theory LamTest
imports "../Nominal2"
begin
atom_decl name
nominal_datatype lam =
Var "name"
| App "lam" "lam"
| Lam x::"name" l::"lam" bind x in l
ML {*
val trace = Unsynchronized.ref false
fun trace_msg msg = if ! trace then tracing (msg ()) else ()
val boolT = HOLogic.boolT
val mk_eq = HOLogic.mk_eq
open Function_Lib
open Function_Common
datatype globals = Globals of
{fvar: term,
domT: typ,
ranT: typ,
h: term,
y: term,
x: term,
z: term,
a: term,
P: term,
D: term,
Pbool:term}
datatype rec_call_info = RCInfo of
{RIvs: (string * typ) list, (* Call context: fixes and assumes *)
CCas: thm list,
rcarg: term, (* The recursive argument *)
llRI: thm,
h_assum: term}
datatype clause_context = ClauseContext of
{ctxt : Proof.context,
qs : term list,
gs : term list,
lhs: term,
rhs: term,
cqs: cterm list,
ags: thm list,
case_hyp : thm}
fun transfer_clause_ctx thy (ClauseContext { ctxt, qs, gs, lhs, rhs, cqs, ags, case_hyp }) =
ClauseContext { ctxt = ProofContext.transfer thy ctxt,
qs = qs, gs = gs, lhs = lhs, rhs = rhs, cqs = cqs, ags = ags, case_hyp = case_hyp }
datatype clause_info = ClauseInfo of
{no: int,
qglr : ((string * typ) list * term list * term * term),
cdata : clause_context,
tree: Function_Ctx_Tree.ctx_tree,
lGI: thm,
RCs: rec_call_info list}
*}
thm accp_induct_rule
ML {*
(* Theory dependencies. *)
val acc_induct_rule = @{thm accp_induct_rule}
val ex1_implies_ex = @{thm FunDef.fundef_ex1_existence}
val ex1_implies_un = @{thm FunDef.fundef_ex1_uniqueness}
val ex1_implies_iff = @{thm FunDef.fundef_ex1_iff}
val acc_downward = @{thm accp_downward}
val accI = @{thm accp.accI}
val case_split = @{thm HOL.case_split}
val fundef_default_value = @{thm FunDef.fundef_default_value}
val not_acc_down = @{thm not_accp_down}
*}
ML {*
fun find_calls tree =
let
fun add_Ri (fixes,assumes) (_ $ arg) _ (_, xs) =
([], (fixes, assumes, arg) :: xs)
| add_Ri _ _ _ _ = raise Match
in
rev (Function_Ctx_Tree.traverse_tree add_Ri tree [])
end
*}
ML {*
fun mk_eqvt_at (f_trm, arg_trm) =
let
val f_ty = fastype_of f_trm
val arg_ty = domain_type f_ty
in
Const (@{const_name eqvt_at}, [f_ty, arg_ty] ---> @{typ bool}) $ f_trm $ arg_trm
|> HOLogic.mk_Trueprop
end
*}
ML {*
fun find_calls2 f t =
let
fun aux (g $ arg) = aux g #> aux arg #> (if f = g then cons ((g, arg)) else I)
| aux (Abs (_, _, t)) = aux t
| aux _ = I
in
aux t []
end
*}
ML {*
(** building proof obligations *)
fun mk_compat_proof_obligations domT ranT fvar f glrs =
let
fun mk_impl ((qs, gs, lhs, rhs), (qs', gs', lhs', rhs')) =
let
val shift = incr_boundvars (length qs')
val RCs_rhs = find_calls2 fvar rhs
val RCs_rhs' = find_calls2 fvar rhs'
in
Logic.mk_implies
(HOLogic.mk_Trueprop (HOLogic.eq_const domT $ shift lhs $ lhs'),
HOLogic.mk_Trueprop (HOLogic.eq_const ranT $ shift rhs $ rhs'))
|> fold_rev (curry Logic.mk_implies) (map shift gs @ gs')
|> fold_rev (curry Logic.mk_implies) (map (shift o mk_eqvt_at) RCs_rhs) (* HERE *)
(*|> fold_rev (curry Logic.mk_implies) (map mk_eqvt_at RCs_rhs')*) (* HERE *)
|> fold_rev (fn (n,T) => fn b => Term.all T $ Abs(n,T,b)) (qs @ qs')
|> curry abstract_over fvar
|> curry subst_bound f
end
in
map mk_impl (unordered_pairs glrs)
end
*}
ML {*
fun mk_completeness (Globals {x, Pbool, ...}) clauses qglrs =
let
fun mk_case (ClauseContext {qs, gs, lhs, ...}, (oqs, _, _, _)) =
HOLogic.mk_Trueprop Pbool
|> curry Logic.mk_implies (HOLogic.mk_Trueprop (mk_eq (x, lhs)))
|> fold_rev (curry Logic.mk_implies) gs
|> fold_rev mk_forall_rename (map fst oqs ~~ qs)
in
HOLogic.mk_Trueprop Pbool
|> fold_rev (curry Logic.mk_implies o mk_case) (clauses ~~ qglrs)
|> mk_forall_rename ("x", x)
|> mk_forall_rename ("P", Pbool)
end
*}
(** making a context with it's own local bindings **)
ML {*
fun mk_clause_context x ctxt (pre_qs,pre_gs,pre_lhs,pre_rhs) =
let
val (qs, ctxt') = Variable.variant_fixes (map fst pre_qs) ctxt
|>> map2 (fn (_, T) => fn n => Free (n, T)) pre_qs
val thy = ProofContext.theory_of ctxt'
fun inst t = subst_bounds (rev qs, t)
val gs = map inst pre_gs
val lhs = inst pre_lhs
val rhs = inst pre_rhs
val cqs = map (cterm_of thy) qs
val ags = map (Thm.assume o cterm_of thy) gs
val case_hyp = Thm.assume (cterm_of thy (HOLogic.mk_Trueprop (mk_eq (x, lhs))))
in
ClauseContext { ctxt = ctxt', qs = qs, gs = gs, lhs = lhs, rhs = rhs,
cqs = cqs, ags = ags, case_hyp = case_hyp }
end
*}
ML {*
(* lowlevel term function. FIXME: remove *)
fun abstract_over_list vs body =
let
fun abs lev v tm =
if v aconv tm then Bound lev
else
(case tm of
Abs (a, T, t) => Abs (a, T, abs (lev + 1) v t)
| t $ u => abs lev v t $ abs lev v u
| t => t)
in
