(* Title: Tools/induct.ML
Author: Markus Wenzel, TU Muenchen
Proof by cases, induction, and coinduction.
*)
signature INDUCT_ARGS =
sig
val cases_default: thm
val atomize: thm list
val rulify: thm list
val rulify_fallback: thm list
val equal_def: thm
val dest_def: term -> (term * term) option
val trivial_tac: Proof.context -> int -> tactic
end;
signature INDUCT =
sig
(*rule declarations*)
val vars_of: term -> term list
val dest_rules: Proof.context ->
{type_cases: (string * thm) list, pred_cases: (string * thm) list,
type_induct: (string * thm) list, pred_induct: (string * thm) list,
type_coinduct: (string * thm) list, pred_coinduct: (string * thm) list}
val print_rules: Proof.context -> unit
val lookup_casesT: Proof.context -> string -> thm option
val lookup_casesP: Proof.context -> string -> thm option
val lookup_inductT: Proof.context -> string -> thm option
val lookup_inductP: Proof.context -> string -> thm option
val lookup_coinductT: Proof.context -> string -> thm option
val lookup_coinductP: Proof.context -> string -> thm option
val find_casesT: Proof.context -> typ -> thm list
val find_casesP: Proof.context -> term -> thm list
val find_inductT: Proof.context -> typ -> thm list
val find_inductP: Proof.context -> term -> thm list
val find_coinductT: Proof.context -> typ -> thm list
val find_coinductP: Proof.context -> term -> thm list
val cases_type: string -> attribute
val cases_pred: string -> attribute
val cases_del: attribute
val induct_type: string -> attribute
val induct_pred: string -> attribute
val induct_del: attribute
val coinduct_type: string -> attribute
val coinduct_pred: string -> attribute
val coinduct_del: attribute
val map_simpset: (Proof.context -> Proof.context) -> Context.generic -> Context.generic
val induct_simp_add: attribute
val induct_simp_del: attribute
val no_simpN: string
val casesN: string
val inductN: string
val coinductN: string
val typeN: string
val predN: string
val setN: string
(*proof methods*)
val arbitrary_tac: Proof.context -> int -> (string * typ) list -> int -> tactic
val add_defs: (binding option * (term * bool)) option list -> Proof.context ->
(term option list * thm list) * Proof.context
val atomize_term: Proof.context -> term -> term
val atomize_cterm: Proof.context -> conv
val atomize_tac: Proof.context -> int -> tactic
val inner_atomize_tac: Proof.context -> int -> tactic
val rulified_term: Proof.context -> thm -> term
val rulify_tac: Proof.context -> int -> tactic
val simplified_rule: Proof.context -> thm -> thm
val simplify_tac: Proof.context -> int -> tactic
val trivial_tac: Proof.context -> int -> tactic
val rotate_tac: int -> int -> int -> tactic
val internalize: Proof.context -> int -> thm -> thm
val guess_instance: Proof.context -> thm -> int -> thm -> thm Seq.seq
val cases_tac: Proof.context -> bool -> term option list list -> thm option ->
thm list -> int -> cases_tactic
val get_inductT: Proof.context -> term option list list -> thm list list
val gen_induct_tac: ((Rule_Cases.info * int) * thm -> (Rule_Cases.info * int) * thm) ->
Proof.context -> bool -> (binding option * (term * bool)) option list list ->
(string * typ) list list -> term option list -> thm list option ->
thm list -> int -> cases_tactic
val induct_tac: Proof.context -> bool -> (binding option * (term * bool)) option list list ->
(string * typ) list list -> term option list -> thm list option ->
thm list -> int -> cases_tactic
val coinduct_tac: Proof.context -> term option list -> term option list -> thm option ->
thm list -> int -> cases_tactic
val gen_induct_setup: binding ->
(Proof.context -> bool -> (binding option * (term * bool)) option list list ->
