src/HOL/ex/Tarski.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Tarski.thy	Tue Jul 27 22:34:11 1999 +0200
@@ -0,0 +1,141 @@
+(*  Title:      HOL/ex/Tarski
+    ID:         $Id$
+    Author:     Florian Kammueller, Cambridge University Computer Laboratory
+    Copyright   1999  University of Cambridge
+
+Minimal version of lattice theory plus the full theorem of Tarski:
+   The fixedpoints of a complete lattice themselves form a complete lattice.
+
+Illustrates first-class theories, using the Sigma representation of structures
+*)
+
+Tarski = Main + 
+
+
+record 'a potype = 
+  pset  :: "'a set"
+  order :: "('a * 'a) set"
+
+syntax
+  "@pset" :: "'a potype => 'a set"             ("_ .<A>"  [90] 90)
+  "@order" :: "'a potype => ('a *'a)set"       ("_ .<r>"  [90] 90) 
+
+translations
+  "po.<A>" == "pset po"
+  "po.<r>" == "order po"
+
+constdefs
+  monotone :: "['a => 'a, 'a set, ('a *'a)set] => bool"
+    "monotone f A r == ! x: A. ! y: A. (x, y): r --> ((f x), (f y)) : r"
+
+  least :: "['a => bool, 'a potype] => 'a"
+   "least P po == @ x. x: po.<A> & P x &
+                       (! y: po.<A>. P y --> (x,y): po.<r>)"
+
+  greatest :: "['a => bool, 'a potype] => 'a"
+   "greatest P po == @ x. x: po.<A> & P x &
+                          (! y: po.<A>. P y --> (y,x): po.<r>)"
+
+  lub  :: "['a set, 'a potype] => 'a"
+   "lub S po == least (%x. ! y: S. (y,x): po.<r>) po"
+
+  glb  :: "['a set, 'a potype] => 'a"
+   "glb S po == greatest (%x. ! y: S. (x,y): po.<r>) po"
+
+  islub :: "['a set, 'a potype, 'a] => bool"
+   "islub S po == %L. (L: po.<A> & (! y: S. (y,L): po.<r>) &
+                      (! z:po.<A>. (! y: S. (y,z): po.<r>) --> (L,z): po.<r>))"
+
+  isglb :: "['a set, 'a potype, 'a] => bool"
+   "isglb S po == %G. (G: po.<A> & (! y: S. (G,y): po.<r>) &
+                     (! z: po.<A>. (! y: S. (z,y): po.<r>) --> (z,G): po.<r>))"
+
+  fix    :: "[('a => 'a), 'a set] => 'a set"
+   "fix f A  == {x. x: A & f x = x}"
+
+  interval :: "[('a*'a) set,'a, 'a ] => 'a set"
+   "interval r a b == {x. (a,x): r & (x,b): r}"
+
+
+constdefs
+  Bot :: "'a potype => 'a"
+   "Bot po == least (%x. True) po"
+
+  Top :: "'a potype => 'a"
+   "Top po == greatest (%x. True) po"
+
+  PartialOrder :: "('a potype) set"
+   "PartialOrder == {P. refl (P.<A>) (P.<r>) & antisym (P.<r>) &
+		        trans (P.<r>)}"
+
+  CompleteLattice :: "('a potype) set"
+   "CompleteLattice == {cl. cl: PartialOrder & 
+			(! S. S <= cl.<A> --> (? L. islub S cl L)) &
+			(! S. S <= cl.<A> --> (? G. isglb S cl G))}"
+
+  CLF :: "('a potype * ('a => 'a)) set"
+   "CLF == SIGMA cl: CompleteLattice.
+             {f. f: cl.<A> funcset cl.<A> & monotone f (cl.<A>) (cl.<r>)}"
+  
+  induced :: "['a set, ('a * 'a) set] => ('a *'a)set"
+   "induced A r == {(a,b). a : A & b: A & (a,b): r}"
+
+
+
+
+constdefs
+  sublattice :: "('a potype * 'a set)set"
+   "sublattice == 
+      SIGMA cl: CompleteLattice.
+          {S. S <= cl.<A> &
+	   (| pset = S, order = induced S (cl.<r>) |): CompleteLattice }"
+
+syntax
+  "@SL"  :: "['a set, 'a potype] => bool" ("_ <<= _" [51,50]50)
+
+translations
+  "S <<= cl" == "S : sublattice ^^ {cl}"
+
+constdefs
+  dual :: "'a potype => 'a potype"
+   "dual po == (| pset = po.<A>, order = converse (po.<r>) |)"
+
+locale PO = 
+fixes 
+  cl :: "'a potype"
+  A  :: "'a set"
+  r  :: "('a * 'a) set"
+assumes 
+  cl_po  "cl : PartialOrder"
+defines
+  A_def "A == cl.<A>"
+  r_def "r == cl.<r>"
+
+locale CL = PO +
+fixes 
+assumes 
+  cl_co  "cl : CompleteLattice"
+
+locale CLF = CL +
+fixes
+  f :: "'a => 'a"
+  P :: "'a set"
+assumes 
+  f_cl "f : CLF ^^{cl}"
+defines
+  P_def "P == fix f A"
+
+
+locale Tarski = CLF + 
+fixes
+  Y :: "'a set"
+  intY1 :: "'a set"
+  v     :: "'a"
+assumes
+  Y_ss "Y <= P"
+defines
+  intY1_def "intY1 == interval r (lub Y cl) (Top cl)"
+  v_def "v == glb {x. ((lam x: intY1. f x) x, x): induced intY1 r & x: intY1}
+	          (| pset=intY1, order=induced intY1 r|)"
+
+end