src/HOL/Import/HOL4Compat.thy
 author haftmann Tue Sep 01 15:39:33 2009 +0200 (2009-09-01) changeset 32479 521cc9bf2958 parent 30971 7fbebf75b3ef child 32960 69916a850301 permissions -rw-r--r--
some reorganization of number theory
```     1 (*  Title:      HOL/Import/HOL4Compat.thy
```
```     2     Author:     Sebastian Skalberg (TU Muenchen)
```
```     3 *)
```
```     4
```
```     5 theory HOL4Compat
```
```     6 imports HOL4Setup Complex_Main "~~/src/HOL/Old_Number_Theory/Primes" ContNotDenum
```
```     7 begin
```
```     8
```
```     9 no_notation differentiable (infixl "differentiable" 60)
```
```    10 no_notation sums (infixr "sums" 80)
```
```    11
```
```    12 lemma EXISTS_UNIQUE_DEF: "(Ex1 P) = (Ex P & (ALL x y. P x & P y --> (x = y)))"
```
```    13   by auto
```
```    14
```
```    15 lemma COND_DEF:"(If b t f) = (@x. ((b = True) --> (x = t)) & ((b = False) --> (x = f)))"
```
```    16   by auto
```
```    17
```
```    18 constdefs
```
```    19   LET :: "['a \<Rightarrow> 'b,'a] \<Rightarrow> 'b"
```
```    20   "LET f s == f s"
```
```    21
```
```    22 lemma [hol4rew]: "LET f s = Let s f"
```
```    23   by (simp add: LET_def Let_def)
```
```    24
```
```    25 lemmas [hol4rew] = ONE_ONE_rew
```
```    26
```
```    27 lemma bool_case_DEF: "(bool_case x y b) = (if b then x else y)"
```
```    28   by simp
```
```    29
```
```    30 lemma INR_INL_11: "(ALL y x. (Inl x = Inl y) = (x = y)) & (ALL y x. (Inr x = Inr y) = (x = y))"
```
```    31   by safe
```
```    32
```
```    33 (*lemma INL_neq_INR: "ALL v1 v2. Sum_Type.Inr v2 ~= Sum_Type.Inl v1"
```
```    34   by simp*)
```
```    35
```
```    36 consts
```
```    37   ISL :: "'a + 'b => bool"
```
```    38   ISR :: "'a + 'b => bool"
```
```    39
```
```    40 primrec ISL_def:
```
```    41   "ISL (Inl x) = True"
```
```    42   "ISL (Inr x) = False"
```
```    43
```
```    44 primrec ISR_def:
```
```    45   "ISR (Inl x) = False"
```
```    46   "ISR (Inr x) = True"
```
```    47
```
```    48 lemma ISL: "(ALL x. ISL (Inl x)) & (ALL y. ~ISL (Inr y))"
```
```    49   by simp
```
```    50
```
```    51 lemma ISR: "(ALL x. ISR (Inr x)) & (ALL y. ~ISR (Inl y))"
```
```    52   by simp
```
```    53
```
```    54 consts
```
```    55   OUTL :: "'a + 'b => 'a"
```
```    56   OUTR :: "'a + 'b => 'b"
```
```    57
```
```    58 primrec OUTL_def:
```
```    59   "OUTL (Inl x) = x"
```
```    60
```
```    61 primrec OUTR_def:
```
```    62   "OUTR (Inr x) = x"
```
```    63
```
```    64 lemma OUTL: "OUTL (Inl x) = x"
```
```    65   by simp
```
```    66
```
```    67 lemma OUTR: "OUTR (Inr x) = x"
```
```    68   by simp
```
```    69
```
```    70 lemma sum_case_def: "(ALL f g x. sum_case f g (Inl x) = f x) & (ALL f g y. sum_case f g (Inr y) = g y)"
```
```    71   by simp;
```
```    72
```
```    73 lemma one: "ALL v. v = ()"
```
```    74   by simp;
```
```    75
```
```    76 lemma option_case_def: "(!u f. option_case u f None = u) & (!u f x. option_case u f (Some x) = f x)"
```
```    77   by simp
```
```    78
```
```    79 lemma OPTION_MAP_DEF: "(!f x. Option.map f (Some x) = Some (f x)) & (!f. Option.map f None = None)"
```
```    80   by simp
```
```    81
```
```    82 consts
```
```    83   IS_SOME :: "'a option => bool"
```
```    84   IS_NONE :: "'a option => bool"
```
```    85
```
```    86 primrec IS_SOME_def:
```
```    87   "IS_SOME (Some x) = True"
```
```    88   "IS_SOME None = False"
```
```    89
```
```    90 primrec IS_NONE_def:
```