(string * typ) list list -> term option list -> thm list option ->
thm list -> int -> cases_tactic) ->
local_theory -> local_theory
end;
functor Induct(Induct_Args: INDUCT_ARGS): INDUCT =
struct
(** variables -- ordered left-to-right, preferring right **)
fun vars_of tm =
rev (distinct (op =) (Term.fold_aterms (fn t as Var _ => cons t | _ => I) tm []));
local
val mk_var = Net.encode_type o #2 o Term.dest_Var;
fun concl_var which thm = mk_var (which (vars_of (Thm.concl_of thm))) handle List.Empty =>
raise THM ("No variables in conclusion of rule", 0, [thm]);
in
fun left_var_prem thm = mk_var (hd (vars_of (hd (Thm.prems_of thm)))) handle List.Empty =>
raise THM ("No variables in major premise of rule", 0, [thm]);
val left_var_concl = concl_var hd;
val right_var_concl = concl_var List.last;
end;
(** constraint simplification **)
(* rearrange parameters and premises to allow application of one-point-rules *)
fun swap_params_conv ctxt i j cv =
let
fun conv1 0 ctxt = Conv.forall_conv (cv o snd) ctxt
| conv1 k ctxt =
Conv.rewr_conv @{thm swap_params} then_conv
Conv.forall_conv (conv1 (k - 1) o snd) ctxt;
fun conv2 0 ctxt = conv1 j ctxt
| conv2 k ctxt = Conv.forall_conv (conv2 (k - 1) o snd) ctxt;
in conv2 i ctxt end;
fun swap_prems_conv 0 = Conv.all_conv
| swap_prems_conv i =
Conv.implies_concl_conv (swap_prems_conv (i - 1)) then_conv
Conv.rewr_conv Drule.swap_prems_eq;
fun find_eq ctxt t =
let
val l = length (Logic.strip_params t);
val Hs = Logic.strip_assums_hyp t;
fun find (i, t) =
(case Induct_Args.dest_def (Object_Logic.drop_judgment ctxt t) of
SOME (Bound j, _) => SOME (i, j)
| SOME (_, Bound j) => SOME (i, j)
| _ => NONE);
in
(case get_first find (map_index I Hs) of
NONE => NONE
| SOME (0, 0) => NONE
| SOME (i, j) => SOME (i, l - j - 1, j))
end;
fun mk_swap_rrule ctxt ct =
(case find_eq ctxt (Thm.term_of ct) of
NONE => NONE
| SOME (i, k, j) => SOME (swap_params_conv ctxt k j (K (swap_prems_conv i)) ct));
val rearrange_eqs_simproc =
Simplifier.simproc_global Pure.thy "rearrange_eqs" ["Pure.all t"]
(fn ctxt => fn t => mk_swap_rrule ctxt (Thm.cterm_of ctxt t));
(* rotate k premises to the left by j, skipping over first j premises *)
fun rotate_conv 0 _ 0 = Conv.all_conv
| rotate_conv 0 j k = swap_prems_conv j then_conv rotate_conv 1 j (k - 1)
| rotate_conv i j k = Conv.implies_concl_conv (rotate_conv (i - 1) j k);
fun rotate_tac _ 0 = K all_tac
| rotate_tac j k = SUBGOAL (fn (goal, i) =>
CONVERSION (rotate_conv
j (length (Logic.strip_assums_hyp goal) - j - k) k) i);
(* rulify operators around definition *)
fun rulify_defs_conv ctxt ct =
if exists_subterm (is_some o Induct_Args.dest_def) (Thm.term_of ct) andalso
not (is_some (Induct_Args.dest_def (Object_Logic.drop_judgment ctxt (Thm.term_of ct))))
then
(Conv.forall_conv (rulify_defs_conv o snd) ctxt else_conv
Conv.implies_conv (Conv.try_conv (rulify_defs_conv ctxt))
(Conv.try_conv (rulify_defs_conv ctxt)) else_conv
Conv.first_conv (map Conv.rewr_conv Induct_Args.rulify) then_conv
Conv.try_conv (rulify_defs_conv ctxt)) ct
else Conv.no_conv ct;
(** induct data **)
(* rules *)
type rules = (string * thm) Item_Net.T;
fun init_rules index : rules =
Item_Net.init
(fn ((s1, th1), (s2, th2)) => s1 = s2 andalso Thm.eq_thm_prop (th1, th2))
(single o index);
fun filter_rules (rs: rules) th =
filter (fn (_, th') => Thm.eq_thm_prop (th, th')) (Item_Net.content rs);
fun lookup_rule (rs: rules) = AList.lookup (op =) (Item_Net.content rs);
fun pretty_rules ctxt kind rs =
let val thms = map snd (Item_Net.content rs)
in Pretty.big_list kind (map (Display.pretty_thm_item ctxt) thms) end;
(* context data *)
structure Data = Generic_Data
(
