--- a/src/HOL/Tools/Function/induction_scheme.ML Fri Nov 06 09:50:37 2009 -0800
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,405 +0,0 @@
-(* Title: HOL/Tools/Function/induction_scheme.ML
- Author: Alexander Krauss, TU Muenchen
-
-A method to prove induction schemes.
-*)
-
-signature INDUCTION_SCHEME =
-sig
- val mk_ind_tac : (int -> tactic) -> (int -> tactic) -> (int -> tactic)
- -> Proof.context -> thm list -> tactic
- val induct_scheme_tac : Proof.context -> thm list -> tactic
- val setup : theory -> theory
-end
-
-
-structure Induction_Scheme : INDUCTION_SCHEME =
-struct
-
-open Function_Lib
-
-
-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 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
- {
- 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))
-
-fun dest_hhf ctxt t =
- let
- val (ctxt', vars, imp) = dest_all_all_ctx ctxt t
- in
- (ctxt', vars, Logic.strip_imp_prems imp, Logic.strip_imp_concl imp)
- end
-
-
-fun mk_scheme' ctxt cases concl =
- let
- fun mk_branch concl =
- let
- val (ctxt', ws, Cs, _ $ Pxs) = dest_hhf ctxt concl
- val (P, xs) = strip_comb Pxs
- in
- SchemeBranch { P=P, xs=map dest_Free xs, ws=ws, Cs=Cs }
- end
-
- 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 = Balanced_Tree.make (uncurry SumTree.mk_sumT) (map PT_of branches)
- in
- IndScheme {T=ST, cases=map mk_case cases', branches=branches }
- end
-
-
-
-fun mk_completeness ctxt (IndScheme {cases, branches, ...}) bidx =
- let
- 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, oqnames, gs, lhs, ...}) =
- HOLogic.mk_Trueprop Pbool
- |> 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_rename (oqnames ~~ map Free qs)
- in
- HOLogic.mk_Trueprop Pbool
- |> fold_rev (curry Logic.mk_implies o mk_case) relevant_cases
- |> fold_rev (curry Logic.mk_implies) Cs'
- |> fold_rev (Logic.all 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, branches}) =
- let
- 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
- 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 (Logic.all 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 (Logic.all 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
- |> Logic.all thesis)
- end
- in
- map g rs
- end
- in
- map f cases
- end
-
-
-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 (Logic.all 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 = Term.all T $ Abs ("z", T,
- Logic.mk_implies
- (HOLogic.mk_Trueprop (
- 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_comp $ Bound 0)))
- |> cert
-
- val aihyp = assume ihyp
-
- (* 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 = Pat_Completeness.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 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 (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
-
- 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 (Logic.all 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
- (res, (cidx, step))
- end
-
- 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 (Conv.fconv_rule Drule.beta_eta_conversion bstep)
- |> Goal.finish ctxt
- |> implies_intr (cprop_of branch_hyp)
- |> fold_rev (forall_intr o cert) fxs
- in
- (Pxs, steps)
- end
-
- val (branches, steps) = split_list (map_index prove_branch (branches ~~ (complete_thms ~~ pats)))
- |> apsnd flat
-
- 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" *)
-
- val induct_rule =
- @{thm "wf_induct_rule"}
- |> (curry op COMP) wf_thm
- |> (curry op COMP) istep
-
- val steps_sorted = map snd (sort (int_ord o pairself fst) steps)
- in
- (steps_sorted, induct_rule)
- end
-
-
-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
- val scheme as IndScheme {T=ST, branches, ...} = mk_scheme' ctxt' cases concl
-(* val _ = 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)
- val cert = cterm_of (ProofContext.theory_of ctxt)
-
- val ineqss = mk_ineqs R scheme
- |> map (map (pairself (assume o cert)))
- val complete = map_range (mk_completeness ctxt scheme #> cert #> assume) (length branches)
- val wf_thm = mk_wf ctxt R scheme |> cert |> assume
-
- val (descent, pres) = split_list (flat ineqss)
- val newgoals = complete @ pres @ wf_thm :: descent
-
- val (steps, indthm) = mk_induct_rule ctxt'' R x complete wf_thm ineqss scheme
-
- fun project (i, SchemeBranch {xs, ...}) =
- let
- val inst = cert (SumTree.mk_inj ST (length branches) (i + 1) (foldr1 HOLogic.mk_prod (map Free xs)))
- in
- indthm |> Drule.instantiate' [] [SOME inst]
- |> simplify SumTree.sumcase_split_ss
- |> Conv.fconv_rule ind_rulify
-(* |> (fn thm => (tracing (makestring thm); thm))*)
- end
-
- 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))
-
-
-fun induct_scheme_tac ctxt =
- mk_ind_tac (K all_tac) (assume_tac APPEND' Goal.assume_rule_tac ctxt) (K all_tac) ctxt;
-
-val setup =
- Method.setup @{binding induct_scheme} (Scan.succeed (RAW_METHOD o induct_scheme_tac))
- "proves an induction principle"
-
-end