--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/nominal_induct.ML Thu Dec 30 10:00:09 2010 +0000
@@ -0,0 +1,185 @@
+(* Author: Christian Urban and Makarius
+
+The nominal induct proof method.
+*)
+
+structure NominalInduct:
+sig
+ val nominal_induct_tac: Proof.context -> bool -> (binding option * (term * bool)) option list list ->
+ (string * typ) list -> (string * typ) list list -> thm list -> thm list -> int -> Rule_Cases.cases_tactic
+
+ val nominal_induct_method: (Proof.context -> Proof.method) context_parser
+end =
+
+struct
+
+(* proper tuples -- nested left *)
+
+fun tupleT Ts = HOLogic.unitT |> fold (fn T => fn U => HOLogic.mk_prodT (U, T)) Ts;
+fun tuple ts = HOLogic.unit |> fold (fn t => fn u => HOLogic.mk_prod (u, t)) ts;
+
+fun tuple_fun Ts (xi, T) =
+ Library.funpow (length Ts) HOLogic.mk_split
+ (Var (xi, (HOLogic.unitT :: Ts) ---> Term.range_type T));
+
+val split_all_tuples =
+ Simplifier.full_simplify (HOL_basic_ss addsimps
+ @{thms split_conv split_paired_all unit_all_eq1})
+(*
+ @{thm fresh_unit_elim}, @{thm fresh_prod_elim}] @
+ @{thms fresh_star_unit_elim} @ @{thms fresh_star_prod_elim})
+*)
+
+
+(* prepare rule *)
+
+fun inst_mutual_rule ctxt insts avoiding rules =
+ let
+ val (nconcls, joined_rule) = Rule_Cases.strict_mutual_rule ctxt rules;
+ val concls = Logic.dest_conjunctions (Thm.concl_of joined_rule);
+ val (cases, consumes) = Rule_Cases.get joined_rule;
+
+ val l = length rules;
+ val _ =
+ if length insts = l then ()
+ else error ("Bad number of instantiations for " ^ string_of_int l ^ " rules");
+
+ fun subst inst concl =
+ let
+ val vars = Induct.vars_of concl;
+ val m = length vars and n = length inst;
+ val _ = if m >= n + 2 then () else error "Too few variables in conclusion of rule";
+ val P :: x :: ys = vars;
+ val zs = drop (m - n - 2) ys;
+ in
+ (P, tuple_fun (map #2 avoiding) (Term.dest_Var P)) ::
+ (x, tuple (map Free avoiding)) ::
+ map_filter (fn (z, SOME t) => SOME (z, t) | _ => NONE) (zs ~~ inst)
+ end;
+ val substs =
+ map2 subst insts concls |> flat |> distinct (op =)
+ |> map (pairself (Thm.cterm_of (ProofContext.theory_of ctxt)));
+ in
+ (((cases, nconcls), consumes), Drule.cterm_instantiate substs joined_rule)
+ end;
+
+fun rename_params_rule internal xs rule =
+ let
+ val tune =
+ if internal then Name.internal
+ else fn x => the_default x (try Name.dest_internal x);
+ val n = length xs;
+ fun rename prem =
+ let
+ val ps = Logic.strip_params prem;
+ val p = length ps;
+ val ys =
+ if p < n then []
+ else map (tune o #1) (take (p - n) ps) @ xs;
+ in Logic.list_rename_params (ys, prem) end;
+ fun rename_prems prop =
+ let val (As, C) = Logic.strip_horn prop
+ in Logic.list_implies (map rename As, C) end;
+ in Thm.equal_elim (Thm.reflexive (Drule.cterm_fun rename_prems (Thm.cprop_of rule))) rule end;
+
+
+(* nominal_induct_tac *)
+
+fun nominal_induct_tac ctxt simp def_insts avoiding fixings rules facts =
+ let
+ val thy = ProofContext.theory_of ctxt;
+ val cert = Thm.cterm_of thy;
+
+ val ((insts, defs), defs_ctxt) = fold_map Induct.add_defs def_insts ctxt |>> split_list;
