(* 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
[@{thm split_conv}, @{thm split_paired_all}, @{thm 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;