fold_index (fn (i, v) => fn t => abs i v t) vs body
end
fun mk_clause_info globals G f no cdata qglr tree RCs GIntro_thm RIntro_thms =
let
val Globals {h, ...} = globals
val ClauseContext { ctxt, qs, cqs, ags, ... } = cdata
val cert = Thm.cterm_of (ProofContext.theory_of ctxt)
(* Instantiate the GIntro thm with "f" and import into the clause context. *)
val lGI = GIntro_thm
|> Thm.forall_elim (cert f)
|> fold Thm.forall_elim cqs
|> fold Thm.elim_implies ags
fun mk_call_info (rcfix, rcassm, rcarg) RI =
let
val llRI = RI
|> fold Thm.forall_elim cqs
|> fold (Thm.forall_elim o cert o Free) rcfix
|> fold Thm.elim_implies ags
|> fold Thm.elim_implies rcassm
val h_assum =
HOLogic.mk_Trueprop (G $ rcarg $ (h $ rcarg))
|> fold_rev (curry Logic.mk_implies o prop_of) rcassm
|> fold_rev (Logic.all o Free) rcfix
|> Pattern.rewrite_term (ProofContext.theory_of ctxt) [(f, h)] []
|> abstract_over_list (rev qs)
in
RCInfo {RIvs=rcfix, rcarg=rcarg, CCas=rcassm, llRI=llRI, h_assum=h_assum}
end
val RC_infos = map2 mk_call_info RCs RIntro_thms
in
ClauseInfo {no=no, cdata=cdata, qglr=qglr, lGI=lGI, RCs=RC_infos,
tree=tree}
end
*}
ML {*
fun store_compat_thms 0 thms = []
| store_compat_thms n thms =
let
val (thms1, thms2) = chop n thms
in
(thms1 :: store_compat_thms (n - 1) thms2)
end
*}
ML {*
(* expects i <= j *)
fun lookup_compat_thm i j cts =
nth (nth cts (i - 1)) (j - i)
(* Returns "Gsi, Gsj, lhs_i = lhs_j |-- rhs_j_f = rhs_i_f" *)
(* if j < i, then turn around *)
fun get_compat_thm thy cts eqvtsi eqvtsj i j ctxi ctxj =
let
val ClauseContext {cqs=cqsi,ags=agsi,lhs=lhsi,...} = ctxi
val ClauseContext {cqs=cqsj,ags=agsj,lhs=lhsj,...} = ctxj
val lhsi_eq_lhsj = cterm_of thy (HOLogic.mk_Trueprop (mk_eq (lhsi, lhsj)))
in if j < i then
let
val compat = lookup_compat_thm j i cts
in
compat (* "!!qj qi. Gsj => Gsi => lhsj = lhsi ==> rhsj = rhsi" *)
|> fold Thm.forall_elim (cqsj @ cqsi) (* "Gsj => Gsi => lhsj = lhsi ==> rhsj = rhsi" *)
|> fold Thm.elim_implies (rev eqvtsj) (* HERE *)
|> fold Thm.elim_implies agsj
|> fold Thm.elim_implies agsi
|> Thm.elim_implies ((Thm.assume lhsi_eq_lhsj) RS sym) (* "Gsj, Gsi, lhsi = lhsj |-- rhsj = rhsi" *)
end
else
let
val compat = lookup_compat_thm i j cts
in
compat (* "!!qi qj. Gsi => Gsj => lhsi = lhsj ==> rhsi = rhsj" *)
|> fold Thm.forall_elim (cqsi @ cqsj) (* "Gsi => Gsj => lhsi = lhsj ==> rhsi = rhsj" *)
|> fold Thm.elim_implies (rev eqvtsi) (* HERE *)
|> fold Thm.elim_implies agsi
|> fold Thm.elim_implies agsj
|> Thm.elim_implies (Thm.assume lhsi_eq_lhsj)
|> (fn thm => thm RS sym) (* "Gsi, Gsj, lhsi = lhsj |-- rhsj = rhsi" *)
end
end
*}
ML {*
(* Generates the replacement lemma in fully quantified form. *)
fun mk_replacement_lemma thy h ih_elim clause =
let
val ClauseInfo {cdata=ClauseContext {qs, cqs, ags, case_hyp, ...},
RCs, tree, ...} = clause
local open Conv in
val ih_conv = arg1_conv o arg_conv o arg_conv
end
val ih_elim_case =
Conv.fconv_rule (ih_conv (K (case_hyp RS eq_reflection))) ih_elim
val Ris = map (fn RCInfo {llRI, ...} => llRI) RCs
val h_assums = map (fn RCInfo {h_assum, ...} =>
Thm.assume (cterm_of thy (subst_bounds (rev qs, h_assum)))) RCs
val (eql, _) =
Function_Ctx_Tree.rewrite_by_tree thy h ih_elim_case (Ris ~~ h_assums) tree
val replace_lemma = (eql RS meta_eq_to_obj_eq)
|> Thm.implies_intr (cprop_of case_hyp)
|> fold_rev (Thm.implies_intr o cprop_of) h_assums
|> fold_rev (Thm.implies_intr o cprop_of) ags
|> fold_rev Thm.forall_intr cqs
|> Thm.close_derivation
in
replace_lemma
end
*}
ML {*
fun mk_eqvt_lemma thy ih_eqvt clause =
let
val ClauseInfo {cdata=ClauseContext {cqs, ags, case_hyp, ...}, RCs, ...} = clause
local open Conv in
val ih_conv = arg1_conv o arg_conv o arg_conv
end
val ih_eqvt_case =
Conv.fconv_rule (ih_conv (K (case_hyp RS eq_reflection))) ih_eqvt
fun prep_eqvt (RCInfo {llRI, RIvs, CCas, ...}) =
(llRI RS ih_eqvt_case)
|> fold_rev (Thm.implies_intr o cprop_of) CCas
|> fold_rev (Thm.forall_intr o cterm_of thy o Free) RIvs
in
map prep_eqvt RCs
|> map (fold_rev (Thm.implies_intr o cprop_of) ags)
|> map (Thm.implies_intr (cprop_of case_hyp))
|> map (fold_rev Thm.forall_intr cqs)
|> map (Thm.close_derivation)
end
*}
ML {*
fun mk_uniqueness_clause thy globals compat_store eqvts clausei clausej RLj =
let
val Globals {h, y, x, fvar, ...} = globals
val ClauseInfo {no=i, cdata=cctxi as ClauseContext {ctxt=ctxti, lhs=lhsi, case_hyp, cqs = cqsi,
ags = agsi, ...}, ...} = clausei
val ClauseInfo {no=j, qglr=cdescj, RCs=RCsj, ...} = clausej
val cctxj as ClauseContext {ags = agsj', lhs = lhsj', rhs = rhsj', qs = qsj', cqs = cqsj', ...} =
