add: inverse_less_imp_less [of x] ygt)
```
```   654 	have "inverse y \<in> A"
```
```   655 	  by (simp add: preal_downwards_closed [OF A x] iyless)}
```
```   656      thus ?thesis by (auto simp add: inverse_set_def)
```
```   657     qed
```
```   658   qed
```
```   659 qed
```
```   660
```
```   661 lemma inverse_set_not_rat_set:
```
```   662    assumes A: "A \<in> preal"  shows "inverse_set A < {r. 0 < r}"
```
```   663 proof
```
```   664   show "inverse_set A \<subseteq> {r. 0 < r}"  by (force simp add: inverse_set_def)
```
```   665 next
```
```   666   show "inverse_set A \<noteq> {r. 0 < r}"
```
```   667     by (insert preal_not_mem_inverse_set_Ex [OF A], blast)
```
```   668 qed
```
```   669
```
```   670 text{*Part 3 of Dedekind sections definition*}
```
```   671 lemma inverse_set_lemma3:
```
```   672      "[|A \<in> preal; u \<in> inverse_set A; 0 < z; z < u|]
```
```   673       ==> z \<in> inverse_set A"
```
```   674 apply (auto simp add: inverse_set_def)
```
```   675 apply (auto intro: order_less_trans)
```
```   676 done
```
```   677
```
```   678 text{*Part 4 of Dedekind sections definition*}
```
```   679 lemma inverse_set_lemma4:
```
```   680      "[|A \<in> preal; y \<in> inverse_set A|] ==> \<exists>u \<in> inverse_set A. y < u"
```
```   681 apply (auto simp add: inverse_set_def)
```
```   682 apply (drule dense [of y])
```
```   683 apply (blast intro: order_less_trans)
```
```   684 done
```
```   685
```
```   686
```
```   687 lemma mem_inverse_set:
```
```   688      "A \<in> preal ==> inverse_set A \<in> preal"
```
```   689 apply (simp (no_asm_simp) add: preal_def cut_def)
```
```   690 apply (blast intro!: inverse_set_not_empty inverse_set_not_rat_set
```
```   691                      inverse_set_lemma3 inverse_set_lemma4)
```
```   692 done
```
```   693
```
```   694
```
```   695 subsection{*Gleason's Lemma 9-3.4, page 122*}
```
```   696
```
```   697 lemma Gleason9_34_exists:
```
```   698   assumes A: "A \<in> preal"
```
```   699       and "\<forall>x\<in>A. x + u \<in> A"
```
```   700       and "0 \<le> z"
```
```   701      shows "\<exists>b\<in>A. b + (of_int z) * u \<in> A"
```
```   702 proof (cases z rule: int_cases)
```
```   703   case (nonneg n)
```
```   704   show ?thesis
```
```   705   proof (simp add: prems, induct n)
```
```   706     case 0
```
```   707       from preal_nonempty [OF A]
```
```   708       show ?case  by force
```
```   709     case (Suc k)
```
```   710       from this obtain b where "b \<in> A" "b + of_nat k * u \<in> A" ..
```
```   711       hence "b + of_int (int k)*u + u \<in> A" by (simp add: prems)
```
```   712       thus ?case by (force simp add: left_distrib add_ac prems)
```
```   713   qed
```
```   714 next
```
```   715   case (neg n)
```
```   716   with prems show ?thesis by simp
```
```   717 qed
```
```   718
```
```   719 lemma Gleason9_34_contra:
```
```   720   assumes A: "A \<in> preal"
```
```   721     shows "[|\<forall>x\<in>A. x + u \<in> A; 0 < u; 0 < y; y \<notin> A|] ==> False"
```
```   722 proof (induct u, induct y)
```
```   723   fix a::int and b::int
```
```   724   fix c::int and d::int
```
```   725   assume bpos [simp]: "0 < b"
```
```   726      and dpos [simp]: "0 < d"
```
```   727      and closed: "\<forall>x\<in>A. x + (Fract c d) \<in> A"
```
```   728      and upos: "0 < Fract c d"
```
```   729      and ypos: "0 < Fract a b"
```
```   730      and notin: "Fract a b \<notin> A"
```
```   731   have cpos [simp]: "0 < c"
```
```   732     by (simp add: zero_less_Fract_iff [OF dpos, symmetric] upos)
```
```   733   have apos [simp]: "0 < a"
```
```   734     by (simp add: zero_less_Fract_iff [OF bpos, symmetric] ypos)
```
```   735   let ?k = "a*d"
```
```   736   have frle: "Fract a b \<le> Fract ?k 1 * (Fract c d)"
```
```   737   proof -
```
```   738     have "?thesis = ((a * d * b * d) \<le> c * b * (a * d * b * d))"
```
```   739       by (simp add: mult_rat le_rat order_less_imp_not_eq2 mult_ac)
```
```   740     moreover
```
```   741     have "(1 * (a * d * b * d)) \<le> c * b * (a * d * b * d)"
```
```   742       by (rule mult_mono,
```
```   743           simp_all add: int_one_le_iff_zero_less zero_less_mult_iff
```
```   744                         order_less_imp_le)
```
```   745     ultimately
```
```   746     show ?thesis by simp
```
```   747   qed
```
```   748   have k: "0 \<le> ?k" by (simp add: order_less_imp_le zero_less_mult_iff)
```
```   749   from Gleason9_34_exists [OF A closed k]
```
```   750   obtain z where z: "z \<in> A"
```
```   751              and mem: "z + of_int ?k * Fract c d \<in> A" ..