type T = (rules * rules) * (rules * rules) * (rules * rules) * simpset;
val empty =
((init_rules (left_var_prem o #2), init_rules (Thm.major_prem_of o #2)),
(init_rules (right_var_concl o #2), init_rules (Thm.major_prem_of o #2)),
(init_rules (left_var_concl o #2), init_rules (Thm.concl_of o #2)),
simpset_of (empty_simpset @{context}
addsimprocs [rearrange_eqs_simproc] addsimps [Drule.norm_hhf_eq]));
val extend = I;
fun merge (((casesT1, casesP1), (inductT1, inductP1), (coinductT1, coinductP1), simpset1),
((casesT2, casesP2), (inductT2, inductP2), (coinductT2, coinductP2), simpset2)) =
((Item_Net.merge (casesT1, casesT2), Item_Net.merge (casesP1, casesP2)),
(Item_Net.merge (inductT1, inductT2), Item_Net.merge (inductP1, inductP2)),
(Item_Net.merge (coinductT1, coinductT2), Item_Net.merge (coinductP1, coinductP2)),
merge_ss (simpset1, simpset2));
);
val get_local = Data.get o Context.Proof;
fun dest_rules ctxt =
let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in
{type_cases = Item_Net.content casesT,
pred_cases = Item_Net.content casesP,
type_induct = Item_Net.content inductT,
pred_induct = Item_Net.content inductP,
type_coinduct = Item_Net.content coinductT,
pred_coinduct = Item_Net.content coinductP}
end;
fun print_rules ctxt =
let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in
[pretty_rules ctxt "coinduct type:" coinductT,
pretty_rules ctxt "coinduct pred:" coinductP,
pretty_rules ctxt "induct type:" inductT,
pretty_rules ctxt "induct pred:" inductP,
pretty_rules ctxt "cases type:" casesT,
pretty_rules ctxt "cases pred:" casesP]
|> Pretty.writeln_chunks
end;
val _ =
Outer_Syntax.command @{command_keyword print_induct_rules}
"print induction and cases rules"
(Scan.succeed (Toplevel.keep (print_rules o Toplevel.context_of)));
(* access rules *)
val lookup_casesT = lookup_rule o #1 o #1 o get_local;
val lookup_casesP = lookup_rule o #2 o #1 o get_local;
val lookup_inductT = lookup_rule o #1 o #2 o get_local;
val lookup_inductP = lookup_rule o #2 o #2 o get_local;
val lookup_coinductT = lookup_rule o #1 o #3 o get_local;
val lookup_coinductP = lookup_rule o #2 o #3 o get_local;
fun find_rules which how ctxt x =
map snd (Item_Net.retrieve (which (get_local ctxt)) (how x));
val find_casesT = find_rules (#1 o #1) Net.encode_type;
val find_casesP = find_rules (#2 o #1) I;
val find_inductT = find_rules (#1 o #2) Net.encode_type;
val find_inductP = find_rules (#2 o #2) I;
val find_coinductT = find_rules (#1 o #3) Net.encode_type;
val find_coinductP = find_rules (#2 o #3) I;
(** attributes **)
local
fun mk_att f g name =
Thm.mixed_attribute (fn (context, thm) =>
let
val thm' = g thm;
val context' = Data.map (f (name, thm')) context;
in (context', thm') end);
fun del_att which =
Thm.declaration_attribute (fn th => Data.map (which (apply2 (fn rs =>
fold Item_Net.remove (filter_rules rs th) rs))));
fun add_casesT rule x = @{apply 4(1)} (apfst (Item_Net.update rule)) x;
fun add_casesP rule x = @{apply 4(1)} (apsnd (Item_Net.update rule)) x;
fun add_inductT rule x = @{apply 4(2)} (apfst (Item_Net.update rule)) x;
fun add_inductP rule x = @{apply 4(2)} (apsnd (Item_Net.update rule)) x;
fun add_coinductT rule x = @{apply 4(3)} (apfst (Item_Net.update rule)) x;
fun add_coinductP rule x = @{apply 4(3)} (apsnd (Item_Net.update rule)) x;
val consumes0 = Rule_Cases.default_consumes 0;
val consumes1 = Rule_Cases.default_consumes 1;
in
val cases_type = mk_att add_casesT consumes0;
val cases_pred = mk_att add_casesP consumes1;
val cases_del = del_att @{apply 4(1)};
val induct_type = mk_att add_inductT consumes0;
val induct_pred = mk_att add_inductP consumes1;
val induct_del = del_att @{apply 4(2)};