mk_clause_context x ctxti cdescj
val rhsj'h = Pattern.rewrite_term thy [(fvar,h)] [] rhsj'
val Ghsj' = map
(fn RCInfo {h_assum, ...} => Thm.assume (cterm_of thy (subst_bounds (rev qsj', h_assum)))) RCsj
val y_eq_rhsj'h = Thm.assume (cterm_of thy (HOLogic.mk_Trueprop (mk_eq (y, rhsj'h))))
val lhsi_eq_lhsj' = Thm.assume (cterm_of thy (HOLogic.mk_Trueprop (mk_eq (lhsi, lhsj'))))
(* lhs_i = lhs_j' |-- lhs_i = lhs_j' *)
val case_hypj' = trans OF [case_hyp, lhsi_eq_lhsj']
val RLj_import = RLj
|> fold Thm.forall_elim cqsj'
|> fold Thm.elim_implies agsj'
|> fold Thm.elim_implies Ghsj'
val eqvtsi = nth eqvts (i - 1)
|> map (fold Thm.forall_elim cqsi)
|> map (fold Thm.elim_implies [case_hyp])
|> map (fold Thm.elim_implies agsi)
val eqvtsj = nth eqvts (j - 1)
|> map (fold Thm.forall_elim cqsj')
|> map (fold Thm.elim_implies [case_hypj'])
|> map (fold Thm.elim_implies agsj')
val compat = get_compat_thm thy compat_store eqvtsi eqvtsj i j cctxi cctxj
in
(trans OF [case_hyp, lhsi_eq_lhsj']) (* lhs_i = lhs_j' |-- x = lhs_j' *)
|> Thm.implies_elim RLj_import
(* Rj1' ... Rjk', lhs_i = lhs_j' |-- rhs_j'_h = rhs_j'_f *)
|> (fn it => trans OF [it, compat])
(* lhs_i = lhs_j', Gj', Rj1' ... Rjk' |-- rhs_j'_h = rhs_i_f *)
|> (fn it => trans OF [y_eq_rhsj'h, it])
(* lhs_i = lhs_j', Gj', Rj1' ... Rjk', y = rhs_j_h' |-- y = rhs_i_f *)
|> fold_rev (Thm.implies_intr o cprop_of) Ghsj'
|> fold_rev (Thm.implies_intr o cprop_of) agsj'
(* lhs_i = lhs_j' , y = rhs_j_h' |-- Gj', Rj1'...Rjk' ==> y = rhs_i_f *)
|> Thm.implies_intr (cprop_of y_eq_rhsj'h)
|> Thm.implies_intr (cprop_of lhsi_eq_lhsj')
|> fold_rev Thm.forall_intr (cterm_of thy h :: cqsj')
end
*}
ML {*
fun mk_uniqueness_case thy globals G f ihyp ih_intro G_cases compat_store clauses replems eqvtlems clausei =
let
val Globals {x, y, ranT, fvar, ...} = globals
val ClauseInfo {cdata = ClauseContext {lhs, rhs, cqs, ags, case_hyp, ...}, lGI, RCs, ...} = clausei
val rhsC = Pattern.rewrite_term thy [(fvar, f)] [] rhs
val ih_intro_case = full_simplify (HOL_basic_ss addsimps [case_hyp]) ih_intro
fun prep_RC (RCInfo {llRI, RIvs, CCas, ...}) =
(llRI RS ih_intro_case)
|> fold_rev (Thm.implies_intr o cprop_of) CCas
|> fold_rev (Thm.forall_intr o cterm_of thy o Free) RIvs
val existence = fold (curry op COMP o prep_RC) RCs lGI
val P = cterm_of thy (mk_eq (y, rhsC))
val G_lhs_y = Thm.assume (cterm_of thy (HOLogic.mk_Trueprop (G $ lhs $ y)))
val unique_clauses =
map2 (mk_uniqueness_clause thy globals compat_store eqvtlems clausei) clauses replems
fun elim_implies_eta A AB =
Thm.compose_no_flatten true (A, 0) 1 AB |> Seq.list_of |> the_single
val uniqueness = G_cases
|> Thm.forall_elim (cterm_of thy lhs)
|> Thm.forall_elim (cterm_of thy y)
|> Thm.forall_elim P
|> Thm.elim_implies G_lhs_y
|> fold elim_implies_eta unique_clauses
|> Thm.implies_intr (cprop_of G_lhs_y)
|> Thm.forall_intr (cterm_of thy y)
val P2 = cterm_of thy (lambda y (G $ lhs $ y)) (* P2 y := (lhs, y): G *)
val exactly_one =
ex1I |> instantiate' [SOME (ctyp_of thy ranT)] [SOME P2, SOME (cterm_of thy rhsC)]
|> curry (op COMP) existence
|> curry (op COMP) uniqueness
|> simplify (HOL_basic_ss addsimps [case_hyp RS sym])
|> Thm.implies_intr (cprop_of case_hyp)
|> fold_rev (Thm.implies_intr o cprop_of) ags
|> fold_rev Thm.forall_intr cqs
val function_value =
existence
|> Thm.implies_intr ihyp
|> Thm.implies_intr (cprop_of case_hyp)
|> Thm.forall_intr (cterm_of thy x)
|> Thm.forall_elim (cterm_of thy lhs)
|> curry (op RS) refl
in
(exactly_one, function_value)
end
*}
ML {*
fun prove_stuff ctxt globals G f R clauses complete compat compat_store G_elim G_eqvt f_def =
let
val Globals {h, domT, ranT, x, ...} = globals
val thy = ProofContext.theory_of ctxt
(* Inductive Hypothesis: !!z. (z,x):R ==> EX!y. (z,y):G *)
val ihyp = Term.all domT $ Abs ("z", domT,
Logic.mk_implies (HOLogic.mk_Trueprop (R $ Bound 0 $ x),
HOLogic.mk_Trueprop (Const (@{const_name Ex1}, (ranT --> boolT) --> boolT) $
Abs ("y", ranT, G $ Bound 1 $ Bound 0))))
|> cterm_of thy
val ihyp_thm = Thm.assume ihyp |> Thm.forall_elim_vars 0
val ih_intro = ihyp_thm RS (f_def RS ex1_implies_ex)
val ih_elim = ihyp_thm RS (f_def RS ex1_implies_un)
|> instantiate' [] [NONE, SOME (cterm_of thy h)]
val ih_eqvt = ihyp_thm RS (G_eqvt RS (f_def RS @{thm fundef_ex1_eqvt_at}))
val _ = trace_msg (K "Proving Replacement lemmas...")
val repLemmas = map (mk_replacement_lemma thy h ih_elim) clauses
val _ = trace_msg (K "Proving Equivariance lemmas...")
val eqvtLemmas = map (mk_eqvt_lemma thy ih_eqvt) clauses
val _ = trace_msg (K "Proving cases for unique existence...")