```
```   752   have less: "z + of_int ?k * Fract c d < Fract a b"
```
```   753     by (rule not_in_preal_ub [OF A notin mem ypos])
```
```   754   have "0<z" by (rule preal_imp_pos [OF A z])
```
```   755   with frle and less show False by (simp add: Fract_of_int_eq)
```
```   756 qed
```
```   757
```
```   758
```
```   759 lemma Gleason9_34:
```
```   760   assumes A: "A \<in> preal"
```
```   761       and upos: "0 < u"
```
```   762     shows "\<exists>r \<in> A. r + u \<notin> A"
```
```   763 proof (rule ccontr, simp)
```
```   764   assume closed: "\<forall>r\<in>A. r + u \<in> A"
```
```   765   from preal_exists_bound [OF A]
```
```   766   obtain y where y: "y \<notin> A" and ypos: "0 < y" by blast
```
```   767   show False
```
```   768     by (rule Gleason9_34_contra [OF A closed upos ypos y])
```
```   769 qed
```
```   770
```
```   771
```
```   772
```
```   773 subsection{*Gleason's Lemma 9-3.6*}
```
```   774
```
```   775 lemma lemma_gleason9_36:
```
```   776   assumes A: "A \<in> preal"
```
```   777       and x: "1 < x"
```
```   778     shows "\<exists>r \<in> A. r*x \<notin> A"
```
```   779 proof -
```
```   780   from preal_nonempty [OF A]
```
```   781   obtain y where y: "y \<in> A" and  ypos: "0<y" ..
```
```   782   show ?thesis
```
```   783   proof (rule classical)
```
```   784     assume "~(\<exists>r\<in>A. r * x \<notin> A)"
```
```   785     with y have ymem: "y * x \<in> A" by blast
```
```   786     from ypos mult_strict_left_mono [OF x]
```
```   787     have yless: "y < y*x" by simp
```
```   788     let ?d = "y*x - y"
```
```   789     from yless have dpos: "0 < ?d" and eq: "y + ?d = y*x" by auto
```
```   790     from Gleason9_34 [OF A dpos]
```
```   791     obtain r where r: "r\<in>A" and notin: "r + ?d \<notin> A" ..
```
```   792     have rpos: "0<r" by (rule preal_imp_pos [OF A r])
```
```   793     with dpos have rdpos: "0 < r + ?d" by arith
```
```   794     have "~ (r + ?d \<le> y + ?d)"
```
```   795     proof
```
```   796       assume le: "r + ?d \<le> y + ?d"
```
```   797       from ymem have yd: "y + ?d \<in> A" by (simp add: eq)
```
```   798       have "r + ?d \<in> A" by (rule preal_downwards_closed' [OF A yd rdpos le])
```
```   799       with notin show False by simp
```
```   800     qed
```
```   801     hence "y < r" by simp
```
```   802     with ypos have  dless: "?d < (r * ?d)/y"
```
```   803       by (simp add: pos_less_divide_eq mult_commute [of ?d]
```
```   804                     mult_strict_right_mono dpos)
```
```   805     have "r + ?d < r*x"
```
```   806     proof -
```
```   807       have "r + ?d < r + (r * ?d)/y" by (simp add: dless)
```
```   808       also with ypos have "... = (r/y) * (y + ?d)"
```
```   809 	by (simp only: right_distrib divide_inverse mult_ac, simp)
```
```   810       also have "... = r*x" using ypos
```
```   811 	by simp
```
```   812       finally show "r + ?d < r*x" .
```
```   813     qed
```
```   814     with r notin rdpos
```
```   815     show "\<exists>r\<in>A. r * x \<notin> A" by (blast dest:  preal_downwards_closed [OF A])
```
```   816   qed
```
```   817 qed
```
```   818
```
```   819 subsection{*Existence of Inverse: Part 2*}
```
```   820
```
```   821 lemma mem_Rep_preal_inverse_iff:
```
```   822       "(z \<in> Rep_preal(inverse R)) =
```
```   823        (0 < z \<and> (\<exists>y. z < y \<and> inverse y \<notin> Rep_preal R))"
```
```   824 apply (simp add: preal_inverse_def mem_inverse_set Rep_preal)
```
```   825 apply (simp add: inverse_set_def)
```
```   826 done
```
```   827
```
```   828 lemma Rep_preal_of_rat:
```
```   829      "0 < q ==> Rep_preal (preal_of_rat q) = {x. 0 < x \<and> x < q}"
```
```   830 by (simp add: preal_of_rat_def rat_mem_preal)
```
```   831
```
```   832 lemma subset_inverse_mult_lemma:
```
```   833       assumes xpos: "0 < x" and xless: "x < 1"
```
```   834          shows "\<exists>r u y. 0 < r & r < y & inverse y \<notin> Rep_preal R &
```
```   835                         u \<in> Rep_preal R & x = r * u"
```
```   836 proof -
```
```   837   from xpos and xless have "1 < inverse x" by (simp add: one_less_inverse_iff)
```
```   838   from lemma_gleason9_36 [OF Rep_preal this]
```
```   839   obtain r where r: "r \<in> Rep_preal R"
```
```   840              and notin: "r * (inverse x) \<notin> Rep_preal R" ..
```
```   841   have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
```
```   842   from preal_exists_greater [OF Rep_preal r]
```
```   843   obtain u where u: "u \<in> Rep_preal R" and rless: "r < u" ..
