src/HOL/TLA/Intensional.thy
 author berghofe Fri, 13 Oct 2006 18:15:18 +0200 changeset 21020 9af9ceb16d58 parent 17309 c43ed29bd197 child 21624 6f79647cf536 permissions -rw-r--r--
Adapted to changes in FixedPoint theory.
```
(*
File:        TLA/Intensional.thy
ID:          \$Id\$
Author:      Stephan Merz

Theory Name: Intensional
Logic Image: HOL

Define a framework for "intensional" (possible-world based) logics
on top of HOL, with lifting of constants and functions.
*)

theory Intensional
imports Main
begin

axclass
world < type

(** abstract syntax **)

types
('w,'a) expr = "'w => 'a"               (* intention: 'w::world, 'a::type *)
'w form = "('w, bool) expr"

consts
Valid    :: "('w::world) form => bool"
const    :: "'a => ('w::world, 'a) expr"
lift     :: "['a => 'b, ('w::world, 'a) expr] => ('w,'b) expr"
lift2    :: "['a => 'b => 'c, ('w::world,'a) expr, ('w,'b) expr] => ('w,'c) expr"
lift3    :: "['a => 'b => 'c => 'd, ('w::world,'a) expr, ('w,'b) expr, ('w,'c) expr] => ('w,'d) expr"

(* "Rigid" quantification (logic level) *)
RAll     :: "('a => ('w::world) form) => 'w form"       (binder "Rall " 10)
REx      :: "('a => ('w::world) form) => 'w form"       (binder "Rex " 10)
REx1     :: "('a => ('w::world) form) => 'w form"       (binder "Rex! " 10)

(** concrete syntax **)

nonterminals
lift
liftargs

syntax
""            :: "id => lift"                          ("_")
""            :: "longid => lift"                      ("_")
""            :: "var => lift"                         ("_")
"_applC"      :: "[lift, cargs] => lift"               ("(1_/ _)" [1000, 1000] 999)
""            :: "lift => lift"                        ("'(_')")
"_lambda"     :: "[idts, 'a] => lift"                  ("(3%_./ _)" [0, 3] 3)
"_constrain"  :: "[lift, type] => lift"                ("(_::_)" [4, 0] 3)
""            :: "lift => liftargs"                    ("_")
"_liftargs"   :: "[lift, liftargs] => liftargs"        ("_,/ _")
"_Valid"      :: "lift => bool"                        ("(|- _)" 5)
"_holdsAt"    :: "['a, lift] => bool"                  ("(_ |= _)" [100,10] 10)

(* Syntax for lifted expressions outside the scope of |- or |= *)
"LIFT"        :: "lift => 'a"                          ("LIFT _")

(* generic syntax for lifted constants and functions *)
"_const"      :: "'a => lift"                          ("(#_)"  999)
"_lift"       :: "['a, lift] => lift"                  ("(_<_>)"  999)
"_lift2"      :: "['a, lift, lift] => lift"            ("(_<_,/ _>)"  999)
"_lift3"      :: "['a, lift, lift, lift] => lift"      ("(_<_,/ _,/ _>)"  999)

(* concrete syntax for common infix functions: reuse same symbol *)
"_liftEqu"    :: "[lift, lift] => lift"                ("(_ =/ _)" [50,51] 50)
"_liftNeq"    :: "[lift, lift] => lift"                ("(_ ~=/ _)" [50,51] 50)
"_liftNot"    :: "lift => lift"                        ("(~ _)"  40)
"_liftAnd"    :: "[lift, lift] => lift"                ("(_ &/ _)" [36,35] 35)