val (ex1s, values) =
split_list (map (mk_uniqueness_case thy globals G f
ihyp ih_intro G_elim compat_store clauses repLemmas eqvtLemmas) clauses)
val _ = trace_msg (K "Proving: Graph is a function")
val graph_is_function = complete
|> Thm.forall_elim_vars 0
|> fold (curry op COMP) ex1s
|> Thm.implies_intr (ihyp)
|> Thm.implies_intr (cterm_of thy (HOLogic.mk_Trueprop (mk_acc domT R $ x)))
|> Thm.forall_intr (cterm_of thy x)
|> (fn it => Drule.compose_single (it, 2, acc_induct_rule)) (* "EX! y. (?x,y):G" *)
|> (fn it => fold (Thm.forall_intr o cterm_of thy o Var) (Term.add_vars (prop_of it) []) it)
val goalstate = Conjunction.intr graph_is_function complete
|> Thm.close_derivation
|> Goal.protect
|> fold_rev (Thm.implies_intr o cprop_of) compat
|> Thm.implies_intr (cprop_of complete)
in
(goalstate, values)
end
*}
ML {*
(* wrapper -- restores quantifiers in rule specifications *)
fun inductive_def eqvt_flag (binding as ((R, T), _)) intrs lthy =
let
val ({intrs = intrs_gen, elims = [elim_gen], preds = [ Rdef ], induct, raw_induct, ...}, lthy) =
lthy
|> Local_Theory.conceal
|> Inductive.add_inductive_i
{quiet_mode = true,
verbose = ! trace,
alt_name = Binding.empty,
coind = false,
no_elim = false,
no_ind = false,
skip_mono = true,
fork_mono = false}
[binding] (* relation *)
[] (* no parameters *)
(map (fn t => (Attrib.empty_binding, t)) intrs) (* intro rules *)
[] (* no special monos *)
||> Local_Theory.restore_naming lthy
val eqvt_thm' =
if eqvt_flag = false then NONE
else
let
val ([eqvt_thm], lthy) = Nominal_Eqvt.raw_equivariance false [Rdef] raw_induct intrs_gen lthy
in
SOME ((Nominal_ThmDecls.eqvt_transform lthy eqvt_thm) RS @{thm eqvtI})
end
val cert = cterm_of (ProofContext.theory_of lthy)
fun requantify orig_intro thm =
let
val (qs, t) = dest_all_all orig_intro
val frees = frees_in_term lthy t |> remove (op =) (Binding.name_of R, T)
val vars = Term.add_vars (prop_of thm) [] |> rev
val varmap = AList.lookup (op =) (frees ~~ map fst vars)
#> the_default ("",0)
in
fold_rev (fn Free (n, T) =>
forall_intr_rename (n, cert (Var (varmap (n, T), T)))) qs thm
end
in
((Rdef, map2 requantify intrs intrs_gen, forall_intr_vars elim_gen, induct, eqvt_thm'), lthy)
end
*}
ML {*
fun define_graph Gname fvar domT ranT clauses RCss lthy =
let
val GT = domT --> ranT --> boolT
val (Gvar as (n, T)) = singleton (Variable.variant_frees lthy []) (Gname, GT)
fun mk_GIntro (ClauseContext {qs, gs, lhs, rhs, ...}) RCs =
let
fun mk_h_assm (rcfix, rcassm, rcarg) =
HOLogic.mk_Trueprop (Free Gvar $ rcarg $ (fvar $ rcarg))
|> fold_rev (curry Logic.mk_implies o prop_of) rcassm
|> fold_rev (Logic.all o Free) rcfix
in
HOLogic.mk_Trueprop (Free Gvar $ lhs $ rhs)
|> fold_rev (curry Logic.mk_implies o mk_h_assm) RCs
|> fold_rev (curry Logic.mk_implies) gs
|> fold_rev Logic.all (fvar :: qs)
end
val G_intros = map2 mk_GIntro clauses RCss
in
inductive_def true ((Binding.name n, T), NoSyn) G_intros lthy
end
*}
ML {*
fun define_function fdefname (fname, mixfix) domT ranT G default lthy =
let
val f_def =
Abs ("x", domT, Const (@{const_name FunDef.THE_default}, ranT --> (ranT --> boolT) --> ranT)
$ (default $ Bound 0) $ Abs ("y", ranT, G $ Bound 1 $ Bound 0))
|> Syntax.check_term lthy
in
Local_Theory.define
((Binding.name (function_name fname), mixfix),
((Binding.conceal (Binding.name fdefname), []), f_def)) lthy
end
fun define_recursion_relation Rname domT qglrs clauses RCss lthy =
let
val RT = domT --> domT --> boolT
val (Rvar as (n, T)) = singleton (Variable.variant_frees lthy []) (Rname, RT)
fun mk_RIntro (ClauseContext {qs, gs, lhs, ...}, (oqs, _, _, _)) (rcfix, rcassm, rcarg) =
HOLogic.mk_Trueprop (Free Rvar $ rcarg $ lhs)
|> fold_rev (curry Logic.mk_implies o prop_of) rcassm
|> fold_rev (curry Logic.mk_implies) gs
|> fold_rev (Logic.all o Free) rcfix
|> fold_rev mk_forall_rename (map fst oqs ~~ qs)
(* "!!qs xs. CS ==> G => (r, lhs) : R" *)
val R_intross = map2 (map o mk_RIntro) (clauses ~~ qglrs) RCss
val ((R, RIntro_thms, R_elim, _, _), lthy) =
inductive_def false ((Binding.name n, T), NoSyn) (flat R_intross) lthy
in
((R, Library.unflat R_intross RIntro_thms, R_elim), lthy)
end
fun fix_globals domT ranT fvar ctxt =
let
val ([h, y, x, z, a, D, P, Pbool],ctxt') = Variable.variant_fixes
["h_fd", "y_fd", "x_fd", "z_fd", "a_fd", "D_fd", "P_fd", "Pb_fd"] ctxt
in
(Globals {h = Free (h, domT --> ranT),
y = Free (y, ranT),
x = Free (x, domT),
z = Free (z, domT),
a = Free (a, domT),
D = Free (D, domT --> boolT),
P = Free (P, domT --> boolT),
Pbool = Free (Pbool, boolT),
fvar = fvar,
domT = domT,
ranT = ranT},
ctxt')
end
fun inst_RC thy fvar f (rcfix, rcassm, rcarg) =
let
fun inst_term t = subst_bound(f, abstract_over (fvar, t))
in
(rcfix, map (Thm.assume o cterm_of thy o inst_term o prop_of) rcassm, inst_term rcarg)
end
(**********************************************************
* PROVING THE RULES
**********************************************************)
fun mk_psimps thy globals R clauses valthms f_iff graph_is_function =
let
val Globals {domT, z, ...} = globals
fun mk_psimp (ClauseInfo {qglr = (oqs, _, _, _), cdata = ClauseContext {cqs, lhs, ags, ...}, ...}) valthm =
let
val lhs_acc = cterm_of thy (HOLogic.mk_Trueprop (mk_acc domT R $ lhs)) (* "acc R lhs" *)
