(* Title: HOL/Tools/Function/induction_schema.ML
Author: Alexander Krauss, TU Muenchen
A method to prove induction schemas.
*)
signature INDUCTION_SCHEMA =
sig
val mk_ind_tac : (int -> tactic) -> (int -> tactic) -> (int -> tactic)
-> Proof.context -> thm list -> tactic
val induction_schema_tac : Proof.context -> thm list -> tactic
val setup : theory -> theory
end
structure Induction_Schema : INDUCTION_SCHEMA =
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 = Raw_Simplifier.rewrite true @{thms induct_atomize}
val ind_rulify = Raw_Simplifier.rewrite true @{thms induct_rulify}
fun meta thm = thm RS eq_reflection
val sum_prod_conv = Raw_Simplifier.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 ((vars, imp), ctxt') = Function_Lib.focus_term t ctxt
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 (_, 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_conjunctions 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 (_, 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 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_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
|> Object_Logic.drop_judgment thy
|> HOLogic.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 (@{const_name Set.member}, 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 = Thm.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 = Thm.assume (cert (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, pat))))
val C_hyps = map (cert #> Thm.assume) Cs
val (relevant_cases, ineqss') =
(scases_idx ~~ ineqss)
|> filter (fn ((_, SchemeCase {bidx=bidx', ...}), _) => bidx' = bidx)
|> split_list
fun prove_case (cidx, SchemeCase {qs, gs, lhs, rs, ...}) ineq_press =
let
val case_hyps =
map (Thm.assume o cert o HOLogic.mk_Trueprop o HOLogic.mk_eq) (fxs ~~ lhs)
val cqs = map (cert o Free) qs
val ags = map (Thm.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 (Thm.assume o cert) Gas
val cGvs = map (cert o Free) Gvs
val import = fold Thm.forall_elim (cqs @ cGvs)
#> fold Thm.elim_implies (ags @ cGas)
val ipres = pres
|> Thm.forall_elim (cert (list_comb (P_of idx, rcargs)))
|> import
in
sih
|> Thm.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 (Thm.implies_intr o cprop_of) cGas
|> fold_rev Thm.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 = Thm.assume step
|> fold Thm.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 (Thm.implies_intr o cprop_of) (ags @ case_hyps)
|> fold_rev Thm.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
|> Thm.forall_elim (cert (list_comb (P, fxs)))
|> fold (Thm.forall_elim o cert) (fxs @ map Free ws)
|> fold Thm.elim_implies C_hyps
|> fold Thm.elim_implies cases (* P xs *)
|> fold_rev (Thm.implies_intr o cprop_of) C_hyps
|> fold_rev (Thm.forall_intr o cert o Free) ws
val Pxs = cert (HOLogic.mk_Trueprop (P_comp $ x))
|> Goal.init
|> (Simplifier.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
|> Thm.implies_intr (cprop_of branch_hyp)
|> fold_rev (Thm.forall_intr o cert) fxs
in
(Pxs, steps)
end
val (branches, steps) =
map_index prove_branch (branches ~~ (complete_thms ~~ pats))
|> split_list |> apsnd flat
val istep = sum_split_rule
|> fold (fn b => fn th => Drule.compose_single (b, 1, th)) branches
|> Thm.implies_intr ihyp
|> Thm.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 ([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 (Thm.assume o cert)))
val complete =
map_range (mk_completeness ctxt scheme #> cert #> Thm.assume) (length branches)
val wf_thm = mk_wf R scheme |> cert |> Thm.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 = (foldr1 HOLogic.mk_prod (map Free xs))
|> SumTree.mk_inj ST (length branches) (i + 1)
|> cert
in
indthm
|> Drule.instantiate' [] [SOME inst]
|> simplify SumTree.sumcase_split_ss
|> Conv.fconv_rule ind_rulify
end
val res = Conjunction.intr_balanced (map_index project branches)
|> fold_rev Thm.implies_intr (map cprop_of newgoals @ steps)
|> Drule.generalize ([], [Rn])
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 induction_schema_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 induction_schema} (Scan.succeed (RAW_METHOD o induction_schema_tac))
"proves an induction principle"
end