--- a/src/HOL/Tools/function_package/induction_scheme.ML Thu Jun 19 00:02:08 2008 +0200
+++ b/src/HOL/Tools/function_package/induction_scheme.ML Thu Jun 19 11:46:14 2008 +0200
@@ -7,7 +7,8 @@
signature INDUCTION_SCHEME =
sig
- val mk_ind_tac : Proof.context -> thm list -> tactic
+ val mk_ind_tac : (int -> tactic) -> (int -> tactic) -> (int -> tactic)
+ -> Proof.context -> thm list -> tactic
val setup : theory -> theory
end
@@ -17,26 +18,48 @@
open FundefLib
-type rec_call_info = (string * typ) list * term list * term
+
+type rec_call_info = int * (string * typ) list * term list * term list
datatype scheme_case =
SchemeCase of
{
+ bidx : int,
qs: (string * typ) list,
+ oqnames: string list,
gs: term list,
- lhs: term,
+ lhs: term list,
rs: rec_call_info list
}
+datatype scheme_branch =
+ SchemeBranch of
+ {
+ P : term,
+ xs: (string * typ) list,
+ ws: (string * typ) list,
+ Cs: term list
+ }
+
datatype ind_scheme =
IndScheme of
{
- (*cvars : (string, typ) list,
- cassms : term list, *) (* additional context for partial rules *)
- T: typ,
+ T: typ, (* sum of products *)
+ branches: scheme_branch list,
cases: scheme_case list
}
+val ind_atomize = MetaSimplifier.rewrite true @{thms induct_atomize}
+val ind_rulify = MetaSimplifier.rewrite true @{thms induct_rulify}
+
+fun meta thm = thm RS eq_reflection
+
+val sum_prod_conv = MetaSimplifier.rewrite true
+ (map meta (@{thm split_conv} :: @{thms sum_cases}))
+
+fun term_conv thy cv t =
+ cv (cterm_of thy t)
+ |> prop_of |> Logic.dest_equals |> snd
fun mk_relT T = HOLogic.mk_setT (HOLogic.mk_prodT (T, T))
@@ -47,57 +70,126 @@
(ctxt', vars, Logic.strip_imp_prems imp, Logic.strip_imp_concl imp)
end
-fun mk_case P ctxt premise =
+
+fun mk_scheme' ctxt cases concl =
let
- val (ctxt', qs, prems, concl) = dest_hhf ctxt premise
- val _ $ (_ $ lhs) = concl
-
- fun mk_rcinfo pr =
+ fun mk_branch concl =
let
- val (ctxt'', Gvs, Gas, _ $ (_ $ rcarg)) = dest_hhf ctxt' pr
+ val (ctxt', ws, Cs, _ $ Pxs) = dest_hhf ctxt concl
+ val (P, xs) = strip_comb Pxs
in
- (Gvs, Gas, rcarg)
+ SchemeBranch { P=P, xs=map dest_Free xs, ws=ws, Cs=Cs }
end
- val (gs, rcprs) = take_prefix (not o exists_aterm (fn Free v => v = P | _ => false)) prems
+ val (branches, cases') = (* correction *)
+ case Logic.dest_conjunction_list concl of
+ [conc] =>
+ let
+ val _ $ Pxs = Logic.strip_assums_concl conc
+ val (P, _) = strip_comb Pxs
+ val (cases', conds) = take_prefix (Term.exists_subterm (curry op aconv P)) cases
+ val concl' = fold_rev (curry Logic.mk_implies) conds conc
+ in
+ ([mk_branch concl'], cases')
+ end
+ | concls => (map mk_branch concls, cases)
+
+ fun mk_case premise =
+ let
+ val (ctxt', qs, prems, _ $ Plhs) = dest_hhf ctxt premise
+ val (P, lhs) = strip_comb Plhs
+
+ fun bidx Q = find_index (fn SchemeBranch {P=P',...} => Q aconv P') branches
+
+ fun mk_rcinfo pr =
+ let
+ val (ctxt'', Gvs, Gas, _ $ Phyp) = dest_hhf ctxt' pr
+ val (P', rcs) = strip_comb Phyp
+ in