```
```   844   have upos: "0<u" by (rule preal_imp_pos [OF Rep_preal u])
```
```   845   show ?thesis
```
```   846   proof (intro exI conjI)
```
```   847     show "0 < x/u" using xpos upos
```
```   848       by (simp add: zero_less_divide_iff)
```
```   849     show "x/u < x/r" using xpos upos rpos
```
```   850       by (simp add: divide_inverse mult_less_cancel_left rless)
```
```   851     show "inverse (x / r) \<notin> Rep_preal R" using notin
```
```   852       by (simp add: divide_inverse mult_commute)
```
```   853     show "u \<in> Rep_preal R" by (rule u)
```
```   854     show "x = x / u * u" using upos
```
```   855       by (simp add: divide_inverse mult_commute)
```
```   856   qed
```
```   857 qed
```
```   858
```
```   859 lemma subset_inverse_mult:
```
```   860      "Rep_preal(preal_of_rat 1) \<subseteq> Rep_preal(inverse R * R)"
```
```   861 apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff
```
```   862                       mem_Rep_preal_mult_iff)
```
```   863 apply (blast dest: subset_inverse_mult_lemma)
```
```   864 done
```
```   865
```
```   866 lemma inverse_mult_subset_lemma:
```
```   867      assumes rpos: "0 < r"
```
```   868          and rless: "r < y"
```
```   869          and notin: "inverse y \<notin> Rep_preal R"
```
```   870          and q: "q \<in> Rep_preal R"
```
```   871      shows "r*q < 1"
```
```   872 proof -
```
```   873   have "q < inverse y" using rpos rless
```
```   874     by (simp add: not_in_preal_ub [OF Rep_preal notin] q)
```
```   875   hence "r * q < r/y" using rpos
```
```   876     by (simp add: divide_inverse mult_less_cancel_left)
```
```   877   also have "... \<le> 1" using rpos rless
```
```   878     by (simp add: pos_divide_le_eq)
```
```   879   finally show ?thesis .
```
```   880 qed
```
```   881
```
```   882 lemma inverse_mult_subset:
```
```   883      "Rep_preal(inverse R * R) \<subseteq> Rep_preal(preal_of_rat 1)"
```
```   884 apply (auto simp add: Bex_def Rep_preal_of_rat mem_Rep_preal_inverse_iff
```
```   885                       mem_Rep_preal_mult_iff)
```
```   886 apply (simp add: zero_less_mult_iff preal_imp_pos [OF Rep_preal])
```
```   887 apply (blast intro: inverse_mult_subset_lemma)
```
```   888 done
```
```   889
```
```   890 lemma preal_mult_inverse:
```
```   891      "inverse R * R = (preal_of_rat 1)"
```
```   892 apply (rule inj_Rep_preal [THEN injD])
```
```   893 apply (rule equalityI [OF inverse_mult_subset subset_inverse_mult])
```
```   894 done
```
```   895
```
```   896 lemma preal_mult_inverse_right:
```
```   897      "R * inverse R = (preal_of_rat 1)"
```
```   898 apply (rule preal_mult_commute [THEN subst])
```
```   899 apply (rule preal_mult_inverse)
```
```   900 done
```
```   901
```
```   902
```
```   903 text{*Theorems needing @{text Gleason9_34}*}
```
```   904
```
```   905 lemma Rep_preal_self_subset: "Rep_preal (R) \<subseteq> Rep_preal(R + S)"
```
```   906 proof
```
```   907   fix r
```
```   908   assume r: "r \<in> Rep_preal R"
```
```   909   have rpos: "0<r" by (rule preal_imp_pos [OF Rep_preal r])
```
```   910   from mem_Rep_preal_Ex
```
```   911   obtain y where y: "y \<in> Rep_preal S" ..
```
```   912   have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
```
```   913   have ry: "r+y \<in> Rep_preal(R + S)" using r y
```
```   914     by (auto simp add: mem_Rep_preal_add_iff)
```
```   915   show "r \<in> Rep_preal(R + S)" using r ypos rpos
```
```   916     by (simp add:  preal_downwards_closed [OF Rep_preal ry])
```
```   917 qed
```
```   918
```
```   919 lemma Rep_preal_sum_not_subset: "~ Rep_preal (R + S) \<subseteq> Rep_preal(R)"
```
```   920 proof -
```
```   921   from mem_Rep_preal_Ex
```
```   922   obtain y where y: "y \<in> Rep_preal S" ..
```
```   923   have ypos: "0<y" by (rule preal_imp_pos [OF Rep_preal y])
```
```   924   from  Gleason9_34 [OF Rep_preal ypos]
```
```   925   obtain r where r: "r \<in> Rep_preal R" and notin: "r + y \<notin> Rep_preal R" ..