```    91   "IS_NONE (Some x) = False"
```
```    92   "IS_NONE None = True"
```
```    93
```
```    94 lemma IS_NONE_DEF: "(!x. IS_NONE (Some x) = False) & (IS_NONE None = True)"
```
```    95   by simp
```
```    96
```
```    97 lemma IS_SOME_DEF: "(!x. IS_SOME (Some x) = True) & (IS_SOME None = False)"
```
```    98   by simp
```
```    99
```
```   100 consts
```
```   101   OPTION_JOIN :: "'a option option => 'a option"
```
```   102
```
```   103 primrec OPTION_JOIN_def:
```
```   104   "OPTION_JOIN None = None"
```
```   105   "OPTION_JOIN (Some x) = x"
```
```   106
```
```   107 lemma OPTION_JOIN_DEF: "(OPTION_JOIN None = None) & (ALL x. OPTION_JOIN (Some x) = x)"
```
```   108   by simp;
```
```   109
```
```   110 lemma PAIR: "(fst x,snd x) = x"
```
```   111   by simp
```
```   112
```
```   113 lemma PAIR_MAP: "prod_fun f g p = (f (fst p),g (snd p))"
```
```   114   by (simp add: prod_fun_def split_def)
```
```   115
```
```   116 lemma pair_case_def: "split = split"
```
```   117   ..;
```
```   118
```
```   119 lemma LESS_OR_EQ: "m <= (n::nat) = (m < n | m = n)"
```
```   120   by auto
```
```   121
```
```   122 constdefs
```
```   123   nat_gt :: "nat => nat => bool"
```
```   124   "nat_gt == %m n. n < m"
```
```   125   nat_ge :: "nat => nat => bool"
```
```   126   "nat_ge == %m n. nat_gt m n | m = n"
```
```   127
```
```   128 lemma [hol4rew]: "nat_gt m n = (n < m)"
```
```   129   by (simp add: nat_gt_def)
```
```   130
```
```   131 lemma [hol4rew]: "nat_ge m n = (n <= m)"
```
```   132   by (auto simp add: nat_ge_def nat_gt_def)
```
```   133
```
```   134 lemma GREATER_DEF: "ALL m n. (n < m) = (n < m)"
```
```   135   by simp
```
```   136
```
```   137 lemma GREATER_OR_EQ: "ALL m n. n <= (m::nat) = (n < m | m = n)"
```
```   138   by auto
```
```   139
```
```   140 lemma LESS_DEF: "m < n = (? P. (!n. P (Suc n) --> P n) & P m & ~P n)"
```
```   141 proof safe
```
```   142   assume "m < n"
```
```   143   def P == "%n. n <= m"
```
```   144   have "(!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
```
```   145   proof (auto simp add: P_def)
```
```   146     assume "n <= m"
```
```   147     from prems
```
```   148     show False
```
```   149       by auto
```
```   150   qed
```
```   151   thus "? P. (!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
```
```   152     by auto
```
```   153 next
```
```   154   fix P
```
```   155   assume alln: "!n. P (Suc n) \<longrightarrow> P n"
```
```   156   assume pm: "P m"
```
```   157   assume npn: "~P n"
```
```   158   have "!k q. q + k = m \<longrightarrow> P q"
```
```   159   proof
```
```   160     fix k
```
```   161     show "!q. q + k = m \<longrightarrow> P q"
```
```   162     proof (induct k,simp_all)
```
```   163       show "P m" by fact
```
```   164     next
```
```   165       fix k
```
```   166       assume ind: "!q. q + k = m \<longrightarrow> P q"
```
```   167       show "!q. Suc (q + k) = m \<longrightarrow> P q"
```
```   168       proof (rule+)
```
```   169 	fix q
```
```   170 	assume "Suc (q + k) = m"
```
```   171 	hence "(Suc q) + k = m"
```
```   172 	  by simp
```
```   173 	with ind
```
```   174 	have psq: "P (Suc q)"
```
```   175 	  by simp
```
```   176 	from alln
```
```   177 	have "P (Suc q) --> P q"
```
```   178 	  ..
```
```   179 	with psq
```
```   180 	show "P q"
```
```   181 	  by simp
```
```   182       qed
```
```   183     qed
```
```   184   qed
```
```   185   hence "!q. q + (m - n) = m \<longrightarrow> P q"
```
```   186     ..
