src/HOL/Ord.thy

added if_def2

(*  Title:      HOL/Ord.thy
    ID:         $Id$
    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

Type classes for order signatures and orders.
*)

Ord = HOL +


axclass
  ord < term

syntax
  "op <"        :: ['a::ord, 'a] => bool             ("op <")
  "op <="       :: ['a::ord, 'a] => bool             ("op <=")

global

consts
  "op <"        :: ['a::ord, 'a] => bool             ("(_/ < _)"  [50, 51] 50)
  "op <="       :: ['a::ord, 'a] => bool             ("(_/ <= _)" [50, 51] 50)

local

syntax (symbols)
  "op <="       :: ['a::ord, 'a] => bool             ("op \\<le>")
  "op <="       :: ['a::ord, 'a] => bool             ("(_/ \\<le> _)"  [50, 51] 50)

consts
  mono          :: ['a::ord => 'b::ord] => bool      (*monotonicity*)
  min, max      :: ['a::ord, 'a] => 'a
  Least         :: ('a::ord=>bool) => 'a             (binder "LEAST " 10)

defs
  mono_def      "mono(f) == (!A B. A <= B --> f(A) <= f(B))"
  min_def       "min a b == (if a <= b then a else b)"
  max_def       "max a b == (if a <= b then b else a)"
  Least_def     "Least P == @x. P(x) & (ALL y. P(y) --> x <= y)"


axclass order < ord
  order_refl    "x <= x"
  order_trans   "[| x <= y; y <= z |] ==> x <= z"
  order_antisym "[| x <= y; y <= x |] ==> x = y"
  order_less_le "x < y = (x <= y & x ~= y)"

axclass linorder < order
  linorder_linear "x <= y | y <= x"


(* bounded quantifiers *)

syntax
  "_lessAll" :: [idt, 'a, bool] => bool   ("(3ALL _<_./ _)"  [0, 0, 10] 10)
  "_lessEx"  :: [idt, 'a, bool] => bool   ("(3EX _<_./ _)"  [0, 0, 10] 10)
  "_leAll"   :: [idt, 'a, bool] => bool   ("(3ALL _<=_./ _)" [0, 0, 10] 10)
  "_leEx"    :: [idt, 'a, bool] => bool   ("(3EX _<=_./ _)" [0, 0, 10] 10)

syntax (symbols)
  "_lessAll" :: [idt, 'a, bool] => bool   ("(3\\<forall>_<_./ _)"  [0, 0, 10] 10)
  "_lessEx"  :: [idt, 'a, bool] => bool   ("(3\\<exists>_<_./ _)"  [0, 0, 10] 10)
  "_leAll"   :: [idt, 'a, bool] => bool   ("(3\\<forall>_\\<le>_./ _)" [0, 0, 10] 10)
  "_leEx"    :: [idt, 'a, bool] => bool   ("(3\\<exists>_\\<le>_./ _)" [0, 0, 10] 10)

syntax (HOL)
  "_lessAll" :: [idt, 'a, bool] => bool   ("(3! _<_./ _)"  [0, 0, 10] 10)
  "_lessEx"  :: [idt, 'a, bool] => bool   ("(3? _<_./ _)"  [0, 0, 10] 10)
  "_leAll"   :: [idt, 'a, bool] => bool   ("(3! _<=_./ _)" [0, 0, 10] 10)
  "_leEx"    :: [idt, 'a, bool] => bool   ("(3? _<=_./ _)" [0, 0, 10] 10)

translations
 "ALL x<y. P"   =>  "ALL x. x < y --> P"
 "EX x<y. P"    =>  "EX x. x < y  & P"
 "ALL x<=y. P"  =>  "ALL x. x <= y --> P"
 "EX x<=y. P"   =>  "EX x. x <= y & P"


end