```
```   926   have "r + y \<in> Rep_preal (R + S)" using r y
```
```   927     by (auto simp add: mem_Rep_preal_add_iff)
```
```   928   thus ?thesis using notin by blast
```
```   929 qed
```
```   930
```
```   931 lemma Rep_preal_sum_not_eq: "Rep_preal (R + S) \<noteq> Rep_preal(R)"
```
```   932 by (insert Rep_preal_sum_not_subset, blast)
```
```   933
```
```   934 text{*at last, Gleason prop. 9-3.5(iii) page 123*}
```
```   935 lemma preal_self_less_add_left: "(R::preal) < R + S"
```
```   936 apply (unfold preal_less_def psubset_def)
```
```   937 apply (simp add: Rep_preal_self_subset Rep_preal_sum_not_eq [THEN not_sym])
```
```   938 done
```
```   939
```
```   940 lemma preal_self_less_add_right: "(R::preal) < S + R"
```
```   941 by (simp add: preal_add_commute preal_self_less_add_left)
```
```   942
```
```   943 lemma preal_not_eq_self: "x \<noteq> x + (y::preal)"
```
```   944 by (insert preal_self_less_add_left [of x y], auto)
```
```   945
```
```   946
```
```   947 subsection{*Subtraction for Positive Reals*}
```
```   948
```
```   949 text{*Gleason prop. 9-3.5(iv), page 123: proving @{term "A < B ==> \<exists>D. A + D =
```
```   950 B"}. We define the claimed @{term D} and show that it is a positive real*}
```
```   951
```
```   952 text{*Part 1 of Dedekind sections definition*}
```
```   953 lemma diff_set_not_empty:
```
```   954      "R < S ==> {} \<subset> diff_set (Rep_preal S) (Rep_preal R)"
```
```   955 apply (auto simp add: preal_less_def diff_set_def elim!: equalityE)
```
```   956 apply (frule_tac x1 = S in Rep_preal [THEN preal_exists_greater])
```
```   957 apply (drule preal_imp_pos [OF Rep_preal], clarify)
```
```   958 apply (cut_tac a=x and b=u in add_eq_exists, force)
```
```   959 done
```
```   960
```
```   961 text{*Part 2 of Dedekind sections definition*}
```
```   962 lemma diff_set_nonempty:
```
```   963      "\<exists>q. 0 < q & q \<notin> diff_set (Rep_preal S) (Rep_preal R)"
```
```   964 apply (cut_tac X = S in Rep_preal_exists_bound)
```
```   965 apply (erule exE)
```
```   966 apply (rule_tac x = x in exI, auto)
```
```   967 apply (simp add: diff_set_def)
```
```   968 apply (auto dest: Rep_preal [THEN preal_downwards_closed])
```
```   969 done
```
```   970
```
```   971 lemma diff_set_not_rat_set:
```
```   972      "diff_set (Rep_preal S) (Rep_preal R) < {r. 0 < r}" (is "?lhs < ?rhs")
```
```   973 proof
```
```   974   show "?lhs \<subseteq> ?rhs" by (auto simp add: diff_set_def)
```
```   975   show "?lhs \<noteq> ?rhs" using diff_set_nonempty by blast
```
```   976 qed
```
```   977
```
```   978 text{*Part 3 of Dedekind sections definition*}
```
```   979 lemma diff_set_lemma3:
```
```   980      "[|R < S; u \<in> diff_set (Rep_preal S) (Rep_preal R); 0 < z; z < u|]
```
```   981       ==> z \<in> diff_set (Rep_preal S) (Rep_preal R)"
```
```   982 apply (auto simp add: diff_set_def)
```
```   983 apply (rule_tac x=x in exI)
```
```   984 apply (drule Rep_preal [THEN preal_downwards_closed], auto)
```
```   985 done
```
```   986
```
```   987 text{*Part 4 of Dedekind sections definition*}
```
```   988 lemma diff_set_lemma4:
```
```   989      "[|R < S; y \<in> diff_set (Rep_preal S) (Rep_preal R)|]
```
```   990       ==> \<exists>u \<in> diff_set (Rep_preal S) (Rep_preal R). y < u"
```
```   991 apply (auto simp add: diff_set_def)
```
```   992 apply (drule Rep_preal [THEN preal_exists_greater], clarify)
```
```   993 apply (cut_tac a="x+y" and b=u in add_eq_exists, clarify)
```
```   994 apply (rule_tac x="y+xa" in exI)
```
```   995 apply (auto simp add: add_ac)
```
```   996 done
```
```   997
```
```   998 lemma mem_diff_set:
```
```   999      "R < S ==> diff_set (Rep_preal S) (Rep_preal R) \<in> preal"
```
```  1000 apply (unfold preal_def cut_def)
```
```  1001 apply (blast intro!: diff_set_not_empty diff_set_not_rat_set
```
```  1002                      diff_set_lemma3 diff_set_lemma4)
```
```  1003 done
```
```  1004
```
```  1005 lemma mem_Rep_preal_diff_iff:
```
```  1006       "R < S ==>
```
```  1007        (z \<in> Rep_preal(S-R)) =
```
```  1008        (\<exists>x. 0 < x & 0 < z & x \<notin> Rep_preal R & x + z \<in> Rep_preal S)"
```
```  1009 apply (simp add: preal_diff_def mem_diff_set Rep_preal)
```
```  1010 apply (force simp add: diff_set_def)
```
```  1011 done
```
```  1012
```
```  1013
```
```  1014 text{*proving that @{term "R + D \<le> S"}*}
```
```  1015