val coinduct_type = mk_att add_coinductT consumes0;
val coinduct_pred = mk_att add_coinductP consumes1;
val coinduct_del = del_att @{apply 4(3)};
fun map_simpset f context =
context |> (Data.map o @{apply 4(4)} o Simplifier.simpset_map (Context.proof_of context)) f;
fun induct_simp f =
Thm.declaration_attribute (fn thm => map_simpset (fn ctxt => f (ctxt, [thm])));
val induct_simp_add = induct_simp (op addsimps);
val induct_simp_del = induct_simp (op delsimps);
end;
(** attribute syntax **)
val no_simpN = "no_simp";
val casesN = "cases";
val inductN = "induct";
val coinductN = "coinduct";
val typeN = "type";
val predN = "pred";
val setN = "set";
local
fun spec k arg =
Scan.lift (Args.$$$ k -- Args.colon) |-- arg ||
Scan.lift (Args.$$$ k) >> K "";
fun attrib add_type add_pred del =
spec typeN (Args.type_name {proper = false, strict = false}) >> add_type ||
spec predN (Args.const {proper = false, strict = false}) >> add_pred ||
spec setN (Args.const {proper = false, strict = false}) >> add_pred ||
Scan.lift Args.del >> K del;
in
val _ =
Theory.local_setup
(Attrib.local_setup @{binding cases} (attrib cases_type cases_pred cases_del)
"declaration of cases rule" #>
Attrib.local_setup @{binding induct} (attrib induct_type induct_pred induct_del)
"declaration of induction rule" #>
Attrib.local_setup @{binding coinduct} (attrib coinduct_type coinduct_pred coinduct_del)
"declaration of coinduction rule" #>
Attrib.local_setup @{binding induct_simp} (Attrib.add_del induct_simp_add induct_simp_del)
"declaration of rules for simplifying induction or cases rules");
end;
(** method utils **)
(* alignment *)
fun align_left msg xs ys =
let val m = length xs and n = length ys
in if m < n then error msg else (take n xs ~~ ys) end;
fun align_right msg xs ys =
let val m = length xs and n = length ys
in if m < n then error msg else (drop (m - n) xs ~~ ys) end;
(* prep_inst *)
fun prep_inst ctxt align tune (tm, ts) =
let
fun prep_var (x, SOME t) =
let
val cx = Thm.cterm_of ctxt x;
val xT = Thm.typ_of_cterm cx;
val ct = Thm.cterm_of ctxt (tune t);
val tT = Thm.typ_of_cterm ct;
in
if Type.could_unify (tT, xT) then SOME (cx, ct)
else error (Pretty.string_of (Pretty.block
[Pretty.str "Ill-typed instantiation:", Pretty.fbrk,
Syntax.pretty_term ctxt (Thm.term_of ct), Pretty.str " ::", Pretty.brk 1,
Syntax.pretty_typ ctxt tT]))
end
| prep_var (_, NONE) = NONE;
val xs = vars_of tm;
in
align "Rule has fewer variables than instantiations given" xs ts
|> map_filter prep_var
end;
(* trace_rules *)
fun trace_rules _ kind [] = error ("Unable to figure out " ^ kind ^ " rule")
| trace_rules ctxt _ rules = Method.trace ctxt rules;
(* mark equality constraints in cases rule *)
val equal_def' = Thm.symmetric Induct_Args.equal_def;
fun mark_constraints n ctxt = Conv.fconv_rule
(Conv.prems_conv ~1 (Conv.params_conv ~1 (K (Conv.prems_conv n
(Raw_Simplifier.rewrite ctxt false [equal_def']))) ctxt));
fun unmark_constraints ctxt =
Conv.fconv_rule (Raw_Simplifier.rewrite ctxt true [Induct_Args.equal_def]);
(* simplify *)
fun simplify_conv' ctxt =
Simplifier.full_rewrite (put_simpset (#4 (get_local ctxt)) ctxt);
fun simplify_conv ctxt ct =
if exists_subterm (is_some o Induct_Args.dest_def) (Thm.term_of ct) then
(Conv.try_conv (rulify_defs_conv ctxt) then_conv simplify_conv' ctxt) ct
else Conv.all_conv ct;
fun gen_simplified_rule cv ctxt =
Conv.fconv_rule (Conv.prems_conv ~1 (cv ctxt));
val simplified_rule' = gen_simplified_rule simplify_conv';
val simplified_rule = gen_simplified_rule simplify_conv;
fun simplify_tac ctxt = CONVERSION (simplify_conv ctxt);
val trivial_tac = Induct_Args.trivial_tac;
(** cases method **)
