(* Authors: Julien Narboux and Christian Urban
This file introduces the infrastructure for the lemma
collection "eqvts".
By attaching [eqvt] or [eqvt_force] to a lemma, it will get
stored in a data-slot in the context. Possible modifiers
are [... add] and [... del] for adding and deleting,
respectively, the lemma from the data-slot.
*)
signature NOMINAL_THMDECLS =
sig
val eqvt_add: attribute
val eqvt_del: attribute
val eqvt_force_add: attribute
val eqvt_force_del: attribute
val setup: theory -> theory
val get_eqvt_thms: Proof.context -> thm list
val NOMINAL_EQVT_DEBUG : bool Unsynchronized.ref
end;
structure NominalThmDecls: NOMINAL_THMDECLS =
struct
structure Data = GenericDataFun
(
type T = thm list
val empty = []:T
val extend = I
fun merge _ (r1:T, r2:T) = Thm.merge_thms (r1, r2)
)
(* Exception for when a theorem does not conform with form of an equivariance lemma. *)
(* There are two forms: one is an implication (for relations) and the other is an *)
(* equality (for functions). In the implication-case, say P ==> Q, Q must be equal *)
(* to P except that every free variable of Q, say x, is replaced by pi o x. In the *)
(* equality case, say lhs = rhs, the lhs must be of the form pi o t and the rhs must *)
(* be equal to t except that every free variable, say x, is replaced by pi o x. In *)
(* the implicational case it is also checked that the variables and permutation fit *)
(* together, i.e. are of the right "pt_class", so that a stronger version of the *)
(* equality-lemma can be derived. *)
exception EQVT_FORM of string
val NOMINAL_EQVT_DEBUG = Unsynchronized.ref false
fun tactic (msg, tac) =
if !NOMINAL_EQVT_DEBUG
then tac THEN' (K (print_tac ("after " ^ msg)))
else tac
fun prove_eqvt_tac ctxt orig_thm pi pi' =
let
val mypi = Thm.cterm_of ctxt pi
val T = fastype_of pi'
val mypifree = Thm.cterm_of ctxt (Const (@{const_name "rev"}, T --> T) $ pi')
val perm_pi_simp = PureThy.get_thms ctxt "perm_pi_simp"
in
EVERY1 [tactic ("iffI applied", rtac @{thm iffI}),
tactic ("remove pi with perm_boolE", dtac @{thm perm_boolE}),
tactic ("solve with orig_thm", etac orig_thm),
tactic ("applies orig_thm instantiated with rev pi",
dtac (Drule.cterm_instantiate [(mypi,mypifree)] orig_thm)),
tactic ("getting rid of the pi on the right", rtac @{thm perm_boolI}),
tactic ("getting rid of all remaining perms",
full_simp_tac (HOL_basic_ss addsimps perm_pi_simp))]
end;
fun get_derived_thm ctxt hyp concl orig_thm pi typi =
let
val thy = ProofContext.theory_of ctxt;
val pi' = Var (pi, typi);
val lhs = Const (@{const_name "perm"}, typi --> HOLogic.boolT --> HOLogic.boolT) $ pi' $ hyp;
val ([goal_term, pi''], ctxt') = Variable.import_terms false
[HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, concl)), pi'] ctxt
val _ = writeln (Syntax.string_of_term ctxt' goal_term);
in
Goal.prove ctxt' [] [] goal_term
(fn _ => prove_eqvt_tac thy orig_thm pi' pi'') |>
singleton (ProofContext.export ctxt' ctxt)
end
(* replaces in t every variable, say x, with pi o x *)
fun apply_pi trm (pi, typi) =
let
fun replace n ty =
let
val c = Const (@{const_name "perm"}, typi --> ty --> ty)
val v1 = Var (pi, typi)
val v2 = Var (n, ty)
in
c $ v1 $ v2
end
in
map_aterms (fn Var (n, ty) => replace n ty | t => t) trm
end
(* returns *the* pi which is in front of all variables, provided there *)
(* exists such a pi; otherwise raises EQVT_FORM *)
fun get_pi t thy =
let fun get_pi_aux s =
(case s of
(Const (@{const_name "perm"} ,typrm) $
(Var (pi,typi as Type(@{type_name "list"}, [Type ("*", [Type (tyatm,[]),_])]))) $
(Var (n,ty))) =>
let
(* FIXME: this should be an operation the library *)
val class_name = (Long_Name.map_base_name (fn s => "pt_"^s) tyatm)
in
if (Sign.of_sort thy (ty,[class_name]))
then [(pi,typi)]
else raise
EQVT_FORM ("Could not find any permutation or an argument is not an instance of "^class_name)
end
| Abs (_,_,t1) => get_pi_aux t1
| (t1 $ t2) => get_pi_aux t1 @ get_pi_aux t2
| _ => [])
in
(* collect first all pi's in front of variables in t and then use distinct *)
(* to ensure that all pi's must have been the same, i.e. distinct returns *)
(* a singleton-list *)
(case (distinct (op =) (get_pi_aux t)) of
[(pi,typi)] => (pi, typi)
| _ => raise EQVT_FORM "All permutation should be the same")
end;
(* Either adds a theorem (orig_thm) to or deletes one from the equivariance *)
(* lemma list depending on flag. To be added the lemma has to satisfy a *)
(* certain form. *)
fun eqvt_add_del_aux flag orig_thm context =
let
val thy = Context.theory_of context
val thms_to_be_added = (case (prop_of orig_thm) of
(* case: eqvt-lemma is of the implicational form *)
(Const("==>", _) $ (Const ("Trueprop",_) $ hyp) $ (Const ("Trueprop",_) $ concl)) =>
let
val (pi,typi) = get_pi concl thy
in
if (apply_pi hyp (pi,typi) = concl)
then
(warning ("equivariance lemma of the relational form");
[orig_thm,
get_derived_thm (Context.proof_of context) hyp concl orig_thm pi typi])
else raise EQVT_FORM "Type Implication"
end
(* case: eqvt-lemma is of the equational form *)
| (Const (@{const_name "Trueprop"}, _) $ (Const (@{const_name "op ="}, _) $
(Const (@{const_name "perm"},typrm) $ Var (pi,typi) $ lhs) $ rhs)) =>
(if (apply_pi lhs (pi,typi)) = rhs
then [orig_thm]
else raise EQVT_FORM "Type Equality")
| _ => raise EQVT_FORM "Type unknown")
in
fold (fn thm => Data.map (flag thm)) thms_to_be_added context
end
handle EQVT_FORM s =>
error (Display.string_of_thm (Context.proof_of context) orig_thm ^
" does not comply with the form of an equivariance lemma (" ^ s ^").")
val eqvt_add = Thm.declaration_attribute (eqvt_add_del_aux (Thm.add_thm));
val eqvt_del = Thm.declaration_attribute (eqvt_add_del_aux (Thm.del_thm));
val eqvt_force_add = Thm.declaration_attribute (Data.map o Thm.add_thm);
val eqvt_force_del = Thm.declaration_attribute (Data.map o Thm.del_thm);
val get_eqvt_thms = Context.Proof #> Data.get;
val setup =
Attrib.setup @{binding eqvt} (Attrib.add_del eqvt_add eqvt_del)
"equivariance theorem declaration"
#> Attrib.setup @{binding eqvt_force} (Attrib.add_del eqvt_force_add eqvt_force_del)
"equivariance theorem declaration (without checking the form of the lemma)"
#> PureThy.add_thms_dynamic (Binding.name "eqvts", Data.get)
end;