val z_smaller = cterm_of thy (HOLogic.mk_Trueprop (R $ z $ lhs)) (* "R z lhs" *)
in
((Thm.assume z_smaller) RS ((Thm.assume lhs_acc) RS acc_downward))
|> (fn it => it COMP graph_is_function)
|> Thm.implies_intr z_smaller
|> Thm.forall_intr (cterm_of thy z)
|> (fn it => it COMP valthm)
|> Thm.implies_intr lhs_acc
|> asm_simplify (HOL_basic_ss addsimps [f_iff])
|> fold_rev (Thm.implies_intr o cprop_of) ags
|> fold_rev forall_intr_rename (map fst oqs ~~ cqs)
end
in
map2 mk_psimp clauses valthms
end
(** Induction rule **)
val acc_subset_induct = @{thm predicate1I} RS @{thm accp_subset_induct}
fun mk_partial_induct_rule thy globals R complete_thm clauses =
let
val Globals {domT, x, z, a, P, D, ...} = globals
val acc_R = mk_acc domT R
val x_D = Thm.assume (cterm_of thy (HOLogic.mk_Trueprop (D $ x)))
val a_D = cterm_of thy (HOLogic.mk_Trueprop (D $ a))
val D_subset = cterm_of thy (Logic.all x
(Logic.mk_implies (HOLogic.mk_Trueprop (D $ x), HOLogic.mk_Trueprop (acc_R $ x))))
val D_dcl = (* "!!x z. [| x: D; (z,x):R |] ==> z:D" *)
Logic.all x (Logic.all z (Logic.mk_implies (HOLogic.mk_Trueprop (D $ x),
Logic.mk_implies (HOLogic.mk_Trueprop (R $ z $ x),
HOLogic.mk_Trueprop (D $ z)))))
|> cterm_of thy
(* Inductive Hypothesis: !!z. (z,x):R ==> P z *)
val ihyp = Term.all domT $ Abs ("z", domT,
Logic.mk_implies (HOLogic.mk_Trueprop (R $ Bound 0 $ x),
HOLogic.mk_Trueprop (P $ Bound 0)))
|> cterm_of thy
val aihyp = Thm.assume ihyp
fun prove_case clause =
let
val ClauseInfo {cdata = ClauseContext {ctxt, qs, cqs, ags, gs, lhs, case_hyp, ...},
RCs, qglr = (oqs, _, _, _), ...} = clause
val case_hyp_conv = K (case_hyp RS eq_reflection)
local open Conv in
val lhs_D = fconv_rule (arg_conv (arg_conv (case_hyp_conv))) x_D
val sih =
fconv_rule (Conv.binder_conv
(K (arg1_conv (arg_conv (arg_conv case_hyp_conv)))) ctxt) aihyp
end
fun mk_Prec (RCInfo {llRI, RIvs, CCas, rcarg, ...}) = sih
|> Thm.forall_elim (cterm_of thy rcarg)
|> Thm.elim_implies llRI
|> fold_rev (Thm.implies_intr o cprop_of) CCas
|> fold_rev (Thm.forall_intr o cterm_of thy o Free) RIvs
val P_recs = map mk_Prec RCs (* [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
|> curry Logic.mk_implies (HOLogic.mk_Trueprop (D $ lhs))
|> fold_rev mk_forall_rename (map fst oqs ~~ qs)
|> cterm_of thy
val P_lhs = Thm.assume step
|> fold Thm.forall_elim cqs
|> Thm.elim_implies lhs_D
|> fold Thm.elim_implies ags
|> fold Thm.elim_implies P_recs
val res = cterm_of thy (HOLogic.mk_Trueprop (P $ x))
|> Conv.arg_conv (Conv.arg_conv case_hyp_conv)
|> Thm.symmetric (* P lhs == P x *)
|> (fn eql => Thm.equal_elim eql P_lhs) (* "P x" *)
|> Thm.implies_intr (cprop_of case_hyp)
|> fold_rev (Thm.implies_intr o cprop_of) ags
|> fold_rev Thm.forall_intr cqs
in
(res, step)
end
val (cases, steps) = split_list (map prove_case clauses)
val istep = complete_thm
|> Thm.forall_elim_vars 0
|> fold (curry op COMP) cases (* P x *)
|> Thm.implies_intr ihyp
|> Thm.implies_intr (cprop_of x_D)
|> Thm.forall_intr (cterm_of thy x)
val subset_induct_rule =
acc_subset_induct
|> (curry op COMP) (Thm.assume D_subset)
|> (curry op COMP) (Thm.assume D_dcl)
|> (curry op COMP) (Thm.assume a_D)
|> (curry op COMP) istep
|> fold_rev Thm.implies_intr steps
|> Thm.implies_intr a_D
|> Thm.implies_intr D_dcl
|> Thm.implies_intr D_subset
val simple_induct_rule =
subset_induct_rule
|> Thm.forall_intr (cterm_of thy D)
|> Thm.forall_elim (cterm_of thy acc_R)
|> assume_tac 1 |> Seq.hd
|> (curry op COMP) (acc_downward
|> (instantiate' [SOME (ctyp_of thy domT)]
(map (SOME o cterm_of thy) [R, x, z]))
|> Thm.forall_intr (cterm_of thy z)
|> Thm.forall_intr (cterm_of thy x))
|> Thm.forall_intr (cterm_of thy a)
|> Thm.forall_intr (cterm_of thy P)
in
simple_induct_rule
end
(* FIXME: broken by design *)
fun mk_domain_intro ctxt (Globals {domT, ...}) R R_cases clause =
let
val thy = ProofContext.theory_of ctxt
val ClauseInfo {cdata = ClauseContext {gs, lhs, cqs, ...},
qglr = (oqs, _, _, _), ...} = clause
val goal = HOLogic.mk_Trueprop (mk_acc domT R $ lhs)
|> fold_rev (curry Logic.mk_implies) gs
|> cterm_of thy
in
Goal.init goal
|> (SINGLE (resolve_tac [accI] 1)) |> the
|> (SINGLE (eresolve_tac [Thm.forall_elim_vars 0 R_cases] 1)) |> the
|> (SINGLE (auto_tac (clasimpset_of ctxt))) |> the
|> Goal.conclude
|> fold_rev forall_intr_rename (map fst oqs ~~ cqs)
end
(** Termination rule **)
val wf_induct_rule = @{thm Wellfounded.wfP_induct_rule}
val wf_in_rel = @{thm FunDef.wf_in_rel}
val in_rel_def = @{thm FunDef.in_rel_def}
fun mk_nest_term_case thy globals R' ihyp clause =
let
val Globals {z, ...} = globals
val ClauseInfo {cdata = ClauseContext {qs, cqs, ags, lhs, case_hyp, ...}, tree,
qglr=(oqs, _, _, _), ...} = clause
val ih_case = full_simplify (HOL_basic_ss addsimps [case_hyp]) ihyp
fun step (fixes, assumes) (_ $ arg) u (sub,(hyps,thms)) =
let
val used = (u @ sub)
|> map (fn (ctx,thm) => Function_Ctx_Tree.export_thm thy ctx thm)
val hyp = HOLogic.mk_Trueprop (R' $ arg $ lhs)
|> fold_rev (curry Logic.mk_implies o prop_of) used (* additional hyps *)
|> Function_Ctx_Tree.export_term (fixes, assumes)
|> fold_rev (curry Logic.mk_implies o prop_of) ags
|> fold_rev mk_forall_rename (map fst oqs ~~ qs)
|> cterm_of thy
val thm = Thm.assume hyp