```
```  1016 lemma less_add_left_lemma:
```
```  1017   assumes Rless: "R < S"
```
```  1018       and a: "a \<in> Rep_preal R"
```
```  1019       and cb: "c + b \<in> Rep_preal S"
```
```  1020       and "c \<notin> Rep_preal R"
```
```  1021       and "0 < b"
```
```  1022       and "0 < c"
```
```  1023   shows "a + b \<in> Rep_preal S"
```
```  1024 proof -
```
```  1025   have "0<a" by (rule preal_imp_pos [OF Rep_preal a])
```
```  1026   moreover
```
```  1027   have "a < c" using prems
```
```  1028     by (blast intro: not_in_Rep_preal_ub )
```
```  1029   ultimately show ?thesis using prems
```
```  1030     by (simp add: preal_downwards_closed [OF Rep_preal cb])
```
```  1031 qed
```
```  1032
```
```  1033 lemma less_add_left_le1:
```
```  1034        "R < (S::preal) ==> R + (S-R) \<le> S"
```
```  1035 apply (auto simp add: Bex_def preal_le_def mem_Rep_preal_add_iff
```
```  1036                       mem_Rep_preal_diff_iff)
```
```  1037 apply (blast intro: less_add_left_lemma)
```
```  1038 done
```
```  1039
```
```  1040 subsection{*proving that @{term "S \<le> R + D"} --- trickier*}
```
```  1041
```
```  1042 lemma lemma_sum_mem_Rep_preal_ex:
```
```  1043      "x \<in> Rep_preal S ==> \<exists>e. 0 < e & x + e \<in> Rep_preal S"
```
```  1044 apply (drule Rep_preal [THEN preal_exists_greater], clarify)
```
```  1045 apply (cut_tac a=x and b=u in add_eq_exists, auto)
```
```  1046 done
```
```  1047
```
```  1048 lemma less_add_left_lemma2:
```
```  1049   assumes Rless: "R < S"
```
```  1050       and x:     "x \<in> Rep_preal S"
```
```  1051       and xnot: "x \<notin>  Rep_preal R"
```
```  1052   shows "\<exists>u v z. 0 < v & 0 < z & u \<in> Rep_preal R & z \<notin> Rep_preal R &
```
```  1053                      z + v \<in> Rep_preal S & x = u + v"
```
```  1054 proof -
```
```  1055   have xpos: "0<x" by (rule preal_imp_pos [OF Rep_preal x])
```
```  1056   from lemma_sum_mem_Rep_preal_ex [OF x]
```
```  1057   obtain e where epos: "0 < e" and xe: "x + e \<in> Rep_preal S" by blast
```
```  1058   from  Gleason9_34 [OF Rep_preal epos]
```
```  1059   obtain r where r: "r \<in> Rep_preal R" and notin: "r + e \<notin> Rep_preal R" ..
```
```  1060   with x xnot xpos have rless: "r < x" by (blast intro: not_in_Rep_preal_ub)
```
```  1061   from add_eq_exists [of r x]
```
```  1062   obtain y where eq: "x = r+y" by auto
```
```  1063   show ?thesis
```
```  1064   proof (intro exI conjI)
```
```  1065     show "r \<in> Rep_preal R" by (rule r)
```
```  1066     show "r + e \<notin> Rep_preal R" by (rule notin)
```
```  1067     show "r + e + y \<in> Rep_preal S" using xe eq by (simp add: add_ac)
```
```  1068     show "x = r + y" by (simp add: eq)
```
```  1069     show "0 < r + e" using epos preal_imp_pos [OF Rep_preal r]
```
```  1070       by simp
```
```  1071     show "0 < y" using rless eq by arith
```
```  1072   qed
```
```  1073 qed
```
```  1074
```
```  1075 lemma less_add_left_le2: "R < (S::preal) ==> S \<le> R + (S-R)"
```
```  1076 apply (auto simp add: preal_le_def)
```
```  1077 apply (case_tac "x \<in> Rep_preal R")
```
```  1078 apply (cut_tac Rep_preal_self_subset [of R], force)
```
```  1079 apply (auto simp add: Bex_def mem_Rep_preal_add_iff mem_Rep_preal_diff_iff)
```
```  1080 apply (blast dest: less_add_left_lemma2)
```
```  1081 done
```
```  1082
```
```  1083 lemma less_add_left: "R < (S::preal) ==> R + (S-R) = S"
```
```  1084 by (blast intro: preal_le_anti_sym [OF less_add_left_le1 less_add_left_le2])
```
```  1085
```
```  1086 lemma less_add_left_Ex: "R < (S::preal) ==> \<exists>D. R + D = S"
```
```  1087 by (fast dest: less_add_left)
```
```  1088
```
```  1089 lemma preal_add_less2_mono1: "R < (S::preal) ==> R + T < S + T"
```
```  1090 apply (auto dest!: less_add_left_Ex simp add: preal_add_assoc)
```
```  1091 apply (rule_tac y1 = D in preal_add_commute [THEN subst])
```
```  1092 apply (auto intro: preal_self_less_add_left simp add: preal_add_assoc [symmetric])
```
```  1093 done
```
```  1094
```
```  1095 lemma preal_add_less2_mono2: "R < (S::preal) ==> T + R < T + S"
```
```  1096 by (auto intro: preal_add_less2_mono1 simp add: preal_add_commute [of T])
```
```  1097
```
```  1098 lemma preal_add_right_less_cancel: "R + T < S + T ==> R < (S::preal)"
```
```  1099 apply (insert linorder_less_linear [of R S], auto)
```
```  1100 apply (drule_tac R = S and T = T in preal_add_less2_mono1)
```