+ (bidx P', Gvs, Gas, rcs)
+ end
+
+ fun is_pred v = exists (fn SchemeBranch {P,...} => v aconv P) branches
+
+ val (gs, rcprs) =
+ take_prefix (not o Term.exists_subterm is_pred) prems
+ in
+ SchemeCase {bidx=bidx P, qs=qs, oqnames=map fst qs(*FIXME*), gs=gs, lhs=lhs, rs=map mk_rcinfo rcprs}
+ end
+
+ fun PT_of (SchemeBranch { xs, ...}) =
+ foldr1 HOLogic.mk_prodT (map snd xs)
+
+ val ST = BalancedTree.make (uncurry SumTree.mk_sumT) (map PT_of branches)
in
- SchemeCase {qs=qs, gs=gs, lhs=lhs, rs=map mk_rcinfo rcprs}
+ IndScheme {T=ST, cases=map mk_case cases', branches=branches }
end
-fun mk_scheme' ctxt cases (Pn, PT) =
- IndScheme {T=domain_type PT, cases=map (mk_case (Pn,PT) ctxt) cases }
+
-fun mk_completeness ctxt (IndScheme {T, cases}) =
+fun mk_completeness ctxt (IndScheme {cases, branches, ...}) bidx =
let
- val allqnames = fold (fn SchemeCase {qs, ...} => fold (insert (op =) o Free) qs) cases []
- val [Pbool, x] = map Free (Variable.variant_frees ctxt allqnames [("P", HOLogic.boolT), ("x", T)])
+ val SchemeBranch { xs, ws, Cs, ... } = nth branches bidx
+ val relevant_cases = filter (fn SchemeCase {bidx=bidx', ...} => bidx' = bidx) cases
+
+ val allqnames = fold (fn SchemeCase {qs, ...} => fold (insert (op =) o Free) qs) relevant_cases []
+ val (Pbool :: xs') = map Free (Variable.variant_frees ctxt allqnames (("P", HOLogic.boolT) :: xs))
+ val Cs' = map (Pattern.rewrite_term (ProofContext.theory_of ctxt) (filter_out (op aconv) (map Free xs ~~ xs')) []) Cs
- fun mk_case (SchemeCase {qs, gs, lhs, ...}) =
+ fun mk_case (SchemeCase {qs, oqnames, gs, lhs, ...}) =
HOLogic.mk_Trueprop Pbool
- |> curry Logic.mk_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, lhs)))
+ |> fold_rev (fn x_l => curry Logic.mk_implies (HOLogic.mk_Trueprop(HOLogic.mk_eq x_l)))
+ (xs' ~~ lhs)
|> fold_rev (curry Logic.mk_implies) gs
- |> fold_rev (mk_forall o Free) qs
+ |> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
in
HOLogic.mk_Trueprop Pbool
- |> fold_rev (curry Logic.mk_implies o mk_case) cases
- |> mk_forall_rename ("x", x)
+ |> fold_rev (curry Logic.mk_implies o mk_case) relevant_cases
+ |> fold_rev (curry Logic.mk_implies) Cs'
+ |> fold_rev (mk_forall o Free) ws
+ |> fold_rev mk_forall_rename (map fst xs ~~ xs')
|> mk_forall_rename ("P", Pbool)
end
fun mk_wf ctxt R (IndScheme {T, ...}) =
HOLogic.Trueprop $ (Const (@{const_name "wf"}, mk_relT T --> HOLogic.boolT) $ R)
-fun mk_ineqs R (IndScheme {T, cases}) =
+fun mk_ineqs R (IndScheme {T, cases, branches}) =
let
- fun f (SchemeCase {qs, gs, lhs, rs, ...}) =
+ fun inject i ts =
+ SumTree.mk_inj T (length branches) (i + 1) (foldr1 HOLogic.mk_prod ts)
+
+ val thesis = Free ("thesis", HOLogic.boolT) (* FIXME *)
+
+ fun mk_pres bdx args =
let
- fun g (Gvs, Gas, rcarg) =
- HOLogic.mk_mem (HOLogic.mk_prod (rcarg, lhs), R)
- |> HOLogic.mk_Trueprop
- |> fold_rev (curry Logic.mk_implies) Gas
- |> fold_rev (curry Logic.mk_implies) gs
