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(* Title: HOL/Tools/Function/mutual.ML
Author: Alexander Krauss, TU Muenchen
Mutual recursive function definitions.
*)
signature FUNCTION_MUTUAL =
sig
val prepare_function_mutual : Function_Common.function_config
-> binding (* defname *)
-> ((binding * typ) * mixfix) list
-> term list
-> local_theory
-> ((thm (* goalstate *)
* (Proof.context -> thm -> Function_Common.function_result) (* proof continuation *)
) * local_theory)
end
structure Function_Mutual: FUNCTION_MUTUAL =
struct
open Function_Lib
open Function_Common
type qgar = string * (string * typ) list * term list * term list * term
datatype mutual_part = MutualPart of
{i : int,
i' : int,
fname : binding,
fT : typ,
cargTs: typ list,
f_def: term,
f: term option,
f_defthm : thm option}
datatype mutual_info = Mutual of
{n : int,
n' : int,
fsum_name : binding,
fsum_type: typ,
ST: typ,
RST: typ,
parts: mutual_part list,
fqgars: qgar list,
qglrs: ((string * typ) list * term list * term * term) list,
fsum : term option}
fun mutual_induct_Pnames n =
if n < 5 then fst (chop n ["P","Q","R","S"])
else map (fn i => "P" ^ string_of_int i) (1 upto n)
fun get_part f =
the o find_first (fn (MutualPart {fname, ...}) => Binding.name_of fname = f)
(* FIXME *)
fun mk_prod_abs e (t1, t2) =
let
val bTs = rev (map snd e)
val T1 = fastype_of1 (bTs, t1)
val T2 = fastype_of1 (bTs, t2)
in
HOLogic.pair_const T1 T2 $ t1 $ t2
end
fun analyze_eqs ctxt defname fs eqs =
let
val num = length fs
val fqgars = map (split_def ctxt (K true)) eqs
fun arity_of fname =
the (get_first (fn (f, _, _, args, _) =>
if f = Binding.name_of fname then SOME (length args) else NONE) fqgars)
fun curried_types (fname, fT) =
let val (caTs, uaTs) = chop (arity_of fname) (binder_types fT)
in (caTs, uaTs ---> body_type fT) end
val (caTss, resultTs) = split_list (map curried_types fs)
val argTs = map (foldr1 HOLogic.mk_prodT) caTss
val dresultTs = distinct (op =) resultTs
val n' = length dresultTs
val RST = Balanced_Tree.make (uncurry Sum_Tree.mk_sumT) dresultTs
val ST = Balanced_Tree.make (uncurry Sum_Tree.mk_sumT) argTs
val fsum_name = derived_name_suffix defname "_sum"
val ([fsum_var_name], _) = Variable.add_fixes_binding [fsum_name] ctxt
val fsum_type = ST --> RST
val fsum_var = (fsum_var_name, fsum_type)
fun define (fname, fT) caTs resultT i =
let
val vars = map_index (fn (j,T) => Free ("x" ^ string_of_int j, T)) caTs (* FIXME: Bind xs properly *)
val i' = find_index (fn Ta => Ta = resultT) dresultTs + 1
val f_exp = Sum_Tree.mk_proj RST n' i' (Free fsum_var $ Sum_Tree.mk_inj ST num i (foldr1 HOLogic.mk_prod vars))
val def = Term.abstract_over (Free fsum_var, fold_rev lambda vars f_exp)
val rew = (Binding.name_of fname, fold_rev lambda vars f_exp)
in
(MutualPart
{i = i, i' = i', fname = fname, fT = fT, cargTs = caTs,
f_def = def, f = NONE, f_defthm = NONE}, rew)
end
val (parts, rews) = split_list (@{map 4} define fs caTss resultTs (1 upto num))
fun convert_eqs (f, qs, gs, args, rhs) =
let
val MutualPart {i, i', ...} = get_part f parts
val rhs' = rhs
|> map_aterms (fn t as Free (n, _) => the_default t (AList.lookup (op =) rews n) | t => t)
in
(qs, gs, Sum_Tree.mk_inj ST num i (foldr1 (mk_prod_abs qs) args),
