--- a/src/HOL/ex/Tarski.thy Tue Jul 27 22:32:22 1999 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,141 +0,0 @@
-(* 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