(* Author: Christian Urban and Makarius The nominal induct proof method. *) structure NominalInduct: sig val nominal_induct_tac: Proof.context -> (binding option * term) option list list -> (string * typ) list -> (string * typ) list list -> thm list -> thm list -> int -> RuleCases.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 [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) = RuleCases.strict_mutual_rule ctxt rules; val concls = Logic.dest_conjunctions (Thm.concl_of joined_rule); val (cases, consumes) = RuleCases.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 = Library.drop (m - n - 2, ys); in (P, tuple_fun (map #2 avoiding) (Term.dest_Var P)) :: (x, tuple (map Free avoiding)) :: List.mapPartial (fn (z, SOME t) => SOME (z, t) | _ => NONE) (zs ~~ inst) end; val substs = map2 subst insts concls |> List.concat |> 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) (Library.take (p - n, ps)) @ xs; in Logic.list_rename_params (ys, prem) end; fun rename_prems prop = let val (As, C) = Logic.strip_horn (Thm.prop_of rule) 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 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 ObjectLogic.atomize)) 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 r = RuleCases.make_nested true (Thm.prop_of r) (Induct.rulified_term r); in (fn i => fn st => rules |> inst_mutual_rule ctxt insts avoiding |> RuleCases.consume (List.concat defs) facts |> Seq.maps (fn (((cases, concls), (more_consumes, more_facts)), rule) => (PRECISE_CONJUNCTS (length concls) (ALLGOALS (fn j => (CONJUNCTS (ALLGOALS (Method.insert_tac (more_facts @ nth_list atomized_defs (j - 1)) THEN' Induct.fix_tac defs_ctxt (nth concls (j - 1) + more_consumes) (nth_list fixings (j - 1)))) 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 rule' cases) (Tactic.rtac (rename_params_rule false [] rule') i THEN PRIMITIVE (singleton (ProofContext.export defs_ctxt ctxt))) st')))) THEN_ALL_NEW_CASES Induct.rulify_tac end; (* concrete syntax *) local structure P = OuterParse; val avoidingN = "avoiding"; val fixingN = "arbitrary"; (* to be consistent with induct; hopefully this changes again *) val ruleN = "rule"; val inst = Scan.lift (Args.$$$ "_") >> K NONE || Args.term >> SOME; val def_inst = ((Scan.lift (Args.binding --| (Args.$$$ "≡" || Args.$$$ "==")) >> SOME) -- Args.term) >> SOME || inst >> Option.map (pair NONE); val free = Args.context -- Args.term >> (fn (_, Free v) => v | (ctxt, t) => error ("Bad free variable: " ^ Syntax.string_of_term ctxt t)); fun unless_more_args scan = Scan.unless (Scan.lift ((Args.$$$ 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) |-- P.and_list' (Scan.repeat (unless_more_args free))) []; val rule_spec = Scan.lift (Args.$$$ "rule" -- Args.colon) |-- Attrib.thms; in val nominal_induct_method = P.and_list' (Scan.repeat (unless_more_args def_inst)) -- avoiding -- fixing -- rule_spec >> (fn (((x, y), z), w) => fn ctxt => RAW_METHOD_CASES (fn facts => HEADGOAL (nominal_induct_tac ctxt x y z w facts))); end; end;