(* 

    File:	 TLA/Intensional.thy

    Author:      Stephan Merz

    Copyright:   1997 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  =  Prod +

classes

    world < logic    (* Type class of "possible worlds". Concrete types

                        will be provided by children theories. *)

types

    ('a,'w) term = "'w => 'a"    (* Intention: 'w::world *)

    'w form = "'w => bool"

consts

  TrueInt  :: "('w::world form) => prop"             ("(_)" 5)

  (* Holds at *)

  holdsAt  :: "['w::world, 'w form] => bool"   ("(_ |= _)" [100,9] 8)

  (* Lifting base functions to "intensional" level *)

  con      :: "'a => ('w::world => 'a)"               ("(#_)" [100] 99)

  lift     :: "['a => 'b, 'w::world => 'a] => ('w => 'b)"  ("(_[_])")

  lift2    :: "['a => ('b => 'c), 'w::world => 'a, 'w => 'b] => ('w => 'c)" ("(_[_,/ _])")

  lift3    :: "['a => 'b => 'c => 'd, 'w::world => 'a, 'w => 'b, 'w => 'c] => ('w => 'd)" ("(_[_,/ _,/ _])")

  (* Lifted infix functions *)

  IntEqu   :: "['w::world => 'a, 'w => 'a] => 'w form"  ("(_ .=/ _)" [50,51] 50)

  IntNeq   :: "['w::world => 'a, 'w => 'a] => 'w form"  ("(_ .~=/ _)" [50,51] 50)

  NotInt   :: "('w::world) form => 'w form"               ("(.~ _)" [40] 40)

  AndInt   :: "[('w::world) form, 'w form] => 'w form"    ("(_ .&/ _)" [36,35] 35)

  OrInt    :: "[('w::world) form, 'w form] => 'w form"    ("(_ .|/ _)" [31,30] 30)

  ImpInt   :: "[('w::world) form, 'w form] => 'w form"    ("(_ .->/ _)" [26,25] 25)

  IfInt    :: "[('w::world) form, ('a,'w) term, ('a,'w) term] => ('a,'w) term" ("(.if (_)/ .then (_)/ .else (_))" 10)

  PlusInt  :: "[('w::world) => ('a::plus), 'w => 'a] => ('w => 'a)"  ("(_ .+/ _)" [66,65] 65)

  MinusInt :: "[('w::world) => ('a::minus), 'w => 'a] => ('w => 'a)"  ("(_ .-/ _)" [66,65] 65)

  TimesInt :: "[('w::world) => ('a::times), 'w => 'a] => ('w => 'a)"  ("(_ .*/ _)" [71,70] 70)

  LessInt  :: "['w::world => 'a::ord, 'w => 'a] => 'w form"        ("(_/ .< _)"  [50, 51] 50)

  LeqInt   :: "['w::world => 'a::ord, 'w => 'a] => 'w form"        ("(_/ .<= _)" [50, 51] 50)

  (* lifted set membership *)

  memInt   :: "[('a,'w::world) term, ('a set,'w) term] => 'w form"  ("(_/ .: _)" [50, 51] 50)

  (* "Rigid" quantification *)

  RAll     :: "('a => 'w::world form) => 'w form"     (binder "RALL " 10)

  REx      :: "('a => 'w::world form) => 'w form"     (binder "REX " 10)

syntax

  "@tupleInt"    :: "args => ('a * 'b, 'w) term"  ("(1{[_]})")

translations

  "{[x,y,z]}"   == "{[x, {[y,z]} ]}"

  "{[x,y]}"     == "Pair [x, y]"

  "{[x]}"       => "x"

  "u .= v" == "op =[u,v]"

  "u .~= v" == ".~(u .= v)"

  ".~ A"   == "Not[A]"

  "A .& B" == "op &[A,B]"

  "A .| B"  == "op |[A,B]"

  "A .-> B" == "op -->[A,B]"

  ".if A .then u .else v" == "If[A,u,v]"

  "u .+ v"  == "op +[u,v]"

  "u .- v" == "op -[u,v]"

  "u .* v" == "op *[u,v]"

  "a .< b"  == "op < [a,b]"

  "a .<= b" == "op <= [a,b]"

  "a .: A"  == "op :[a,A]"

  "holdsAt w (lift f x)"      == "lift f x w"

  "holdsAt w (lift2 f x y)"   == "lift2 f x y w"

  "holdsAt w (lift3 f x y z)" == "lift3 f x y z w"

  "w |= A"              => "A(w)"

syntax (symbols)

  holdsAt  :: "['w::world, 'w form] => bool"   ("(_ \\<Turnstile> _)" [100,9] 8)

rules

  inteq_reflection   "(x .= y) ==> (x == y)"

  int_valid   "TrueInt(A) == (!! w. w |= A)"

  unl_con     "(#c) w == c"             (* constants *)

  unl_lift    "(f[x]) w == f(x w)"

  unl_lift2   "(f[x,y]) w == f (x w) (y w)"

  unl_lift3   "(f[x, y, z]) w == f (x w) (y w) (z w)"

  unl_Rall    "(RALL x. A(x)) w == ALL x. (w |= A(x))"

  unl_Rex     "(REX x. A(x)) w == EX x. (w |= A(x))"

end