(*
rule selection scheme:
cases - default case split
`A t` cases ... - predicate/set cases
cases t - type cases
... cases ... r - explicit rule
*)
local
fun get_casesT ctxt ((SOME t :: _) :: _) = find_casesT ctxt (Term.fastype_of t)
| get_casesT _ _ = [];
fun get_casesP ctxt (fact :: _) = find_casesP ctxt (Thm.concl_of fact)
| get_casesP _ _ = [];
in
fun cases_tac ctxt simp insts opt_rule facts =
let
fun inst_rule r =
(if null insts then r
else
(align_left "Rule has fewer premises than arguments given" (Thm.prems_of r) insts
|> maps (prep_inst ctxt align_left I)
|> Drule.cterm_instantiate) r)
|> simp ? mark_constraints (Rule_Cases.get_constraints r) ctxt
|> pair (Rule_Cases.get r);
val (opt_rule', facts') =
(case (opt_rule, facts) of
(NONE, th :: ths) =>
if is_some (Object_Logic.elim_concl ctxt th) then (SOME th, ths)
else (opt_rule, facts)
| _ => (opt_rule, facts));
val ruleq =
(case opt_rule' of
SOME r => Seq.single (inst_rule r)
| NONE =>
(get_casesP ctxt facts' @ get_casesT ctxt insts @ [Induct_Args.cases_default])
|> tap (trace_rules ctxt casesN)
|> Seq.of_list |> Seq.maps (Seq.try inst_rule));
in
fn i => fn st =>
ruleq
|> Seq.maps (Rule_Cases.consume ctxt [] facts')
|> Seq.maps (fn ((cases, (_, facts'')), rule) =>
let
val rule' = rule
|> simp ? (simplified_rule' ctxt #> unmark_constraints ctxt);
in
CASES (Rule_Cases.make_common ctxt
(Thm.prop_of (Rule_Cases.internalize_params rule')) cases)
((Method.insert_tac facts'' THEN' resolve_tac ctxt [rule'] THEN_ALL_NEW
(if simp then TRY o trivial_tac ctxt else K all_tac)) i) st
end)
end;
end;
(** induct method **)
val conjunction_congs = @{thms Pure.all_conjunction imp_conjunction};
(* atomize *)
fun atomize_term ctxt =
Raw_Simplifier.rewrite_term (Proof_Context.theory_of ctxt) Induct_Args.atomize []
#> Object_Logic.drop_judgment ctxt;
fun atomize_cterm ctxt = Raw_Simplifier.rewrite ctxt true Induct_Args.atomize;
fun atomize_tac ctxt = rewrite_goal_tac ctxt Induct_Args.atomize;
fun inner_atomize_tac ctxt =
rewrite_goal_tac ctxt (map Thm.symmetric conjunction_congs) THEN' atomize_tac ctxt;
(* rulify *)
fun rulify_term thy =
Raw_Simplifier.rewrite_term thy (Induct_Args.rulify @ conjunction_congs) [] #>
Raw_Simplifier.rewrite_term thy Induct_Args.rulify_fallback [];
fun rulified_term ctxt thm =
let
val rulify = rulify_term (Proof_Context.theory_of ctxt);
val (As, B) = Logic.strip_horn (Thm.prop_of thm);
in Logic.list_implies (map rulify As, rulify B) end;
fun rulify_tac ctxt =
rewrite_goal_tac ctxt (Induct_Args.rulify @ conjunction_congs) THEN'
rewrite_goal_tac ctxt Induct_Args.rulify_fallback THEN'
Goal.conjunction_tac THEN_ALL_NEW
(rewrite_goal_tac ctxt @{thms Pure.conjunction_imp} THEN' Goal.norm_hhf_tac ctxt);
(* prepare rule *)
fun rule_instance ctxt inst rule =
Drule.cterm_instantiate (prep_inst ctxt align_left I (Thm.prop_of rule, inst)) rule;
fun internalize ctxt k th =
th |> Thm.permute_prems 0 k
|> Conv.fconv_rule (Conv.concl_conv (Thm.nprems_of th - k) (atomize_cterm ctxt));
(* guess rule instantiation -- cannot handle pending goal parameters *)
local
fun dest_env ctxt env =
let
val pairs = Vartab.dest (Envir.term_env env);
val types = Vartab.dest (Envir.type_env env);
val ts = map (Thm.cterm_of ctxt o Envir.norm_term env o #2 o #2) pairs;
val xs = map2 (curry (Thm.cterm_of ctxt o Var)) (map #1 pairs) (map Thm.typ_of_cterm ts);
in (map (fn (xi, (S, T)) => apply2 (Thm.ctyp_of ctxt) (TVar (xi, S), T)) types, xs ~~ ts) end;
in
fun guess_instance ctxt rule i st =
let
val maxidx = Thm.maxidx_of st;
val goal = Thm.term_of (Thm.cprem_of st i); (*exception Subscript*)
val params = rev (Term.rename_wrt_term goal (Logic.strip_params goal));