```
```   187   hence hehe: "n + (m - n) = m \<longrightarrow> P n"
```
```   188     ..
```
```   189   show "m < n"
```
```   190   proof (rule classical)
```
```   191     assume "~(m<n)"
```
```   192     hence "n <= m"
```
```   193       by simp
```
```   194     with hehe
```
```   195     have "P n"
```
```   196       by simp
```
```   197     with npn
```
```   198     show "m < n"
```
```   199       ..
```
```   200   qed
```
```   201 qed;
```
```   202
```
```   203 constdefs
```
```   204   FUNPOW :: "('a => 'a) => nat => 'a => 'a"
```
```   205   "FUNPOW f n == f ^^ n"
```
```   206
```
```   207 lemma FUNPOW: "(ALL f x. (f ^^ 0) x = x) &
```
```   208   (ALL f n x. (f ^^ Suc n) x = (f ^^ n) (f x))"
```
```   209   by (simp add: funpow_swap1)
```
```   210
```
```   211 lemma [hol4rew]: "FUNPOW f n = f ^^ n"
```
```   212   by (simp add: FUNPOW_def)
```
```   213
```
```   214 lemma ADD: "(!n. (0::nat) + n = n) & (!m n. Suc m + n = Suc (m + n))"
```
```   215   by simp
```
```   216
```
```   217 lemma MULT: "(!n. (0::nat) * n = 0) & (!m n. Suc m * n = m * n + n)"
```
```   218   by simp
```
```   219
```
```   220 lemma SUB: "(!m. (0::nat) - m = 0) & (!m n. (Suc m) - n = (if m < n then 0 else Suc (m - n)))"
```
```   221   by (simp) arith
```
```   222
```
```   223 lemma MAX_DEF: "max (m::nat) n = (if m < n then n else m)"
```
```   224   by (simp add: max_def)
```
```   225
```
```   226 lemma MIN_DEF: "min (m::nat) n = (if m < n then m else n)"
```
```   227   by (simp add: min_def)
```
```   228
```
```   229 lemma DIVISION: "(0::nat) < n --> (!k. (k = k div n * n + k mod n) & k mod n < n)"
```
```   230   by simp
```
```   231
```
```   232 constdefs
```
```   233   ALT_ZERO :: nat
```
```   234   "ALT_ZERO == 0"
```
```   235   NUMERAL_BIT1 :: "nat \<Rightarrow> nat"
```
```   236   "NUMERAL_BIT1 n == n + (n + Suc 0)"
```
```   237   NUMERAL_BIT2 :: "nat \<Rightarrow> nat"
```
```   238   "NUMERAL_BIT2 n == n + (n + Suc (Suc 0))"
```
```   239   NUMERAL :: "nat \<Rightarrow> nat"
```
```   240   "NUMERAL x == x"
```
```   241
```
```   242 lemma [hol4rew]: "NUMERAL ALT_ZERO = 0"
```
```   243   by (simp add: ALT_ZERO_def NUMERAL_def)
```
```   244
```
```   245 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT1 ALT_ZERO) = 1"
```
```   246   by (simp add: ALT_ZERO_def NUMERAL_BIT1_def NUMERAL_def)
```
```   247
```
```   248 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT2 ALT_ZERO) = 2"
```
```   249   by (simp add: ALT_ZERO_def NUMERAL_BIT2_def NUMERAL_def)
```
```   250
```
```   251 lemma EXP: "(!m. m ^ 0 = (1::nat)) & (!m n. m ^ Suc n = m * (m::nat) ^ n)"
```
```   252   by auto
```
```   253
```
```   254 lemma num_case_def: "(!b f. nat_case b f 0 = b) & (!b f n. nat_case b f (Suc n) = f n)"
```
```   255   by simp;
```
```   256
```
```   257 lemma divides_def: "(a::nat) dvd b = (? q. b = q * a)"
```
```   258   by (auto simp add: dvd_def);
```
```   259
```
```   260 lemma list_case_def: "(!v f. list_case v f [] = v) & (!v f a0 a1. list_case v f (a0#a1) = f a0 a1)"
```
```   261   by simp
```
```   262
```
```   263 consts
```
```   264   list_size :: "('a \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> nat"
```
```   265
```
```   266 primrec
```
```   267   "list_size f [] = 0"
```
```   268   "list_size f (a0#a1) = 1 + (f a0 + list_size f a1)"
```
```   269
```
```   270 lemma list_size_def: "(!f. list_size f [] = 0) &
```
```   271          (!f a0 a1. list_size f (a0#a1) = 1 + (f a0 + list_size f a1))"
```
```   272   by simp
```
```   273
```
```   274 lemma list_case_cong: "! M M' v f. M = M' & (M' = [] \<longrightarrow>  v = v') &
```
```   275            (!a0 a1. (M' = a0#a1) \<longrightarrow> (f a0 a1 = f' a0 a1)) -->
```
```   276            (list_case v f M = list_case v' f' M')"
```
```   277 proof clarify
```
```   278   fix M M' v f
```
```   279   assume "M' = [] \<longrightarrow> v = v'"
```
```   280     and "!a0 a1. M' = a0 # a1 \<longrightarrow> f a0 a1 = f' a0 a1"
```
```   281   show "list_case v f M' = list_case v' f' M'"
```
```   282   proof (rule List.list.case_cong)
```
```   283     show "M' = M'"
```
```   284       ..