- |> fold_rev (mk_forall o Free) Gvs
- |> fold_rev (mk_forall o Free) qs
+ val SchemeBranch { xs, ws, Cs, ... } = nth branches bdx
+ fun replace (x, v) t = betapply (lambda (Free x) t, v)
+ val Cs' = map (fold replace (xs ~~ args)) Cs
+ val cse =
+ HOLogic.mk_Trueprop thesis
+ |> fold_rev (curry Logic.mk_implies) Cs'
+ |> fold_rev (mk_forall o Free) ws
+ in
+ Logic.mk_implies (cse, HOLogic.mk_Trueprop thesis)
+ end
+
+ fun f (SchemeCase {bidx, qs, oqnames, gs, lhs, rs, ...}) =
+ let
+ fun g (bidx', Gvs, Gas, rcarg) =
+ let val export =
+ fold_rev (curry Logic.mk_implies) Gas
+ #> fold_rev (curry Logic.mk_implies) gs
+ #> fold_rev (mk_forall o Free) Gvs
+ #> fold_rev mk_forall_rename (oqnames ~~ map Free qs)
+ in
+ (HOLogic.mk_mem (HOLogic.mk_prod (inject bidx' rcarg, inject bidx lhs), R)
+ |> HOLogic.mk_Trueprop
+ |> export,
+ mk_pres bidx' rcarg
+ |> export
+ |> mk_forall thesis)
+ end
in
map g rs
end
@@ -106,11 +198,37 @@
end
-fun mk_induct_rule ctxt R P x complete_thm wf_thm ineqss (IndScheme {T, cases=scases}) =
+fun mk_hol_imp a b = HOLogic.imp $ a $ b
+
+fun mk_ind_goal thy branches =
let
+ fun brnch (SchemeBranch { P, xs, ws, Cs, ... }) =
+ HOLogic.mk_Trueprop (list_comb (P, map Free xs))
+ |> fold_rev (curry Logic.mk_implies) Cs
+ |> fold_rev (mk_forall o Free) ws
+ |> term_conv thy ind_atomize
+ |> ObjectLogic.drop_judgment thy
+ |> tupled_lambda (foldr1 HOLogic.mk_prod (map Free xs))
+ in
+ SumTree.mk_sumcases HOLogic.boolT (map brnch branches)
+ end
+
+
+fun mk_induct_rule ctxt R x complete_thms wf_thm ineqss (IndScheme {T, cases=scases, branches}) =
+ let
+ val n = length branches
+
+ val scases_idx = map_index I scases
+
+ fun inject i ts =
+ SumTree.mk_inj T n (i + 1) (foldr1 HOLogic.mk_prod ts)
+ val P_of = nth (map (fn (SchemeBranch { P, ... }) => P) branches)
+
val thy = ProofContext.theory_of ctxt
val cert = cterm_of thy
+ val P_comp = mk_ind_goal thy branches
+
(* Inductive Hypothesis: !!z. (z,x):R ==> P z *)
val ihyp = all T $ Abs ("z", T,
implies $
@@ -118,63 +236,106 @@
Const ("op :", HOLogic.mk_prodT (T, T) --> mk_relT T --> HOLogic.boolT)
$ (HOLogic.pair_const T T $ Bound 0 $ x)
$ R)
- $ HOLogic.mk_Trueprop (P $ Bound 0))
+ $ HOLogic.mk_Trueprop (P_comp $ Bound 0))
|> cert
val aihyp = assume ihyp
- fun prove_case (SchemeCase {qs, gs, lhs, rs, ...}) ineqs =
+ (* Rule for case splitting along the sum types *)
+ val xss = map (fn (SchemeBranch { xs, ... }) => map Free xs) branches
+ val pats = map_index (uncurry inject) xss
+ val sum_split_rule = FundefDatatype.prove_completeness thy [x] (P_comp $ x) xss (map single pats)
+
+ fun prove_branch (bidx, (SchemeBranch { P, xs, ws, Cs, ... }, (complete_thm, pat))) =
let
- val case_hyp = assume (cert (HOLogic.Trueprop $ (HOLogic.mk_eq (x, lhs))))
+ val fxs = map Free xs
+ val branch_hyp = assume (cert (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, pat))))
+
+ val C_hyps = map (cert #> assume) Cs
+
