(*
File: TLA/Intensional.thy
Author: Stephan Merz
Copyright: 1998 University of Munich
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.
*)
Intensional = Main +
axclass
world < term
(** abstract syntax **)
types
('w,'a) expr = 'w => 'a (* intention: 'w::world, 'a::term *)
'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 ("(#_)" [1000] 999)
"_lift" :: ['a, lift] => lift ("(_<_>)" [1000] 999)
"_lift2" :: ['a, lift, lift] => lift ("(_<_,/ _>)" [1000] 999)
"_lift3" :: ['a, lift, lift, lift] => lift ("(_<_,/ _,/ _>)" [1000] 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] 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) *)
"_RAll" :: [idts, lift] => lift ("(3! _./ _)" [0, 10] 10)
"_REx" :: [idts, lift] => lift ("(3? _./ _)" [0, 10] 10)
"_REx1" :: [idts, lift] => lift ("(3?! _./ _)" [0, 10] 10)
"_ARAll" :: [idts, lift] => lift ("(3ALL _./ _)" [0, 10] 10)
"_AREx" :: [idts, lift] => lift ("(3EX _./ _)" [0, 10] 10)
"_AREx1" :: [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 (op #)"
"_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 |= ! x. A" <= "_RAll x A w"
"w |= ? x. A" <= "_REx x A w"
"w |= ?! x. A" <= "_REx1 x A w"
syntax (symbols)
"_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] 40)
"_liftAnd" :: [lift, lift] => lift (infixr "\\<and>" 35)
"_liftOr" :: [lift, lift] => lift (infixr "\\<or>" 30)
"_liftImp" :: [lift, lift] => lift (infixr "\\<midarrow>\\<rightarrow>" 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)
"_liftNot" :: lift => lift ("\\<not> _" [40] 40)
rules
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 |= ! x. A x == ! x. (w |= A x)"
unl_Rex "w |= ? x. A x == ? x. (w |= A x)"
unl_Rex1 "w |= ?! x. A x == ?! x. (w |= A x)"
end