```  1101 apply (blast dest: order_less_trans)
```
```  1102 done
```
```  1103
```
```  1104 lemma preal_add_left_less_cancel: "T + R < T + S ==> R <  (S::preal)"
```
```  1105 by (auto elim: preal_add_right_less_cancel simp add: preal_add_commute [of T])
```
```  1106
```
```  1107 lemma preal_add_less_cancel_right: "((R::preal) + T < S + T) = (R < S)"
```
```  1108 by (blast intro: preal_add_less2_mono1 preal_add_right_less_cancel)
```
```  1109
```
```  1110 lemma preal_add_less_cancel_left: "(T + (R::preal) < T + S) = (R < S)"
```
```  1111 by (blast intro: preal_add_less2_mono2 preal_add_left_less_cancel)
```
```  1112
```
```  1113 lemma preal_add_le_cancel_right: "((R::preal) + T \<le> S + T) = (R \<le> S)"
```
```  1114 by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_right)
```
```  1115
```
```  1116 lemma preal_add_le_cancel_left: "(T + (R::preal) \<le> T + S) = (R \<le> S)"
```
```  1117 by (simp add: linorder_not_less [symmetric] preal_add_less_cancel_left)
```
```  1118
```
```  1119 lemma preal_add_less_mono:
```
```  1120      "[| x1 < y1; x2 < y2 |] ==> x1 + x2 < y1 + (y2::preal)"
```
```  1121 apply (auto dest!: less_add_left_Ex simp add: preal_add_ac)
```
```  1122 apply (rule preal_add_assoc [THEN subst])
```
```  1123 apply (rule preal_self_less_add_right)
```
```  1124 done
```
```  1125
```
```  1126 lemma preal_add_right_cancel: "(R::preal) + T = S + T ==> R = S"
```
```  1127 apply (insert linorder_less_linear [of R S], safe)
```
```  1128 apply (drule_tac [!] T = T in preal_add_less2_mono1, auto)
```
```  1129 done
```
```  1130
```
```  1131 lemma preal_add_left_cancel: "C + A = C + B ==> A = (B::preal)"
```
```  1132 by (auto intro: preal_add_right_cancel simp add: preal_add_commute)
```
```  1133
```
```  1134 lemma preal_add_left_cancel_iff: "(C + A = C + B) = ((A::preal) = B)"
```
```  1135 by (fast intro: preal_add_left_cancel)
```
```  1136
```
```  1137 lemma preal_add_right_cancel_iff: "(A + C = B + C) = ((A::preal) = B)"
```
```  1138 by (fast intro: preal_add_right_cancel)
```
```  1139
```
```  1140 lemmas preal_cancels =
```
```  1141     preal_add_less_cancel_right preal_add_less_cancel_left
```
```  1142     preal_add_le_cancel_right preal_add_le_cancel_left
```
```  1143     preal_add_left_cancel_iff preal_add_right_cancel_iff
```
```  1144
```
```  1145
```
```  1146 subsection{*Completeness of type @{typ preal}*}
```
```  1147
```
```  1148 text{*Prove that supremum is a cut*}
```
```  1149
```
```  1150 text{*Part 1 of Dedekind sections definition*}
```
```  1151
```
```  1152 lemma preal_sup_set_not_empty:
```
```  1153      "P \<noteq> {} ==> {} \<subset> (\<Union>X \<in> P. Rep_preal(X))"
```
```  1154 apply auto
```
```  1155 apply (cut_tac X = x in mem_Rep_preal_Ex, auto)
```
```  1156 done
```
```  1157
```
```  1158
```
```  1159 text{*Part 2 of Dedekind sections definition*}
```
```  1160
```
```  1161 lemma preal_sup_not_exists:
```
```  1162      "\<forall>X \<in> P. X \<le> Y ==> \<exists>q. 0 < q & q \<notin> (\<Union>X \<in> P. Rep_preal(X))"
```
```  1163 apply (cut_tac X = Y in Rep_preal_exists_bound)
```
```  1164 apply (auto simp add: preal_le_def)
```
```  1165 done
```
```  1166
```
```  1167 lemma preal_sup_set_not_rat_set:
```
```  1168      "\<forall>X \<in> P. X \<le> Y ==> (\<Union>X \<in> P. Rep_preal(X)) < {r. 0 < r}"
```
```  1169 apply (drule preal_sup_not_exists)
```
```  1170 apply (blast intro: preal_imp_pos [OF Rep_preal])
```
```  1171 done
```
```  1172
```
```  1173 text{*Part 3 of Dedekind sections definition*}
```
```  1174 lemma preal_sup_set_lemma3:
```
```  1175      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; u \<in> (\<Union>X \<in> P. Rep_preal(X)); 0 < z; z < u|]
```
```  1176       ==> z \<in> (\<Union>X \<in> P. Rep_preal(X))"
```
```  1177 by (auto elim: Rep_preal [THEN preal_downwards_closed])
```
```  1178
```
```  1179 text{*Part 4 of Dedekind sections definition*}
```
```  1180 lemma preal_sup_set_lemma4:
```
```  1181      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y; y \<in> (\<Union>X \<in> P. Rep_preal(X)) |]
```
```  1182           ==> \<exists>u \<in> (\<Union>X \<in> P. Rep_preal(X)). y < u"
```
```  1183 by (blast dest: Rep_preal [THEN preal_exists_greater])
```
```  1184
```
```  1185 lemma preal_sup:
```
```  1186      "[|P \<noteq> {}; \<forall>X \<in> P. X \<le> Y|] ==> (\<Union>X \<in> P. Rep_preal(X)) \<in> preal"
```