+ val (relevant_cases, ineqss') = filter (fn ((_, SchemeCase {bidx=bidx', ...}), _) => bidx' = bidx) (scases_idx ~~ ineqss)
+ |> split_list
+
+ fun prove_case (cidx, SchemeCase {qs, oqnames, gs, lhs, rs, ...}) ineq_press =
+ let
+ val case_hyps = map (assume o cert o HOLogic.mk_Trueprop o HOLogic.mk_eq) (fxs ~~ lhs)
- val cqs = map (cert o Free) qs
- val ags = map (assume o cert) gs
-
- val replace_x_ss = HOL_basic_ss addsimps [case_hyp]
- val sih = full_simplify replace_x_ss aihyp
+ val cqs = map (cert o Free) qs
+ val ags = map (assume o cert) gs
+
+ val replace_x_ss = HOL_basic_ss addsimps (branch_hyp :: case_hyps)
+ val sih = full_simplify replace_x_ss aihyp
+
+ fun mk_Prec (idx, Gvs, Gas, rcargs) (ineq, pres) =
+ let
+ val cGas = map (assume o cert) Gas
+ val cGvs = map (cert o Free) Gvs
+ val import = fold forall_elim (cqs @ cGvs)
+ #> fold Thm.elim_implies (ags @ cGas)
+ val ipres = pres
+ |> forall_elim (cert (list_comb (P_of idx, rcargs)))
+ |> import
+ in
+ sih |> forall_elim (cert (inject idx rcargs))
+ |> Thm.elim_implies (import ineq) (* Psum rcargs *)
+ |> Conv.fconv_rule sum_prod_conv
+ |> Conv.fconv_rule ind_rulify
+ |> (fn th => th COMP ipres) (* P rs *)
+ |> fold_rev (implies_intr o cprop_of) cGas
+ |> fold_rev forall_intr cGvs
+ end
- fun mk_Prec (Gvs, Gas, rcarg) ineq =
- let
- val cGas = map (assume o cert) Gas
- val cGvs = map (cert o Free) Gvs
- val loc_ineq = ineq
- |> fold forall_elim (cqs @ cGvs)
- |> fold Thm.elim_implies (ags @ cGas)
+ val P_recs = map2 mk_Prec rs ineq_press (* [P rec1, P rec2, ... ] *)
+
+ val step = HOLogic.mk_Trueprop (list_comb (P, lhs))
+ |> fold_rev (curry Logic.mk_implies o prop_of) P_recs
+ |> fold_rev (curry Logic.mk_implies) gs
+ |> fold_rev (mk_forall o Free) qs
+ |> cert
+
+ val Plhs_to_Pxs_conv =
+ foldl1 (uncurry Conv.combination_conv)
+ (Conv.all_conv :: map (fn ch => K (Thm.symmetric (ch RS eq_reflection))) case_hyps)
+
+ val res = assume step
+ |> fold forall_elim cqs
+ |> fold Thm.elim_implies ags
+ |> fold Thm.elim_implies P_recs (* P lhs *)
+ |> Conv.fconv_rule (Conv.arg_conv Plhs_to_Pxs_conv) (* P xs *)
+ |> fold_rev (implies_intr o cprop_of) (ags @ case_hyps)
+ |> fold_rev forall_intr cqs (* !!qs. Gas ==> xs = lhss ==> P xs *)
in
- sih |> forall_elim (cert rcarg)
- |> Thm.elim_implies loc_ineq
- |> fold_rev (implies_intr o cprop_of) cGas
- |> fold_rev forall_intr cGvs
+ (res, (cidx, step))
end
-
- val P_recs = map2 mk_Prec rs ineqs (* [P rec1, P rec2, ... ] *)
-
- val step = HOLogic.mk_Trueprop (P $ lhs)
- |> fold_rev (curry Logic.mk_implies o prop_of) P_recs
- |> fold_rev (curry Logic.mk_implies) gs
- |> fold_rev (mk_forall o Free) qs
- |> cert
-
- val res = assume step
- |> fold forall_elim cqs
- |> fold Thm.elim_implies ags
- |> fold Thm.elim_implies P_recs
- |> Conv.fconv_rule
- (Conv.arg_conv (Conv.arg_conv (K (Thm.symmetric (case_hyp RS eq_reflection)))))
- (* "P x" *)
- |> implies_intr (cprop_of case_hyp)
- |> fold_rev (implies_intr o cprop_of) ags