Envir.beta_norm (Sum_Tree.mk_inj RST n' i' rhs'))
end
val qglrs = map convert_eqs fqgars
in
Mutual {n = num, n' = n', fsum_name = fsum_name, fsum_type = fsum_type,
ST = ST, RST = RST, parts = parts, fqgars = fqgars, qglrs = qglrs, fsum = NONE}
end
fun define_projections fixes mutual fsum lthy =
let
fun def ((MutualPart {i=i, i'=i', fname, fT, cargTs, f_def, ...}), (_, mixfix)) lthy =
let
val def_binding = Thm.make_def_binding (Config.get lthy function_internals) fname
val ((f, (_, f_defthm)), lthy') =
Local_Theory.define
((fname, mixfix), ((def_binding, []), Term.subst_bound (fsum, f_def))) lthy
in
(MutualPart {i = i, i' = i', fname = fname, fT = fT, cargTs = cargTs,
f_def = f_def, f = SOME f, f_defthm = SOME f_defthm}, lthy')
end
val Mutual {n, n', fsum_name, fsum_type, ST, RST, parts, fqgars, qglrs, ...} = mutual
val (parts', lthy') = fold_map def (parts ~~ fixes) lthy
in
(Mutual {n = n, n' = n', fsum_name = fsum_name, fsum_type = fsum_type, ST = ST,
RST = RST, parts = parts', fqgars = fqgars, qglrs = qglrs, fsum = SOME fsum}, lthy')
end
fun in_context ctxt (f, pre_qs, pre_gs, pre_args, pre_rhs) F =
let
val oqnames = map fst pre_qs
val (qs, _) = Variable.variant_fixes oqnames ctxt
|>> map2 (fn (_, T) => fn n => Free (n, T)) pre_qs
fun inst t = subst_bounds (rev qs, t)
val gs = map inst pre_gs
val args = map inst pre_args
val rhs = inst pre_rhs
val cqs = map (Thm.cterm_of ctxt) qs
val ags = map (Thm.assume o Thm.cterm_of ctxt) gs
val import = fold Thm.forall_elim cqs
#> fold Thm.elim_implies ags
val export = fold_rev (Thm.implies_intr o Thm.cprop_of) ags
#> fold_rev forall_intr_rename (oqnames ~~ cqs)
in
F ctxt (f, qs, gs, args, rhs) import export
end
fun recover_mutual_psimp all_orig_fdefs parts ctxt (fname, _, _, args, rhs)
import (export : thm -> thm) sum_psimp_eq =
let
val (MutualPart {f=SOME f, ...}) = get_part fname parts
val psimp = import sum_psimp_eq
val (simp, restore_cond) =
case cprems_of psimp of
[] => (psimp, I)
| [cond] => (Thm.implies_elim psimp (Thm.assume cond), Thm.implies_intr cond)
| _ => raise General.Fail "Too many conditions"
val simp_ctxt = fold Thm.declare_hyps (Thm.chyps_of simp) ctxt
in
Goal.prove simp_ctxt [] []
(HOLogic.Trueprop $ HOLogic.mk_eq (list_comb (f, args), rhs))
(fn _ =>
Local_Defs.unfold0_tac ctxt all_orig_fdefs
THEN EqSubst.eqsubst_tac ctxt [0] [simp] 1
THEN (simp_tac ctxt) 1)
|> restore_cond
|> export
end
fun mk_applied_form ctxt caTs thm =
let
val xs =
map_index (fn (i, T) =>
Thm.cterm_of ctxt
(Free ("x" ^ string_of_int i, T))) caTs (* FIXME: Bind xs properly *)
in
fold (fn x => fn thm => Thm.combination thm (Thm.reflexive x)) xs thm
|> Conv.fconv_rule (Thm.beta_conversion true)
|> fold_rev Thm.forall_intr xs
|> Thm.forall_elim_vars 0
end
fun mutual_induct_rules ctxt induct all_f_defs (Mutual {n, ST, parts, ...}) =
let
val newPs =
map2 (fn Pname => fn MutualPart {cargTs, ...} =>
Free (Pname, cargTs ---> HOLogic.boolT))
(mutual_induct_Pnames (length parts)) parts
fun mk_P (MutualPart {cargTs, ...}) P =
let
val avars = map_index (fn (i,T) => Var (("a", i), T)) cargTs
val atup = foldr1 HOLogic.mk_prod avars
in
HOLogic.tupled_lambda atup (list_comb (P, avars))
end
val Ps = map2 mk_P parts newPs