```
```   285   next
```
```   286     assume "M' = []"
```
```   287     with prems
```
```   288     show "v = v'"
```
```   289       by auto
```
```   290   next
```
```   291     fix a0 a1
```
```   292     assume "M' = a0 # a1"
```
```   293     with prems
```
```   294     show "f a0 a1 = f' a0 a1"
```
```   295       by auto
```
```   296   qed
```
```   297 qed
```
```   298
```
```   299 lemma list_Axiom: "ALL f0 f1. EX fn. (fn [] = f0) & (ALL a0 a1. fn (a0#a1) = f1 a0 a1 (fn a1))"
```
```   300 proof safe
```
```   301   fix f0 f1
```
```   302   def fn == "list_rec f0 f1"
```
```   303   have "fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
```
```   304     by (simp add: fn_def)
```
```   305   thus "EX fn. fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
```
```   306     by auto
```
```   307 qed
```
```   308
```
```   309 lemma list_Axiom_old: "EX! fn. (fn [] = x) & (ALL h t. fn (h#t) = f (fn t) h t)"
```
```   310 proof safe
```
```   311   def fn == "list_rec x (%h t r. f r h t)"
```
```   312   have "fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
```
```   313     by (simp add: fn_def)
```
```   314   thus "EX fn. fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
```
```   315     by auto
```
```   316 next
```
```   317   fix fn1 fn2
```
```   318   assume "ALL h t. fn1 (h # t) = f (fn1 t) h t"
```
```   319   assume "ALL h t. fn2 (h # t) = f (fn2 t) h t"
```
```   320   assume "fn2 [] = fn1 []"
```
```   321   show "fn1 = fn2"
```
```   322   proof
```
```   323     fix xs
```
```   324     show "fn1 xs = fn2 xs"
```
```   325       by (induct xs,simp_all add: prems)
```
```   326   qed
```
```   327 qed
```
```   328
```
```   329 lemma NULL_DEF: "(null [] = True) & (!h t. null (h # t) = False)"
```
```   330   by simp
```
```   331
```
```   332 constdefs
```
```   333   sum :: "nat list \<Rightarrow> nat"
```
```   334   "sum l == foldr (op +) l 0"
```
```   335
```
```   336 lemma SUM: "(sum [] = 0) & (!h t. sum (h#t) = h + sum t)"
```
```   337   by (simp add: sum_def)
```
```   338
```
```   339 lemma APPEND: "(!l. [] @ l = l) & (!l1 l2 h. (h#l1) @ l2 = h# l1 @ l2)"
```
```   340   by simp
```
```   341
```
```   342 lemma FLAT: "(concat [] = []) & (!h t. concat (h#t) = h @ (concat t))"
```
```   343   by simp
```
```   344
```
```   345 lemma LENGTH: "(length [] = 0) & (!h t. length (h#t) = Suc (length t))"
```
```   346   by simp
```
```   347
```
```   348 lemma MAP: "(!f. map f [] = []) & (!f h t. map f (h#t) = f h#map f t)"
```
```   349   by simp
```
```   350
```
```   351 lemma MEM: "(!x. x mem [] = False) & (!x h t. x mem (h#t) = ((x = h) | x mem t))"
```
```   352   by auto
```
```   353
```
```   354 lemma FILTER: "(!P. filter P [] = []) & (!P h t.