- |> fold_rev forall_intr cqs
+
+ val (cases, steps) = split_list (map2 prove_case relevant_cases ineqss')
+
+ val bstep = complete_thm
+ |> forall_elim (cert (list_comb (P, fxs)))
+ |> fold (forall_elim o cert) (fxs @ map Free ws)
+ |> fold Thm.elim_implies C_hyps (* FIXME: optimization using rotate_prems *)
+ |> fold Thm.elim_implies cases (* P xs *)
+ |> fold_rev (implies_intr o cprop_of) C_hyps
+ |> fold_rev (forall_intr o cert o Free) ws
+
+ val Pxs = cert (HOLogic.mk_Trueprop (P_comp $ x))
+ |> Goal.init
+ |> (MetaSimplifier.rewrite_goals_tac (map meta (branch_hyp :: @{thm split_conv} :: @{thms sum_cases}))
+ THEN CONVERSION ind_rulify 1)
+ |> Seq.hd
+ |> Thm.elim_implies bstep
+ |> Goal.finish
+ |> implies_intr (cprop_of branch_hyp)
+ |> fold_rev (forall_intr o cert) fxs
in
- (res, step)
+ (Pxs, steps)
end
-
- val (cases, steps) = split_list (map2 prove_case scases ineqss)
+
+ val (branches, steps) = split_list (map_index prove_branch (branches ~~ (complete_thms ~~ pats)))
+ |> apsnd flat
- val istep = complete_thm
- |> forall_elim (cert (P $ x))
- |> forall_elim (cert x)
- |> fold (Thm.elim_implies) cases
+ val istep = sum_split_rule
+ |> fold (fn b => fn th => Drule.compose_single (b, 1, th)) branches
|> implies_intr ihyp
|> forall_intr (cert x) (* "!!x. (!!y<x. P y) ==> P x" *)
@@ -182,100 +343,60 @@
@{thm "wf_induct_rule"}
|> (curry op COMP) wf_thm
|> (curry op COMP) istep
- |> fold_rev implies_intr steps
- |> forall_intr (cert P)
+
+ val steps_sorted = map snd (sort (int_ord o pairself fst) steps)
in
- induct_rule
+ (steps_sorted, induct_rule)
end
-fun mk_ind_tac ctxt facts = (ALLGOALS (Method.insert_tac facts)) THEN HEADGOAL
+
+fun mk_ind_tac comp_tac pres_tac term_tac ctxt facts = (ALLGOALS (Method.insert_tac facts)) THEN HEADGOAL
(SUBGOAL (fn (t, i) =>
let
val (ctxt', _, cases, concl) = dest_hhf ctxt t
-
- fun get_types t =
- let
- val (P, vs) = strip_comb (HOLogic.dest_Trueprop t)
- val Ts = map fastype_of vs
- val tupT = foldr1 HOLogic.mk_prodT Ts
- in
- ((P, Ts), tupT)
- end
-
- val concls = Logic.dest_conjunction_list (Logic.strip_imp_concl concl)
- val (PTss, tupTs) = split_list (map get_types concls)
-
- val n = length tupTs
- val ST = BalancedTree.make (uncurry SumTree.mk_sumT) tupTs
-
- val ([Psn, Rn, xn], ctxt'') = Variable.variant_fixes ["Psum", "R", "x"] ctxt'
- val Psum = (Psn, ST --> HOLogic.boolT)
+ val scheme as IndScheme {T=ST, branches, ...} = mk_scheme' ctxt' cases concl
+(* val _ = Output.tracing (makestring scheme)*)
+ val ([Rn,xn], ctxt'') = Variable.variant_fixes ["R","x"] ctxt'
val R = Free (Rn, mk_relT ST)
val x = Free (xn, ST)
-
- fun mk_rews (i, (P, Ts)) =
- let
- val vs = map_index (fn (j,T) => Free ("x" ^ string_of_int j, T)) Ts
- val t = Free Psum $ SumTree.mk_inj ST n (i + 1) (foldr1 HOLogic.mk_prod vs)
- |> fold_rev lambda vs
- in
- (P, t)
- end
-
- val rews = map_index mk_rews PTss
- val thy = ProofContext.theory_of ctxt''