"_liftOr"     :: "[lift, lift] => lift"                ("(_ |/ _)" [31,30] 30)
"_liftImp"    :: "[lift, lift] => lift"                ("(_ -->/ _)" [26,25] 25)
"_liftIf"     :: "[lift, lift, lift] => lift"          ("(if (_)/ then (_)/ else (_))" 10)
"_liftPlus"   :: "[lift, lift] => lift"                ("(_ +/ _)" [66,65] 65)
"_liftMinus"  :: "[lift, lift] => lift"                ("(_ -/ _)" [66,65] 65)
"_liftTimes"  :: "[lift, lift] => lift"                ("(_ */ _)" [71,70] 70)
"_liftDiv"    :: "[lift, lift] => lift"                ("(_ div _)" [71,70] 70)
"_liftMod"    :: "[lift, lift] => lift"                ("(_ mod _)" [71,70] 70)
"_liftLess"   :: "[lift, lift] => lift"                ("(_/ < _)"  [50, 51] 50)
"_liftLeq"    :: "[lift, lift] => lift"                ("(_/ <= _)" [50, 51] 50)
"_liftMem"    :: "[lift, lift] => lift"                ("(_/ : _)" [50, 51] 50)
"_liftNotMem" :: "[lift, lift] => lift"                ("(_/ ~: _)" [50, 51] 50)
"_liftFinset" :: "liftargs => lift"                    ("{(_)}")
(** TODO: syntax for lifted collection / comprehension **)
"_liftPair"   :: "[lift,liftargs] => lift"                   ("(1'(_,/ _'))")
(* infix syntax for list operations *)
"_liftCons" :: "[lift, lift] => lift"                  ("(_ #/ _)" [65,66] 65)
"_liftApp"  :: "[lift, lift] => lift"                  ("(_ @/ _)" [65,66] 65)
"_liftList" :: "liftargs => lift"                      ("[(_)]")

(* Rigid quantification (syntax level) *)
"_ARAll"  :: "[idts, lift] => lift"                    ("(3! _./ _)" [0, 10] 10)
"_AREx"   :: "[idts, lift] => lift"                    ("(3? _./ _)" [0, 10] 10)
"_AREx1"  :: "[idts, lift] => lift"                    ("(3?! _./ _)" [0, 10] 10)
"_RAll" :: "[idts, lift] => lift"                      ("(3ALL _./ _)" [0, 10] 10)
"_REx"  :: "[idts, lift] => lift"                      ("(3EX _./ _)" [0, 10] 10)
"_REx1" :: "[idts, lift] => lift"                      ("(3EX! _./ _)" [0, 10] 10)

translations
"_const"        == "const"
"_lift"         == "lift"
"_lift2"        == "lift2"
"_lift3"        == "lift3"
"_Valid"        == "Valid"
"_RAll x A"     == "Rall x. A"
"_REx x  A"     == "Rex x. A"
"_REx1 x  A"    == "Rex! x. A"
"_ARAll"        => "_RAll"
"_AREx"         => "_REx"
"_AREx1"        => "_REx1"

"w |= A"        => "A w"
"LIFT A"        => "A::_=>_"

"_liftEqu"      == "_lift2 (op =)"
"_liftNeq u v"  == "_liftNot (_liftEqu u v)"
"_liftNot"      == "_lift Not"
"_liftAnd"      == "_lift2 (op &)"
"_liftOr"       == "_lift2 (op | )"
"_liftImp"      == "_lift2 (op -->)"
"_liftIf"       == "_lift3 If"
"_liftPlus"     == "_lift2 (op +)"
"_liftMinus"    == "_lift2 (op -)"
"_liftTimes"    == "_lift2 (op *)"
"_liftDiv"      == "_lift2 (op div)"
"_liftMod"      == "_lift2 (op mod)"
"_liftLess"     == "_lift2 (op <)"
"_liftLeq"      == "_lift2 (op <=)"
"_liftMem"      == "_lift2 (op :)"
"_liftNotMem x xs"   == "_liftNot (_liftMem x xs)"
"_liftFinset (_liftargs x xs)"  == "_lift2 insert x (_liftFinset xs)"