|> fold Thm.forall_elim cqs
|> fold Thm.elim_implies ags
|> Function_Ctx_Tree.import_thm thy (fixes, assumes)
|> fold Thm.elim_implies used (* "(arg, lhs) : R'" *)
val z_eq_arg = HOLogic.mk_Trueprop (mk_eq (z, arg))
|> cterm_of thy |> Thm.assume
val acc = thm COMP ih_case
val z_acc_local = acc
|> Conv.fconv_rule
(Conv.arg_conv (Conv.arg_conv (K (Thm.symmetric (z_eq_arg RS eq_reflection)))))
val ethm = z_acc_local
|> Function_Ctx_Tree.export_thm thy (fixes,
z_eq_arg :: case_hyp :: ags @ assumes)
|> fold_rev forall_intr_rename (map fst oqs ~~ cqs)
val sub' = sub @ [(([],[]), acc)]
in
(sub', (hyp :: hyps, ethm :: thms))
end
| step _ _ _ _ = raise Match
in
Function_Ctx_Tree.traverse_tree step tree
end
fun mk_nest_term_rule thy globals R R_cases clauses =
let
val Globals { domT, x, z, ... } = globals
val acc_R = mk_acc domT R
val R' = Free ("R", fastype_of R)
val Rrel = Free ("R", HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)))
val inrel_R = Const (@{const_name FunDef.in_rel},
HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)) --> fastype_of R) $ Rrel
val wfR' = HOLogic.mk_Trueprop (Const (@{const_name Wellfounded.wfP},
(domT --> domT --> boolT) --> boolT) $ R')
|> cterm_of thy (* "wf R'" *)
(* Inductive Hypothesis: !!z. (z,x):R' ==> z : acc R *)
val ihyp = Term.all domT $ Abs ("z", domT,
Logic.mk_implies (HOLogic.mk_Trueprop (R' $ Bound 0 $ x),
HOLogic.mk_Trueprop (acc_R $ Bound 0)))
|> cterm_of thy
val ihyp_a = Thm.assume ihyp |> Thm.forall_elim_vars 0
val R_z_x = cterm_of thy (HOLogic.mk_Trueprop (R $ z $ x))
val (hyps, cases) = fold (mk_nest_term_case thy globals R' ihyp_a) clauses ([], [])
in
R_cases
|> Thm.forall_elim (cterm_of thy z)
|> Thm.forall_elim (cterm_of thy x)
|> Thm.forall_elim (cterm_of thy (acc_R $ z))
|> curry op COMP (Thm.assume R_z_x)
|> fold_rev (curry op COMP) cases
|> Thm.implies_intr R_z_x
|> Thm.forall_intr (cterm_of thy z)
|> (fn it => it COMP accI)
|> Thm.implies_intr ihyp
|> Thm.forall_intr (cterm_of thy x)
|> (fn it => Drule.compose_single(it,2,wf_induct_rule))
|> curry op RS (Thm.assume wfR')
|> forall_intr_vars
|> (fn it => it COMP allI)
|> fold Thm.implies_intr hyps
|> Thm.implies_intr wfR'
|> Thm.forall_intr (cterm_of thy R')
|> Thm.forall_elim (cterm_of thy (inrel_R))
|> curry op RS wf_in_rel
|> full_simplify (HOL_basic_ss addsimps [in_rel_def])
|> Thm.forall_intr (cterm_of thy Rrel)
end
(* Tail recursion (probably very fragile)
*
* FIXME:
* - Need to do forall_elim_vars on psimps: Unneccesary, if psimps would be taken from the same context.
* - Must we really replace the fvar by f here?
* - Splitting is not configured automatically: Problems with case?
*)
fun mk_trsimps octxt globals f G R f_def R_cases G_induct clauses psimps =
let
val Globals {domT, ranT, fvar, ...} = globals
val R_cases = Thm.forall_elim_vars 0 R_cases (* FIXME: Should be already in standard form. *)
val graph_implies_dom = (* "G ?x ?y ==> dom ?x" *)
Goal.prove octxt ["x", "y"] [HOLogic.mk_Trueprop (G $ Free ("x", domT) $ Free ("y", ranT))]
(HOLogic.mk_Trueprop (mk_acc domT R $ Free ("x", domT)))
(fn {prems=[a], ...} =>
((rtac (G_induct OF [a]))
THEN_ALL_NEW rtac accI
THEN_ALL_NEW etac R_cases
THEN_ALL_NEW asm_full_simp_tac (simpset_of octxt)) 1)
val default_thm =
forall_intr_vars graph_implies_dom COMP (f_def COMP fundef_default_value)
fun mk_trsimp clause psimp =
let
val ClauseInfo {qglr = (oqs, _, _, _), cdata =
ClauseContext {ctxt, cqs, gs, lhs, rhs, ...}, ...} = clause
val thy = ProofContext.theory_of ctxt
val rhs_f = Pattern.rewrite_term thy [(fvar, f)] [] rhs
val trsimp = Logic.list_implies(gs,
HOLogic.mk_Trueprop (HOLogic.mk_eq(f $ lhs, rhs_f))) (* "f lhs = rhs" *)
val lhs_acc = (mk_acc domT R $ lhs) (* "acc R lhs" *)
fun simp_default_tac ss =
asm_full_simp_tac (ss addsimps [default_thm, Let_def])
in
Goal.prove ctxt [] [] trsimp (fn _ =>
rtac (instantiate' [] [SOME (cterm_of thy lhs_acc)] case_split) 1
THEN (rtac (Thm.forall_elim_vars 0 psimp) THEN_ALL_NEW assume_tac) 1
THEN (simp_default_tac (simpset_of ctxt) 1)
THEN TRY ((etac not_acc_down 1)
THEN ((etac R_cases)
THEN_ALL_NEW (simp_default_tac (simpset_of ctxt))) 1))
|> fold_rev forall_intr_rename (map fst oqs ~~ cqs)
end
in
map2 mk_trsimp clauses psimps
end
*}
ML {*
fun prepare_function config defname [((fname, fT), mixfix)] abstract_qglrs lthy =
let
val FunctionConfig {domintros, tailrec, default=default_opt, ...} = config
val default_str = the_default "%x. undefined" default_opt (*FIXME dynamic scoping*)
val fvar = Free (fname, fT)
val domT = domain_type fT
val ranT = range_type fT
val default = Syntax.parse_term lthy default_str
|> Type.constraint fT |> Syntax.check_term lthy
val (globals, ctxt') = fix_globals domT ranT fvar lthy
val Globals { x, h, ... } = globals
val clauses = map (mk_clause_context x ctxt') abstract_qglrs
val n = length abstract_qglrs
fun build_tree (ClauseContext { ctxt, rhs, ...}) =
Function_Ctx_Tree.mk_tree (fname, fT) h ctxt rhs
val trees = map build_tree clauses
val RCss = map find_calls trees
val ((G, GIntro_thms, G_elim, G_induct, SOME G_eqvt), lthy) =
PROFILE "def_graph" (define_graph (graph_name defname) fvar domT ranT clauses RCss) lthy
val ((f, (_, f_defthm)), lthy) =
