Theory Phantom_Type

(*  Title:      HOL/Library/Phantom_Type.thy
    Author:     Andreas Lochbihler
*)

section ‹A generic phantom type›

theory Phantom_Type
imports Main
begin

datatype ('a, 'b) phantom = phantom (of_phantom: 'b)

lemma type_definition_phantom': "type_definition of_phantom phantom UNIV"
by(unfold_locales) simp_all

lemma phantom_comp_of_phantom [simp]: "phantom ∘ of_phantom = id"
  and of_phantom_comp_phantom [simp]: "of_phantom ∘ phantom = id"
by(simp_all add: o_def id_def)

syntax "_Phantom" :: "type ⇒ logic" ("(1Phantom/(1'(_')))")
translations
  "Phantom('t)" => "CONST phantom :: _ ⇒ ('t, _) phantom"

typed_print_translation ‹
  let
    fun phantom_tr' ctxt (Type (\<^type_name>‹fun›, [_, Type (\<^type_name>‹phantom›, [T, _])])) ts =
          list_comb
            (Syntax.const \<^syntax_const>‹_Phantom› $ Syntax_Phases.term_of_typ ctxt T, ts)
      | phantom_tr' _ _ _ = raise Match;
  in [(\<^const_syntax>‹phantom›, phantom_tr')] end
›

lemma of_phantom_inject [simp]:
  "of_phantom x = of_phantom y ⟷ x = y"
by(cases x y rule: phantom.exhaust[case_product phantom.exhaust]) simp

end