```  1187 apply (unfold preal_def cut_def)
```
```  1188 apply (blast intro!: preal_sup_set_not_empty preal_sup_set_not_rat_set
```
```  1189                      preal_sup_set_lemma3 preal_sup_set_lemma4)
```
```  1190 done
```
```  1191
```
```  1192 lemma preal_psup_le:
```
```  1193      "[| \<forall>X \<in> P. X \<le> Y;  x \<in> P |] ==> x \<le> psup P"
```
```  1194 apply (simp (no_asm_simp) add: preal_le_def)
```
```  1195 apply (subgoal_tac "P \<noteq> {}")
```
```  1196 apply (auto simp add: psup_def preal_sup)
```
```  1197 done
```
```  1198
```
```  1199 lemma psup_le_ub: "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> psup P \<le> Y"
```
```  1200 apply (simp (no_asm_simp) add: preal_le_def)
```
```  1201 apply (simp add: psup_def preal_sup)
```
```  1202 apply (auto simp add: preal_le_def)
```
```  1203 done
```
```  1204
```
```  1205 text{*Supremum property*}
```
```  1206 lemma preal_complete:
```
```  1207      "[| P \<noteq> {}; \<forall>X \<in> P. X \<le> Y |] ==> (\<exists>X \<in> P. Z < X) = (Z < psup P)"
```
```  1208 apply (simp add: preal_less_def psup_def preal_sup)
```
```  1209 apply (auto simp add: preal_le_def)
```
```  1210 apply (rename_tac U)
```
```  1211 apply (cut_tac x = U and y = Z in linorder_less_linear)
```
```  1212 apply (auto simp add: preal_less_def)
```
```  1213 done
```
```  1214
```
```  1215
```
```  1216 subsection{*The Embadding from @{typ rat} into @{typ preal}*}
```
```  1217
```
```  1218 lemma preal_of_rat_add_lemma1:
```
```  1219      "[|x < y + z; 0 < x; 0 < y|] ==> x * y * inverse (y + z) < (y::rat)"
```
```  1220 apply (frule_tac c = "y * inverse (y + z) " in mult_strict_right_mono)
```
```  1221 apply (simp add: zero_less_mult_iff)
```
```  1222 apply (simp add: mult_ac)
```
```  1223 done
```
```  1224
```
```  1225 lemma preal_of_rat_add_lemma2:
```
```  1226   assumes "u < x + y"
```
```  1227       and "0 < x"
```
```  1228       and "0 < y"
```
```  1229       and "0 < u"
```
```  1230   shows "\<exists>v w::rat. w < y & 0 < v & v < x & 0 < w & u = v + w"
```
```  1231 proof (intro exI conjI)
```
```  1232   show "u * x * inverse(x+y) < x" using prems
```
```  1233     by (simp add: preal_of_rat_add_lemma1)
```
```  1234   show "u * y * inverse(x+y) < y" using prems
```
```  1235     by (simp add: preal_of_rat_add_lemma1 add_commute [of x])
```
```  1236   show "0 < u * x * inverse (x + y)" using prems
```
```  1237     by (simp add: zero_less_mult_iff)
```
```  1238   show "0 < u * y * inverse (x + y)" using prems
```
```  1239     by (simp add: zero_less_mult_iff)
```
```  1240   show "u = u * x * inverse (x + y) + u * y * inverse (x + y)" using prems
```
```  1241     by (simp add: left_distrib [symmetric] right_distrib [symmetric] mult_ac)
```
```  1242 qed
```
```  1243
```
```  1244 lemma preal_of_rat_add:
```
```  1245      "[| 0 < x; 0 < y|]
```
```  1246       ==> preal_of_rat ((x::rat) + y) = preal_of_rat x + preal_of_rat y"
```
```  1247 apply (unfold preal_of_rat_def preal_add_def)
```
```  1248 apply (simp add: rat_mem_preal)
```
```  1249 apply (rule_tac f = Abs_preal in arg_cong)
```
```  1250 apply (auto simp add: add_set_def)
```
```  1251 apply (blast dest: preal_of_rat_add_lemma2)
```
```  1252 done
```
```  1253
```
```  1254 lemma preal_of_rat_mult_lemma1:
```
```  1255      "[|x < y; 0 < x; 0 < z|] ==> x * z * inverse y < (z::rat)"
```
```  1256 apply (frule_tac c = "z * inverse y" in mult_strict_right_mono)
```
```  1257 apply (simp add: zero_less_mult_iff)
```
```  1258 apply (subgoal_tac "y * (z * inverse y) = z * (y * inverse y)")
```
```  1259 apply (simp_all add: mult_ac)
```
```  1260 done
```
```  1261
```
```  1262 lemma preal_of_rat_mult_lemma2:
```
```  1263   assumes xless: "x < y * z"
```
```  1264       and xpos: "0 < x"
```
```  1265       and ypos: "0 < y"
```
```  1266   shows "x * z * inverse y * inverse z < (z::rat)"
```
```  1267 proof -
```
```  1268   have "0 < y * z" using prems by simp
```
```  1269   hence zpos:  "0 < z" using prems by (simp add: zero_less_mult_iff)
```
```  1270   have "x * z * inverse y * inverse z = x * inverse y * (z * inverse z)"
```
```  1271     by (simp add: mult_ac)
```
```  1272   also have "... = x/y" using zpos
```
```  1273     by (simp add: divide_inverse)
```
```  1274   also have "... < z"
```
```  1275     by (simp add: pos_divide_less_eq [OF ypos] mult_commute)
```
```  1276   finally show ?thesis .