PROFILE "def_fun" (define_function (defname ^ "_sumC_def") (fname, mixfix) domT ranT G default) lthy
val RCss = map (map (inst_RC (ProofContext.theory_of lthy) fvar f)) RCss
val trees = map (Function_Ctx_Tree.inst_tree (ProofContext.theory_of lthy) fvar f) trees
val ((R, RIntro_thmss, R_elim), lthy) =
PROFILE "def_rel" (define_recursion_relation (rel_name defname) domT abstract_qglrs clauses RCss) lthy
val (_, lthy) =
Local_Theory.abbrev Syntax.mode_default ((Binding.name (dom_name defname), NoSyn), mk_acc domT R) lthy
val newthy = ProofContext.theory_of lthy
val clauses = map (transfer_clause_ctx newthy) clauses
val cert = cterm_of (ProofContext.theory_of lthy)
val xclauses = PROFILE "xclauses"
(map7 (mk_clause_info globals G f) (1 upto n) clauses abstract_qglrs trees
RCss GIntro_thms) RIntro_thmss
val complete =
mk_completeness globals clauses abstract_qglrs |> cert |> Thm.assume
val compat =
mk_compat_proof_obligations domT ranT fvar f abstract_qglrs
|> map (cert #> Thm.assume)
val compat_store = store_compat_thms n compat
val (goalstate, values) = PROFILE "prove_stuff"
(prove_stuff lthy globals G f R xclauses complete compat
compat_store G_elim G_eqvt) f_defthm
val mk_trsimps =
mk_trsimps lthy globals f G R f_defthm R_elim G_induct xclauses
fun mk_partial_rules provedgoal =
let
val newthy = theory_of_thm provedgoal (*FIXME*)
val (graph_is_function, complete_thm) =
provedgoal
|> Conjunction.elim
|> apfst (Thm.forall_elim_vars 0)
val f_iff = graph_is_function RS (f_defthm RS ex1_implies_iff)
val psimps = PROFILE "Proving simplification rules"
(mk_psimps newthy globals R xclauses values f_iff) graph_is_function
val simple_pinduct = PROFILE "Proving partial induction rule"
(mk_partial_induct_rule newthy globals R complete_thm) xclauses
val total_intro = PROFILE "Proving nested termination rule"
(mk_nest_term_rule newthy globals R R_elim) xclauses
val dom_intros =
if domintros then SOME (PROFILE "Proving domain introduction rules"
(map (mk_domain_intro lthy globals R R_elim)) xclauses)
else NONE
val trsimps = if tailrec then SOME (mk_trsimps psimps) else NONE
in
FunctionResult {fs=[f], G=G, R=R, cases=complete_thm,
psimps=psimps, simple_pinducts=[simple_pinduct],
termination=total_intro, trsimps=trsimps,
domintros=dom_intros}
end
in
((f, goalstate, mk_partial_rules), lthy)
end
*}
ML {*
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,
fvar : string * typ,
cargTs: typ list,
f_def: term,
f: term option,
f_defthm : thm option}
datatype mutual_info = Mutual of
{n : int,
n' : int,
fsum_var : string * 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 fname =
the o find_first (fn (MutualPart {fvar=(n,_), ...}) => n = fname)
(* 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
val arity_of = map (fn (fname,_,_,args,_) => (fname, length args)) fqgars
|> AList.lookup (op =) #> the
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 SumTree.mk_sumT) dresultTs
val ST = Balanced_Tree.make (uncurry SumTree.mk_sumT) argTs
val fsum_type = ST --> RST
val ([fsum_var_name], _) = Variable.add_fixes [ defname ^ "_sum" ] ctxt
val fsum_var = (fsum_var_name, fsum_type)
fun define (fvar as (n, _)) 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 = SumTree.mk_proj RST n' i'
(Free fsum_var $ SumTree.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 = (n, fold_rev lambda vars f_exp)
in
(MutualPart {i=i, i'=i', fvar=fvar,cargTs=caTs,f_def=def,f=NONE,f_defthm=NONE}, rew)
end
val (parts, rews) = split_list (map4 define fs caTss resultTs (1 upto num))
fun convert_eqs (f, qs, gs, args, rhs) =
let
val MutualPart {i, i', ...} = get_part f parts
in
(qs, gs, SumTree.mk_inj ST num i (foldr1 (mk_prod_abs qs) args),
SumTree.mk_inj RST n' i' (replace_frees rews rhs)
|> Envir.beta_norm)
end
val qglrs = map convert_eqs fqgars
in
Mutual {n=num, n'=n', fsum_var=fsum_var, ST=ST, RST=RST,
parts=parts, fqgars=fqgars, qglrs=qglrs, fsum=NONE}
end
*}
ML {*
fun define_projections fixes mutual fsum lthy =
let
fun def ((MutualPart {i=i, i'=i', fvar=(fname, fT), cargTs, f_def, ...}), (_, mixfix)) lthy =
let
val ((f, (_, f_defthm)), lthy') =
Local_Theory.define
((Binding.name fname, mixfix),
((Binding.conceal (Binding.name (fname ^ "_def")), []),
Term.subst_bound (fsum, f_def))) lthy
in
(MutualPart {i=i, i'=i', fvar=(fname, fT), cargTs=cargTs, f_def=f_def,
f=SOME f, f_defthm=SOME f_defthm },
lthy')
end
val Mutual { n, n', fsum_var, ST, RST, parts, fqgars, qglrs, ... } = mutual
val (parts', lthy') = fold_map def (parts ~~ fixes) lthy
in
(Mutual { n=n, n'=n', fsum_var=fsum_var, 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 thy = ProofContext.theory_of ctxt
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 (cterm_of thy) qs
val ags = map (Thm.assume o cterm_of thy) gs
val import = fold Thm.forall_elim cqs
#> fold Thm.elim_implies ags
val export = fold_rev (Thm.implies_intr o 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"
in
Goal.prove ctxt [] []
(HOLogic.Trueprop $ HOLogic.mk_eq (list_comb (f, args), rhs))
(fn _ => (Local_Defs.unfold_tac ctxt all_orig_fdefs)
THEN EqSubst.eqsubst_tac ctxt [0] [simp] 1
THEN (simp_tac (simpset_of ctxt)) 1) (* FIXME: global simpset?!! *)
|> restore_cond
|> export
end
fun mk_applied_form ctxt caTs thm =
let