val case_exp = Sum_Tree.mk_sumcases HOLogic.boolT Ps
val induct_inst =
Thm.forall_elim (Thm.cterm_of ctxt case_exp) induct
|> full_simplify (put_simpset Sum_Tree.sumcase_split_ss ctxt)
|> full_simplify (put_simpset HOL_basic_ss ctxt addsimps all_f_defs)
fun project rule (MutualPart {cargTs, i, ...}) k =
let
val afs = map_index (fn (j,T) => Free ("a" ^ string_of_int (j + k), T)) cargTs (* FIXME! *)
val inj = Sum_Tree.mk_inj ST n i (foldr1 HOLogic.mk_prod afs)
in
(rule
|> Thm.forall_elim (Thm.cterm_of ctxt inj)
|> full_simplify (put_simpset Sum_Tree.sumcase_split_ss ctxt)
|> fold_rev (Thm.forall_intr o Thm.cterm_of ctxt) (afs @ newPs),
k + length cargTs)
end
in
fst (fold_map (project induct_inst) parts 0)
end
fun mutual_cases_rule ctxt cases_rule n ST (MutualPart {i, cargTs = Ts, ...}) =
let
val [P, x] =
Variable.variant_frees ctxt [] [("P", \<^typ>\<open>bool\<close>), ("x", HOLogic.mk_tupleT Ts)]
|> map (Thm.cterm_of ctxt o Free);
val sumtree_inj = Thm.cterm_of ctxt (Sum_Tree.mk_inj ST n i (Thm.term_of x));
fun prep_subgoal_tac i =
REPEAT (eresolve_tac ctxt
@{thms Pair_inject Inl_inject [elim_format] Inr_inject [elim_format]} i)
THEN REPEAT (eresolve_tac ctxt
@{thms HOL.notE [OF Sum_Type.sum.distinct(1)] HOL.notE [OF Sum_Type.sum.distinct(2)]} i);
in
cases_rule
|> Thm.forall_elim P
|> Thm.forall_elim sumtree_inj
|> Tactic.rule_by_tactic ctxt (ALLGOALS prep_subgoal_tac)
|> Thm.forall_intr x
|> Thm.forall_intr P
|> Thm.solve_constraints
end
fun mk_partial_rules_mutual lthy inner_cont (m as Mutual {parts, fqgars, n, ST, ...}) proof =
let
val result = inner_cont proof
val FunctionResult {G, R, cases=[cases_rule], psimps, simple_pinducts=[simple_pinduct],
termination, domintros, dom, pelims, ...} = result
val (all_f_defs, fs) =
map (fn MutualPart {f_defthm = SOME f_def, f = SOME f, cargTs, ...} =>
(mk_applied_form lthy cargTs (Thm.symmetric f_def), f))
parts
|> split_list
val all_orig_fdefs =
map (fn MutualPart {f_defthm = SOME f_def, ...} => f_def) parts
fun mk_mpsimp fqgar sum_psimp =
in_context lthy fqgar (recover_mutual_psimp all_orig_fdefs parts) sum_psimp
val rew_simpset = put_simpset HOL_basic_ss lthy addsimps all_f_defs
val mpsimps = map2 mk_mpsimp fqgars psimps
val minducts = mutual_induct_rules lthy simple_pinduct all_f_defs m
val mcases = map (mutual_cases_rule lthy cases_rule n ST) parts
val mtermination = full_simplify rew_simpset termination
val mdomintros = Option.map (map (full_simplify rew_simpset)) domintros
in
FunctionResult { fs=fs, G=G, R=R, dom=dom,
psimps=mpsimps, simple_pinducts=minducts,
cases=mcases, pelims=pelims, termination=mtermination,
domintros=mdomintros}
end
fun prepare_function_mutual config defname fixes eqss lthy =
let
val mutual as Mutual {fsum_name, fsum_type, qglrs, ...} =
analyze_eqs lthy defname (map fst fixes) (map Envir.beta_eta_contract eqss)
val ((fsum, goalstate, cont), lthy') =
Function_Core.prepare_function config defname [((fsum_name, fsum_type), NoSyn)] qglrs lthy
val (mutual', lthy'') = define_projections fixes mutual fsum lthy'
fun cont' ctxt = mk_partial_rules_mutual lthy'' (cont ctxt) mutual'
in
((goalstate, cont'), lthy'')
end
end
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