```
```   355            filter P (h#t) = (if P h then h#filter P t else filter P t))"
```
```   356   by simp
```
```   357
```
```   358 lemma REPLICATE: "(ALL x. replicate 0 x = []) &
```
```   359   (ALL n x. replicate (Suc n) x = x # replicate n x)"
```
```   360   by simp
```
```   361
```
```   362 constdefs
```
```   363   FOLDR :: "[['a,'b]\<Rightarrow>'b,'b,'a list] \<Rightarrow> 'b"
```
```   364   "FOLDR f e l == foldr f l e"
```
```   365
```
```   366 lemma [hol4rew]: "FOLDR f e l = foldr f l e"
```
```   367   by (simp add: FOLDR_def)
```
```   368
```
```   369 lemma FOLDR: "(!f e. foldr f [] e = e) & (!f e x l. foldr f (x#l) e = f x (foldr f l e))"
```
```   370   by simp
```
```   371
```
```   372 lemma FOLDL: "(!f e. foldl f e [] = e) & (!f e x l. foldl f e (x#l) = foldl f (f e x) l)"
```
```   373   by simp
```
```   374
```
```   375 lemma EVERY_DEF: "(!P. list_all P [] = True) & (!P h t. list_all P (h#t) = (P h & list_all P t))"
```
```   376   by simp
```
```   377
```
```   378 consts
```
```   379   list_exists :: "['a \<Rightarrow> bool,'a list] \<Rightarrow> bool"
```
```   380
```
```   381 primrec
```
```   382   list_exists_Nil: "list_exists P Nil = False"
```
```   383   list_exists_Cons: "list_exists P (x#xs) = (if P x then True else list_exists P xs)"
```
```   384
```
```   385 lemma list_exists_DEF: "(!P. list_exists P [] = False) &
```
```   386          (!P h t. list_exists P (h#t) = (P h | list_exists P t))"
```
```   387   by simp
```
```   388
```
```   389 consts
```
```   390   map2 :: "[['a,'b]\<Rightarrow>'c,'a list,'b list] \<Rightarrow> 'c list"
```
```   391
```
```   392 primrec
```
```   393   map2_Nil: "map2 f [] l2 = []"
```
```   394   map2_Cons: "map2 f (x#xs) l2 = f x (hd l2) # map2 f xs (tl l2)"
```
```   395
```
```   396 lemma MAP2: "(!f. map2 f [] [] = []) & (!f h1 t1 h2 t2. map2 f (h1#t1) (h2#t2) = f h1 h2#map2 f t1 t2)"
```
```   397   by simp
```
```   398
```
```   399 lemma list_INDUCT: "\<lbrakk> P [] ; !t. P t \<longrightarrow> (!h. P (h#t)) \<rbrakk> \<Longrightarrow> !l. P l"
```
```   400 proof
```
```   401   fix l
```
```   402   assume "P []"
```
```   403   assume allt: "!t. P t \<longrightarrow> (!h. P (h # t))"
```
```   404   show "P l"
```
```   405   proof (induct l)
```
```   406     show "P []" by fact
```
```   407   next
```
```   408     fix h t
```
```   409     assume "P t"
```
```   410     with allt
```
```   411     have "!h. P (h # t)"
```
```   412       by auto
```
```   413     thus "P (h # t)"
```
```   414       ..
```
```   415   qed
```
```   416 qed
```
```   417
```
```   418 lemma list_CASES: "(l = []) | (? t h. l = h#t)"
```
```   419   by (induct l,auto)
```
```   420
```
```   421 constdefs
```
```   422   ZIP :: "'a list * 'b list \<Rightarrow> ('a * 'b) list"
```
```   423   "ZIP == %(a,b). zip a b"
```
```   424
```
```   425 lemma ZIP: "(zip [] [] = []) &
```
```   426   (!x1 l1 x2 l2. zip (x1#l1) (x2#l2) = (x1,x2)#zip l1 l2)"
```
```   427   by simp
```
```   428
```
```   429 lemma [hol4rew]: "ZIP (a,b) = zip a b"
```
```   430   by (simp add: ZIP_def)
```
```   431
```
```   432 consts
```
```   433   unzip :: "('a * 'b) list \<Rightarrow> 'a list * 'b list"
```
```   434
```
```   435 primrec
```
```   436   unzip_Nil: "unzip [] = ([],[])"
```
```   437   unzip_Cons: "unzip (xy#xys) = (let zs = unzip xys in (fst xy # fst zs,snd xy # snd zs))"
```
```   438
```
```   439 lemma UNZIP: "(unzip [] = ([],[])) &
```
```   440          (!x l. unzip (x#l) = (fst x#fst (unzip l),snd x#snd (unzip l)))"
```
```   441   by (simp add: Let_def)
```
```   442
```
```   443 lemma REVERSE: "(rev [] = []) & (!h t. rev (h#t) = (rev t) @ [h])"
```
```   444   by simp;
```
```   445
```
```   446 lemma REAL_SUP_ALLPOS: "\<lbrakk> ALL x. P (x::real) \<longrightarrow> 0 < x ; EX x. P x; EX z. ALL x. P x \<longrightarrow> x < z \<rbrakk> \<Longrightarrow> EX s. ALL y. (EX x. P x & y < x) = (y < s)"
```
```   447 proof safe
```
```   448   fix x z
```
```   449   assume allx: "ALL x. P x \<longrightarrow> 0 < x"
```
```   450   assume px: "P x"
```
```   451   assume allx': "ALL x. P x \<longrightarrow> x < z"
```
```   452   have "EX s. ALL y. (EX x : Collect P. y < x) = (y < s)"
```
```   453   proof (rule posreal_complete)
```
```   454     show "ALL x : Collect P. 0 < x"
```
```   455     proof safe
```
```   456       fix x
```
```   457       assume "P x"
```
```   458       from allx
```
```   459       have "P x \<longrightarrow> 0 < x"
```
```   460 	..