```
```  1277 qed
```
```  1278
```
```  1279 lemma preal_of_rat_mult_lemma3:
```
```  1280   assumes uless: "u < x * y"
```
```  1281       and "0 < x"
```
```  1282       and "0 < y"
```
```  1283       and "0 < u"
```
```  1284   shows "\<exists>v w::rat. v < x & w < y & 0 < v & 0 < w & u = v * w"
```
```  1285 proof -
```
```  1286   from dense [OF uless]
```
```  1287   obtain r where "u < r" "r < x * y" by blast
```
```  1288   thus ?thesis
```
```  1289   proof (intro exI conjI)
```
```  1290   show "u * x * inverse r < x" using prems
```
```  1291     by (simp add: preal_of_rat_mult_lemma1)
```
```  1292   show "r * y * inverse x * inverse y < y" using prems
```
```  1293     by (simp add: preal_of_rat_mult_lemma2)
```
```  1294   show "0 < u * x * inverse r" using prems
```
```  1295     by (simp add: zero_less_mult_iff)
```
```  1296   show "0 < r * y * inverse x * inverse y" using prems
```
```  1297     by (simp add: zero_less_mult_iff)
```
```  1298   have "u * x * inverse r * (r * y * inverse x * inverse y) =
```
```  1299         u * (r * inverse r) * (x * inverse x) * (y * inverse y)"
```
```  1300     by (simp only: mult_ac)
```
```  1301   thus "u = u * x * inverse r * (r * y * inverse x * inverse y)" using prems
```
```  1302     by simp
```
```  1303   qed
```
```  1304 qed
```
```  1305
```
```  1306 lemma preal_of_rat_mult:
```
```  1307      "[| 0 < x; 0 < y|]
```
```  1308       ==> preal_of_rat ((x::rat) * y) = preal_of_rat x * preal_of_rat y"
```
```  1309 apply (unfold preal_of_rat_def preal_mult_def)
```
```  1310 apply (simp add: rat_mem_preal)
```
```  1311 apply (rule_tac f = Abs_preal in arg_cong)
```
```  1312 apply (auto simp add: zero_less_mult_iff mult_strict_mono mult_set_def)
```
```  1313 apply (blast dest: preal_of_rat_mult_lemma3)
```
```  1314 done
```
```  1315
```
```  1316 lemma preal_of_rat_less_iff:
```
```  1317       "[| 0 < x; 0 < y|] ==> (preal_of_rat x < preal_of_rat y) = (x < y)"
```
```  1318 by (force simp add: preal_of_rat_def preal_less_def rat_mem_preal)
```
```  1319
```
```  1320 lemma preal_of_rat_le_iff:
```
```  1321       "[| 0 < x; 0 < y|] ==> (preal_of_rat x \<le> preal_of_rat y) = (x \<le> y)"
```
```  1322 by (simp add: preal_of_rat_less_iff linorder_not_less [symmetric])
```
```  1323
```
```  1324 lemma preal_of_rat_eq_iff:
```
```  1325       "[| 0 < x; 0 < y|] ==> (preal_of_rat x = preal_of_rat y) = (x = y)"
```
```  1326 by (simp add: preal_of_rat_le_iff order_eq_iff)
```
```  1327
```
```  1328
```
```  1329 ML
```
```  1330 {*
```
```  1331 val inj_on_Abs_preal = thm"inj_on_Abs_preal";
```
```  1332 val inj_Rep_preal = thm"inj_Rep_preal";
```
```  1333 val mem_Rep_preal_Ex = thm"mem_Rep_preal_Ex";
```
```  1334 val preal_add_commute = thm"preal_add_commute";
```
```  1335 val preal_add_assoc = thm"preal_add_assoc";
```
```  1336 val preal_add_left_commute = thm"preal_add_left_commute";
```
```  1337 val preal_mult_commute = thm"preal_mult_commute";
```
```  1338 val preal_mult_assoc = thm"preal_mult_assoc";
```
```  1339 val preal_mult_left_commute = thm"preal_mult_left_commute";
```
```  1340 val preal_add_mult_distrib2 = thm"preal_add_mult_distrib2";
```
```  1341 val preal_add_mult_distrib = thm"preal_add_mult_distrib";
```
```  1342 val preal_self_less_add_left = thm"preal_self_less_add_left";
```
```  1343 val preal_self_less_add_right = thm"preal_self_less_add_right";
```
```  1344 val less_add_left = thm"less_add_left";
```
```  1345 val preal_add_less2_mono1 = thm"preal_add_less2_mono1";
```
```  1346 val preal_add_less2_mono2 = thm"preal_add_less2_mono2";
```
```  1347 val preal_add_right_less_cancel = thm"preal_add_right_less_cancel";
```
```  1348 val preal_add_left_less_cancel = thm"preal_add_left_less_cancel";
```
```  1349 val preal_add_right_cancel = thm"preal_add_right_cancel";
```
```  1350 val preal_add_left_cancel = thm"preal_add_left_cancel";
```
```  1351 val preal_add_left_cancel_iff = thm"preal_add_left_cancel_iff";
```
```  1352 val preal_add_right_cancel_iff = thm"preal_add_right_cancel_iff";
```
```  1353 val preal_psup_le = thm"preal_psup_le";
```
```  1354 val psup_le_ub = thm"psup_le_ub";
```
```  1355 val preal_complete = thm"preal_complete";
```
```  1356 val preal_of_rat_add = thm"preal_of_rat_add";
```
```  1357 val preal_of_rat_mult = thm"preal_of_rat_mult";
```
```  1358
```
```  1359 val preal_add_ac = thms"preal_add_ac";
```
```  1360 val preal_mult_ac = thms"preal_mult_ac";
```
```  1361 *}
```
```  1362
```
```  1363 end
```