in
if not (null params) then
(warning ("Cannot determine rule instantiation due to pending parameter(s): " ^
commas_quote (map (Syntax.string_of_term ctxt o Syntax_Trans.mark_bound_abs) params));
Seq.single rule)
else
let
val rule' = Thm.incr_indexes (maxidx + 1) rule;
val concl = Logic.strip_assums_concl goal;
in
Unify.smash_unifiers (Context.Proof ctxt)
[(Thm.concl_of rule', concl)] (Envir.empty (Thm.maxidx_of rule'))
|> Seq.map (fn env => Drule.instantiate_normalize (dest_env ctxt env) rule')
end
end
handle General.Subscript => Seq.empty;
end;
(* special renaming of rule parameters *)
fun special_rename_params ctxt [[SOME (Free (z, Type (T, _)))]] [thm] =
let
val x = Name.clean (Variable.revert_fixed ctxt z);
fun index _ [] = []
| index i (y :: ys) =
if x = y then x ^ string_of_int i :: index (i + 1) ys
else y :: index i ys;
fun rename_params [] = []
| rename_params ((y, Type (U, _)) :: ys) =
(if U = T then x else y) :: rename_params ys
| rename_params ((y, _) :: ys) = y :: rename_params ys;
fun rename_asm A =
let
val xs = rename_params (Logic.strip_params A);
val xs' =
(case filter (fn x' => x' = x) xs of
[] => xs
| [_] => xs
| _ => index 1 xs);
in Logic.list_rename_params xs' A end;
fun rename_prop prop =
let val (As, C) = Logic.strip_horn prop
in Logic.list_implies (map rename_asm As, C) end;
val thm' = Thm.renamed_prop (rename_prop (Thm.prop_of thm)) thm;
in [Rule_Cases.save thm thm'] end
| special_rename_params _ _ ths = ths;
(* arbitrary_tac *)
local
fun goal_prefix k ((c as Const (@{const_name Pure.all}, _)) $ Abs (a, T, B)) =
c $ Abs (a, T, goal_prefix k B)
| goal_prefix 0 _ = Term.dummy_prop
| goal_prefix k ((c as Const (@{const_name Pure.imp}, _)) $ A $ B) =
c $ A $ goal_prefix (k - 1) B
| goal_prefix _ _ = Term.dummy_prop;
fun goal_params k (Const (@{const_name Pure.all}, _) $ Abs (_, _, B)) = goal_params k B + 1
| goal_params 0 _ = 0
| goal_params k (Const (@{const_name Pure.imp}, _) $ _ $ B) = goal_params (k - 1) B
| goal_params _ _ = 0;
fun meta_spec_tac ctxt n (x, T) = SUBGOAL (fn (goal, i) =>
let
val v = Free (x, T);
fun spec_rule prfx (xs, body) =
@{thm Pure.meta_spec}
|> Thm.rename_params_rule ([Name.clean (Variable.revert_fixed ctxt x)], 1)
|> Thm.lift_rule (Thm.cterm_of ctxt prfx)
|> `(Thm.prop_of #> Logic.strip_assums_concl)
|-> (fn pred $ arg =>
Drule.cterm_instantiate
(map (apply2 (Thm.cterm_of ctxt))
[(Term.head_of pred, Logic.rlist_abs (xs, body)),
(Term.head_of arg, Logic.rlist_abs (xs, v))]));
fun goal_concl k xs (Const (@{const_name Pure.all}, _) $ Abs (a, T, B)) =
goal_concl k ((a, T) :: xs) B
| goal_concl 0 xs B =
if not (Term.exists_subterm (fn t => t aconv v) B) then NONE
else SOME (xs, absfree (x, T) (Term.incr_boundvars 1 B))
| goal_concl k xs (Const (@{const_name Pure.imp}, _) $ _ $ B) =
goal_concl (k - 1) xs B
| goal_concl _ _ _ = NONE;
in
(case goal_concl n [] goal of
SOME concl =>
(compose_tac ctxt (false, spec_rule (goal_prefix n goal) concl, 1) THEN'
resolve_tac ctxt [asm_rl]) i
| NONE => all_tac)
end);
fun miniscope_tac p =
CONVERSION o
Conv.params_conv p (fn ctxt =>
Raw_Simplifier.rewrite ctxt true [Thm.symmetric Drule.norm_hhf_eq]);
in
fun arbitrary_tac _ _ [] = K all_tac
| arbitrary_tac ctxt n xs = SUBGOAL (fn (goal, i) =>
(EVERY' (map (meta_spec_tac ctxt n) xs) THEN'
(miniscope_tac (goal_params n goal) ctxt)) i);
end;
(* add_defs *)
fun add_defs def_insts =
let
fun add (SOME (_, (t, true))) ctxt = ((SOME t, []), ctxt)
| add (SOME (SOME x, (t, _))) ctxt =
let val ([(lhs, (_, th))], ctxt') =
Local_Defs.add_defs [((x, NoSyn), ((Thm.def_binding x, []), t))] ctxt
in ((SOME lhs, [th]), ctxt') end