```
```   461       thus "0 < x"
```
```   462 	by (simp add: prems)
```
```   463     qed
```
```   464   next
```
```   465     from px
```
```   466     show "EX x. x : Collect P"
```
```   467       by auto
```
```   468   next
```
```   469     from allx'
```
```   470     show "EX y. ALL x : Collect P. x < y"
```
```   471       apply simp
```
```   472       ..
```
```   473   qed
```
```   474   thus "EX s. ALL y. (EX x. P x & y < x) = (y < s)"
```
```   475     by simp
```
```   476 qed
```
```   477
```
```   478 lemma REAL_10: "~((1::real) = 0)"
```
```   479   by simp
```
```   480
```
```   481 lemma REAL_ADD_ASSOC: "(x::real) + (y + z) = x + y + z"
```
```   482   by simp
```
```   483
```
```   484 lemma REAL_MUL_ASSOC: "(x::real) * (y * z) = x * y * z"
```
```   485   by simp
```
```   486
```
```   487 lemma REAL_ADD_LINV:  "-x + x = (0::real)"
```
```   488   by simp
```
```   489
```
```   490 lemma REAL_MUL_LINV: "x ~= (0::real) ==> inverse x * x = 1"
```
```   491   by simp
```
```   492
```
```   493 lemma REAL_LT_TOTAL: "((x::real) = y) | x < y | y < x"
```
```   494   by auto;
```
```   495
```
```   496 lemma [hol4rew]: "real (0::nat) = 0"
```
```   497   by simp
```
```   498
```
```   499 lemma [hol4rew]: "real (1::nat) = 1"
```
```   500   by simp
```
```   501
```
```   502 lemma [hol4rew]: "real (2::nat) = 2"
```
```   503   by simp
```
```   504
```
```   505 lemma real_lte: "((x::real) <= y) = (~(y < x))"
```
```   506   by auto
```
```   507
```
```   508 lemma real_of_num: "((0::real) = 0) & (!n. real (Suc n) = real n + 1)"
```
```   509   by (simp add: real_of_nat_Suc)
```
```   510
```
```   511 lemma abs: "abs (x::real) = (if 0 <= x then x else -x)"
```
```   512   by (simp add: abs_if)
```
```   513
```
```   514 lemma pow: "(!x::real. x ^ 0 = 1) & (!x::real. ALL n. x ^ (Suc n) = x * x ^ n)"
```
```   515   by simp
```
```   516
```
```   517 constdefs
```
```   518   real_gt :: "real => real => bool"
```
```   519   "real_gt == %x y. y < x"
```
```   520
```
```   521 lemma [hol4rew]: "real_gt x y = (y < x)"
```
```   522   by (simp add: real_gt_def)
```
```   523
```
```   524 lemma real_gt: "ALL x (y::real). (y < x) = (y < x)"
```
```   525   by simp
```
```   526
```
```   527 constdefs
```
```   528   real_ge :: "real => real => bool"
```
```   529   "real_ge x y == y <= x"
```
```   530
```
```   531 lemma [hol4rew]: "real_ge x y = (y <= x)"
```
```   532   by (simp add: real_ge_def)
```
```   533
```
```   534 lemma real_ge: "ALL x y. (y <= x) = (y <= x)"
```
```   535   by simp
```
```   536
```
```   537 end
```