+ val atomized_defs = map (map (Conv.fconv_rule Induct.atomize_cterm)) defs;
+
+ val finish_rule =
+ split_all_tuples
+ #> rename_params_rule true
+ (map (Name.clean o ProofContext.revert_skolem defs_ctxt o fst) avoiding);
+
+ fun rule_cases ctxt r =
+ let val r' = if simp then Induct.simplified_rule ctxt r else r
+ in Rule_Cases.make_nested (Thm.prop_of r') (Induct.rulified_term r') end;
+ in
+ (fn i => fn st =>
+ rules
+ |> inst_mutual_rule ctxt insts avoiding
+ |> Rule_Cases.consume (flat defs) facts
+ |> Seq.maps (fn (((cases, concls), (more_consumes, more_facts)), rule) =>
+ (PRECISE_CONJUNCTS (length concls) (ALLGOALS (fn j =>
+ (CONJUNCTS (ALLGOALS
+ let
+ val adefs = nth_list atomized_defs (j - 1);
+ val frees = fold (Term.add_frees o prop_of) adefs [];
+ val xs = nth_list fixings (j - 1);
+ val k = nth concls (j - 1) + more_consumes
+ in
+ Method.insert_tac (more_facts @ adefs) THEN'
+ (if simp then
+ Induct.rotate_tac k (length adefs) THEN'
+ Induct.fix_tac defs_ctxt k
+ (List.partition (member op = frees) xs |> op @)
+ else
+ Induct.fix_tac defs_ctxt k xs)
+ end)
+ THEN' Induct.inner_atomize_tac) j))
+ THEN' Induct.atomize_tac) i st |> Seq.maps (fn st' =>
+ Induct.guess_instance ctxt
+ (finish_rule (Induct.internalize more_consumes rule)) i st'
+ |> Seq.maps (fn rule' =>
+ CASES (rule_cases ctxt rule' cases)
+ (Tactic.rtac (rename_params_rule false [] rule') i THEN
+ PRIMITIVE (singleton (ProofContext.export defs_ctxt ctxt))) st'))))
+ THEN_ALL_NEW_CASES
+ ((if simp then Induct.simplify_tac ctxt THEN' (TRY o Induct.trivial_tac)
+ else K all_tac)
+ THEN_ALL_NEW Induct.rulify_tac)
+ end;
+
+
+(* concrete syntax *)
+
+local
+
+val avoidingN = "avoiding";
+val fixingN = "arbitrary"; (* to be consistent with induct; hopefully this changes again *)
+val ruleN = "rule";
+
+val inst = Scan.lift (Args.$$$ "_") >> K NONE ||
+ Args.term >> (SOME o rpair false) ||
+ Scan.lift (Args.$$$ "(") |-- (Args.term >> (SOME o rpair true)) --|
+ Scan.lift (Args.$$$ ")");
+
+val def_inst =
+ ((Scan.lift (Args.binding --| (Args.$$$ "\<equiv>" || Args.$$$ "==")) >> SOME)
+ -- (Args.term >> rpair false)) >> SOME ||
+ inst >> Option.map (pair NONE);
+
+val free = Args.context -- Args.term >> (fn (_, Free v) => v | (ctxt, t) =>
+ error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
+
+fun unless_more_args scan = Scan.unless (Scan.lift
+ ((Args.$$$ avoidingN || Args.$$$ fixingN || Args.$$$ ruleN) -- Args.colon)) scan;
+
+
+val avoiding = Scan.optional (Scan.lift (Args.$$$ avoidingN -- Args.colon) |--
+ Scan.repeat (unless_more_args free)) [];
+
+val fixing = Scan.optional (Scan.lift (Args.$$$ fixingN -- Args.colon) |--
+ Parse.and_list' (Scan.repeat (unless_more_args free))) [];
+
+val rule_spec = Scan.lift (Args.$$$ "rule" -- Args.colon) |-- Attrib.thms;
+
+in
+
+val nominal_induct_method =
+ Args.mode Induct.no_simpN -- (Parse.and_list' (Scan.repeat (unless_more_args def_inst)) --
+ avoiding -- fixing -- rule_spec) >>
+ (fn (no_simp, (((x, y), z), w)) => fn ctxt =>
+ RAW_METHOD_CASES (fn facts =>
+ HEADGOAL (nominal_induct_tac ctxt (not no_simp) x y z w facts)));
+
+end
+
+end;