| add (SOME (NONE, (t as Free _, _))) ctxt = ((SOME t, []), ctxt)
| add (SOME (NONE, (t, _))) ctxt =
let
val (s, _) = Name.variant "x" (Variable.names_of ctxt);
val x = Binding.name s;
val ([(lhs, (_, th))], ctxt') = ctxt
|> Local_Defs.add_defs [((x, NoSyn), ((Thm.def_binding x, []), t))];
in ((SOME lhs, [th]), ctxt') end
| add NONE ctxt = ((NONE, []), ctxt);
in fold_map add def_insts #> apfst (split_list #> apsnd flat) end;
(* induct_tac *)
(*
rule selection scheme:
`A x` induct ... - predicate/set induction
induct x - type induction
... induct ... r - explicit rule
*)
fun get_inductT ctxt insts =
fold_rev (map_product cons) (insts |> map
((fn [] => NONE | ts => List.last ts) #>
(fn NONE => TVar (("'a", 0), []) | SOME t => Term.fastype_of t) #>
find_inductT ctxt)) [[]]
|> filter_out (forall Rule_Cases.is_inner_rule);
fun get_inductP ctxt (fact :: _) = map single (find_inductP ctxt (Thm.concl_of fact))
| get_inductP _ _ = [];
fun gen_induct_tac mod_cases ctxt simp def_insts arbitrary taking opt_rule facts =
SUBGOAL_CASES (fn (_, i) =>
let
val ((insts, defs), defs_ctxt) = fold_map add_defs def_insts ctxt |>> split_list;
val atomized_defs = map (map (Conv.fconv_rule (atomize_cterm defs_ctxt))) defs;
fun inst_rule (concls, r) =
(if null insts then `Rule_Cases.get r
else (align_left "Rule has fewer conclusions than arguments given"
(map Logic.strip_imp_concl (Logic.dest_conjunctions (Thm.concl_of r))) insts
|> maps (prep_inst ctxt align_right (atomize_term ctxt))
|> Drule.cterm_instantiate) r |> pair (Rule_Cases.get r))
|> mod_cases
|> (fn ((cases, consumes), th) => (((cases, concls), consumes), th));
val ruleq =
(case opt_rule of
SOME rs => Seq.single (inst_rule (Rule_Cases.strict_mutual_rule ctxt rs))
| NONE =>
(get_inductP ctxt facts @
map (special_rename_params defs_ctxt insts) (get_inductT ctxt insts))
|> map_filter (Rule_Cases.mutual_rule ctxt)
|> tap (trace_rules ctxt inductN o map #2)
|> Seq.of_list |> Seq.maps (Seq.try inst_rule));
fun rule_cases ctxt rule cases =
let
val rule' = Rule_Cases.internalize_params rule;
val rule'' = rule' |> simp ? simplified_rule ctxt;
val nonames = map (fn ((cn, _), cls) => ((cn, []), cls));
val cases' = if Thm.eq_thm_prop (rule', rule'') then cases else nonames cases;
in Rule_Cases.make_nested ctxt (Thm.prop_of rule'') (rulified_term ctxt rule'') cases' end;
in
fn st =>
ruleq
|> Seq.maps (Rule_Cases.consume defs_ctxt (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 Thm.prop_of) adefs [];
val xs = nth_list arbitrary (j - 1);
val k = nth concls (j - 1) + more_consumes;
in
Method.insert_tac (more_facts @ adefs) THEN'
(if simp then
rotate_tac k (length adefs) THEN'
arbitrary_tac defs_ctxt k (List.partition (member op = frees) xs |> op @)
else
arbitrary_tac defs_ctxt k xs)
end)
THEN' inner_atomize_tac defs_ctxt) j))
THEN' atomize_tac defs_ctxt) i st |> Seq.maps (fn st' =>
guess_instance ctxt (internalize ctxt more_consumes rule) i st'
|> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))
|> Seq.maps (fn rule' =>
CASES (rule_cases ctxt rule' cases)
(resolve_tac ctxt [rule'] i THEN
PRIMITIVE (singleton (Proof_Context.export defs_ctxt ctxt))) st')))
end)
THEN_ALL_NEW_CASES
((if simp then simplify_tac ctxt THEN' (TRY o trivial_tac ctxt) else K all_tac)
THEN_ALL_NEW rulify_tac ctxt);
val induct_tac = gen_induct_tac I;
(** coinduct method **)
(*
rule selection scheme:
goal "A x" coinduct ... - predicate/set coinduction
coinduct x - type coinduction
coinduct ... r - explicit rule
*)
local
fun get_coinductT ctxt (SOME t :: _) = find_coinductT ctxt (Term.fastype_of t)
| get_coinductT _ _ = [];
fun get_coinductP ctxt goal = find_coinductP ctxt (Logic.strip_assums_concl goal);