- val cases' = map (Pattern.rewrite_term thy rews []) cases
-
- val scheme = mk_scheme' ctxt'' cases' Psum
-
- val cert = cterm_of thy
+ val cert = cterm_of (ProofContext.theory_of ctxt)
val ineqss = mk_ineqs R scheme
- |> map (map (assume o cert))
- val complete = mk_completeness ctxt scheme |> cert |> assume
+ |> map (map (pairself (assume o cert)))
+ val complete = map (mk_completeness ctxt scheme #> cert #> assume) (0 upto (length branches - 1))
val wf_thm = mk_wf ctxt R scheme |> cert |> assume
- val indthm = mk_induct_rule ctxt'' R (Free Psum) x complete wf_thm ineqss scheme
+ val (descent, pres) = split_list (flat ineqss)
+ val newgoals = complete @ pres @ wf_thm :: descent
- fun mk_P (P, Ts) =
+ val (steps, indthm) = mk_induct_rule ctxt'' R x complete wf_thm ineqss scheme
+
+ fun project (i, SchemeBranch {xs, ...}) =
let
- val avars = map_index (fn (i,T) => Var (("a", i), T)) Ts
- val atup = foldr1 HOLogic.mk_prod avars
+ val inst = cert (SumTree.mk_inj ST (length branches) (i + 1) (foldr1 HOLogic.mk_prod (map Free xs)))
in
- tupled_lambda atup (list_comb (P, avars))
- end
-
- val case_exp = cert (SumTree.mk_sumcases HOLogic.boolT (map mk_P PTss))
- val acases = map (assume o cert) cases
- val indthm' = indthm |> forall_elim case_exp
- |> full_simplify SumTree.sumcase_split_ss
- |> fold Thm.elim_implies acases
-
- fun project (i,t) =
- let
- val (P, vs) = strip_comb (HOLogic.dest_Trueprop t)
- val inst = cert (SumTree.mk_inj ST n (i + 1) (foldr1 HOLogic.mk_prod vs))
- in
- indthm' |> Drule.instantiate' [] [SOME inst]
- |> simplify SumTree.sumcase_split_ss
+ indthm |> Drule.instantiate' [] [SOME inst]
+ |> simplify SumTree.sumcase_split_ss
+ |> Conv.fconv_rule ind_rulify
+(* |> (fn thm => (Output.tracing (makestring thm); thm))*)
end
- val res = Conjunction.intr_balanced (map_index project concls)
- |> fold_rev (implies_intr o cprop_of) acases
- |> Thm.forall_elim_vars 0
- in
- (fn st =>
- Drule.compose_single (res, i, st)
- |> fold_rev (implies_intr o cprop_of) (complete :: wf_thm :: flat ineqss)
- |> forall_intr (cert R)
- |> Thm.forall_elim_vars 0
- |> Seq.single
- )
+ val res = Conjunction.intr_balanced (map_index project branches)
+ |> fold_rev implies_intr (map cprop_of newgoals @ steps)
+ |> (fn thm => Thm.generalize ([], [Rn]) (Thm.maxidx_of thm + 1) thm)
+
+ val nbranches = length branches
+ val npres = length pres
+ in
+ Thm.compose_no_flatten false (res, length newgoals) i
+ THEN term_tac (i + nbranches + npres)
+ THEN (EVERY (map (TRY o pres_tac) ((i + nbranches + npres - 1) downto (i + nbranches))))
+ THEN (EVERY (map (TRY o comp_tac) ((i + nbranches - 1) downto i)))
end))
val setup = Method.add_methods
- [("induct_scheme", Method.ctxt_args (Method.RAW_METHOD o mk_ind_tac),
+ [("induct_scheme", Method.ctxt_args (Method.RAW_METHOD o (fn ctxt => mk_ind_tac (K all_tac) (assume_tac APPEND' Goal.assume_rule_tac ctxt) (K all_tac) ctxt)),
"proves an induction principle")]
end