"_liftFinset x" == "_lift2 insert x (_const {})"
"_liftPair x (_liftargs y z)"       == "_liftPair x (_liftPair y z)"
"_liftPair"     == "_lift2 Pair"
"_liftCons"     == "lift2 Cons"
"_liftApp"      == "lift2 (op @)"
"_liftList (_liftargs x xs)"  == "_liftCons x (_liftList xs)"
"_liftList x"   == "_liftCons x (_const [])"

"w |= ~A"       <= "_liftNot A w"
"w |= A & B"    <= "_liftAnd A B w"
"w |= A | B"    <= "_liftOr A B w"
"w |= A --> B"  <= "_liftImp A B w"
"w |= u = v"    <= "_liftEqu u v w"
"w |= ALL x. A"   <= "_RAll x A w"
"w |= EX x. A"   <= "_REx x A w"
"w |= EX! x. A"  <= "_REx1 x A w"

syntax (xsymbols)
"_Valid"      :: "lift => bool"                        ("(\<turnstile> _)" 5)
"_holdsAt"    :: "['a, lift] => bool"                  ("(_ \<Turnstile> _)" [100,10] 10)
"_liftNeq"    :: "[lift, lift] => lift"                (infixl "\<noteq>" 50)
"_liftNot"    :: "lift => lift"                        ("\<not> _"  40)
"_liftAnd"    :: "[lift, lift] => lift"                (infixr "\<and>" 35)
"_liftOr"     :: "[lift, lift] => lift"                (infixr "\<or>" 30)
"_liftImp"    :: "[lift, lift] => lift"                (infixr "\<longrightarrow>" 25)
"_RAll"       :: "[idts, lift] => lift"                ("(3\<forall>_./ _)" [0, 10] 10)
"_REx"        :: "[idts, lift] => lift"                ("(3\<exists>_./ _)" [0, 10] 10)
"_REx1"       :: "[idts, lift] => lift"                ("(3\<exists>!_./ _)" [0, 10] 10)
"_liftLeq"    :: "[lift, lift] => lift"                ("(_/ \<le> _)" [50, 51] 50)
"_liftMem"    :: "[lift, lift] => lift"                ("(_/ \<in> _)" [50, 51] 50)
"_liftNotMem" :: "[lift, lift] => lift"                ("(_/ \<notin> _)" [50, 51] 50)

syntax (HTML output)
"_liftNeq"    :: "[lift, lift] => lift"                (infixl "\<noteq>" 50)
"_liftNot"    :: "lift => lift"                        ("\<not> _"  40)
"_liftAnd"    :: "[lift, lift] => lift"                (infixr "\<and>" 35)
"_liftOr"     :: "[lift, lift] => lift"                (infixr "\<or>" 30)
"_RAll"       :: "[idts, lift] => lift"                ("(3\<forall>_./ _)" [0, 10] 10)
"_REx"        :: "[idts, lift] => lift"                ("(3\<exists>_./ _)" [0, 10] 10)
"_REx1"       :: "[idts, lift] => lift"                ("(3\<exists>!_./ _)" [0, 10] 10)
"_liftLeq"    :: "[lift, lift] => lift"                ("(_/ \<le> _)" [50, 51] 50)
"_liftMem"    :: "[lift, lift] => lift"                ("(_/ \<in> _)" [50, 51] 50)
"_liftNotMem" :: "[lift, lift] => lift"                ("(_/ \<notin> _)" [50, 51] 50)

axioms
Valid_def:   "|- A    ==  ALL w. w |= A"

unl_con:     "LIFT #c w  ==  c"
unl_lift:    "lift f x w == f (x w)"
unl_lift2:   "LIFT f<x, y> w == f (x w) (y w)"
unl_lift3:   "LIFT f<x, y, z> w == f (x w) (y w) (z w)"

unl_Rall:    "w |= ALL x. A x  ==  ALL x. (w |= A x)"
unl_Rex:     "w |= EX x. A x   ==  EX x. (w |= A x)"
unl_Rex1:    "w |= EX! x. A x  ==  EX! x. (w |= A x)"

ML {* use_legacy_bindings (the_context ()) *}

end
```