fun main_prop_of th =
if Rule_Cases.get_consumes th > 0 then Thm.major_prem_of th else Thm.concl_of th;
in
fun coinduct_tac ctxt inst taking opt_rule facts = SUBGOAL_CASES (fn (goal, i) =>
let
fun inst_rule r =
if null inst then `Rule_Cases.get r
else Drule.cterm_instantiate (prep_inst ctxt align_right I (main_prop_of r, inst)) r
|> pair (Rule_Cases.get r);
in
fn st =>
(case opt_rule of
SOME r => Seq.single (inst_rule r)
| NONE =>
(get_coinductP ctxt goal @ get_coinductT ctxt inst)
|> tap (trace_rules ctxt coinductN)
|> Seq.of_list |> Seq.maps (Seq.try inst_rule))
|> Seq.maps (Rule_Cases.consume ctxt [] facts)
|> Seq.maps (fn ((cases, (_, more_facts)), rule) =>
guess_instance ctxt rule i st
|> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))
|> Seq.maps (fn rule' =>
CASES (Rule_Cases.make_common ctxt
(Thm.prop_of (Rule_Cases.internalize_params rule')) cases)
(Method.insert_tac more_facts i THEN resolve_tac ctxt [rule'] i) st))
end);
end;
(** concrete syntax **)
val arbitraryN = "arbitrary";
val takingN = "taking";
val ruleN = "rule";
local
fun single_rule [rule] = rule
| single_rule _ = error "Single rule expected";
fun named_rule k arg get =
Scan.lift (Args.$$$ k -- Args.colon) |-- Scan.repeat arg :|--
(fn names => Scan.peek (fn context => Scan.succeed (names |> map (fn name =>
(case get (Context.proof_of context) name of SOME x => x
| NONE => error ("No rule for " ^ k ^ " " ^ quote name))))));
fun rule get_type get_pred =
named_rule typeN (Args.type_name {proper = false, strict = false}) get_type ||
named_rule predN (Args.const {proper = false, strict = false}) get_pred ||
named_rule setN (Args.const {proper = false, strict = false}) get_pred ||
Scan.lift (Args.$$$ ruleN -- Args.colon) |-- Attrib.thms;
val cases_rule = rule lookup_casesT lookup_casesP >> single_rule;
val induct_rule = rule lookup_inductT lookup_inductP;
val coinduct_rule = rule lookup_coinductT lookup_coinductP >> single_rule;
val inst = Scan.lift (Args.$$$ "_") >> K NONE || Args.term >> SOME;
val inst' = Scan.lift (Args.$$$ "_") >> 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.$$$ arbitraryN || Args.$$$ takingN || Args.$$$ typeN ||
Args.$$$ predN || Args.$$$ setN || Args.$$$ ruleN) -- Args.colon)) scan;
val arbitrary = Scan.optional (Scan.lift (Args.$$$ arbitraryN -- Args.colon) |--
Parse.and_list1' (Scan.repeat (unless_more_args free))) [];
val taking = Scan.optional (Scan.lift (Args.$$$ takingN -- Args.colon) |--
Scan.repeat1 (unless_more_args inst)) [];
in
fun gen_induct_setup binding tac =
Method.local_setup binding
(Scan.lift (Args.mode no_simpN) --
(Parse.and_list' (Scan.repeat (unless_more_args def_inst)) --
(arbitrary -- taking -- Scan.option induct_rule)) >>
(fn (no_simp, (insts, ((arbitrary, taking), opt_rule))) => fn ctxt => fn facts =>
Seq.DETERM (HEADGOAL (tac ctxt (not no_simp) insts arbitrary taking opt_rule facts))))
"induction on types or predicates/sets";
val _ =
Theory.local_setup
(Method.local_setup @{binding cases}
(Scan.lift (Args.mode no_simpN) --
(Parse.and_list' (Scan.repeat (unless_more_args inst)) -- Scan.option cases_rule) >>
(fn (no_simp, (insts, opt_rule)) => fn ctxt =>
METHOD_CASES (fn facts => Seq.DETERM (HEADGOAL
(cases_tac ctxt (not no_simp) insts opt_rule facts)))))
"case analysis on types or predicates/sets" #>
gen_induct_setup @{binding induct} induct_tac #>
Method.local_setup @{binding coinduct}
(Scan.repeat (unless_more_args inst) -- taking -- Scan.option coinduct_rule >>
(fn ((insts, taking), opt_rule) => fn ctxt => fn facts =>
Seq.DETERM (HEADGOAL (coinduct_tac ctxt insts taking opt_rule facts))))
"coinduction on types or predicates/sets");
end;
end;