val thy = ProofContext.theory_of ctxt
val xs = map_index (fn (i,T) => cterm_of thy (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 lthy induct all_f_defs (Mutual {n, ST, parts, ...}) =
let
val cert = cterm_of (ProofContext.theory_of lthy)
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 = SumTree.mk_sumcases HOLogic.boolT Ps
val induct_inst =
Thm.forall_elim (cert case_exp) induct
|> full_simplify SumTree.sumcase_split_ss
|> full_simplify (HOL_basic_ss 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 = SumTree.mk_inj ST n i (foldr1 HOLogic.mk_prod afs)
in
(rule
|> Thm.forall_elim (cert inj)
|> full_simplify SumTree.sumcase_split_ss
|> fold_rev (Thm.forall_intr o cert) (afs @ newPs),
k + length cargTs)
end
in
fst (fold_map (project induct_inst) parts 0)
end
fun mk_partial_rules_mutual lthy inner_cont (m as Mutual {parts, fqgars, ...}) proof =
let
val result = inner_cont proof
val FunctionResult {G, R, cases, psimps, trsimps, simple_pinducts=[simple_pinduct],
termination, domintros, ...} = 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_ss = HOL_basic_ss addsimps all_f_defs
val mpsimps = map2 mk_mpsimp fqgars psimps
val mtrsimps = Option.map (map2 mk_mpsimp fqgars) trsimps
val minducts = mutual_induct_rules lthy simple_pinduct all_f_defs m
val mtermination = full_simplify rew_ss termination
val mdomintros = Option.map (map (full_simplify rew_ss)) domintros
in
FunctionResult { fs=fs, G=G, R=R,
psimps=mpsimps, simple_pinducts=minducts,
cases=cases, termination=mtermination,
domintros=mdomintros, trsimps=mtrsimps}
end
fun prepare_function_mutual config defname fixes eqss lthy =
let
val mutual as Mutual {fsum_var=(n, T), qglrs, ...} =
analyze_eqs lthy defname (map fst fixes) (map Envir.beta_eta_contract eqss)
val ((fsum, goalstate, cont), lthy') =
prepare_function config defname [((n, T), NoSyn)] qglrs lthy
val (mutual', lthy'') = define_projections fixes mutual fsum lthy'
val mutual_cont = mk_partial_rules_mutual lthy'' cont mutual'
in
((goalstate, mutual_cont), lthy'')
end
*}
ML {*
open Function_Lib
open Function_Common
val simp_attribs = map (Attrib.internal o K)
[Simplifier.simp_add,
Code.add_default_eqn_attribute,
Nitpick_Simps.add]
val psimp_attribs = map (Attrib.internal o K)
[Nitpick_Psimps.add]
fun mk_defname fixes = fixes |> map (fst o fst) |> space_implode "_"
fun add_simps fnames post sort extra_qualify label mod_binding moreatts
simps lthy =
let
val spec = post simps
|> map (apfst (apsnd (fn ats => moreatts @ ats)))
|> map (apfst (apfst extra_qualify))
val (saved_spec_simps, lthy) =
fold_map Local_Theory.note spec lthy
val saved_simps = maps snd saved_spec_simps
val simps_by_f = sort saved_simps
fun add_for_f fname simps =
Local_Theory.note
((mod_binding (Binding.qualify true fname (Binding.name label)), []), simps)
#> snd
in
(saved_simps, fold2 add_for_f fnames simps_by_f lthy)
end
fun prepare_function is_external prep default_constraint fixspec eqns config lthy =
let
val constrn_fxs = map (fn (b, T, mx) => (b, SOME (the_default default_constraint T), mx))
val ((fixes0, spec0), ctxt') = prep (constrn_fxs fixspec) eqns lthy
val fixes = map (apfst (apfst Binding.name_of)) fixes0;
val spec = map (fn (bnd, prop) => (bnd, [prop])) spec0;
val (eqs, post, sort_cont, cnames) = get_preproc lthy config ctxt' fixes spec
val defname = mk_defname fixes
val FunctionConfig {partials, tailrec, default, ...} = config
val _ =
if tailrec then Output.legacy_feature
"'function (tailrec)' command. Use 'partial_function (tailrec)'."
else ()
val _ =
if is_some default then Output.legacy_feature
"'function (default)'. Use 'partial_function'."
else ()
val ((goal_state, cont), lthy') =
prepare_function_mutual config defname fixes eqs lthy
fun afterqed [[proof]] lthy =
let
val FunctionResult {fs, R, psimps, trsimps, simple_pinducts,
termination, domintros, cases, ...} =
cont (Thm.close_derivation proof)
val fnames = map (fst o fst) fixes
fun qualify n = Binding.name n
|> Binding.qualify true defname
val conceal_partial = if partials then I else Binding.conceal
val addsmps = add_simps fnames post sort_cont
val (((psimps', pinducts'), (_, [termination'])), lthy) =
lthy
|> addsmps (conceal_partial o Binding.qualify false "partial")
"psimps" conceal_partial psimp_attribs psimps
||> (case trsimps of NONE => I | SOME thms =>
addsmps I "simps" I simp_attribs thms #> snd
#> Spec_Rules.add Spec_Rules.Equational (fs, thms))
||>> Local_Theory.note ((conceal_partial (qualify "pinduct"),
[Attrib.internal (K (Rule_Cases.case_names cnames)),
Attrib.internal (K (Rule_Cases.consumes 1)),
Attrib.internal (K (Induct.induct_pred ""))]), simple_pinducts)
||>> Local_Theory.note ((Binding.conceal (qualify "termination"), []), [termination])
||> (snd o Local_Theory.note ((qualify "cases",
[Attrib.internal (K (Rule_Cases.case_names cnames))]), [cases]))
||> (case domintros of NONE => I | SOME thms =>
Local_Theory.note ((qualify "domintros", []), thms) #> snd)
val info = { add_simps=addsmps, case_names=cnames, psimps=psimps',
pinducts=snd pinducts', simps=NONE, inducts=NONE, termination=termination',
fs=fs, R=R, defname=defname, is_partial=true }
val _ =
if not is_external then ()
else Specification.print_consts lthy (K false) (map fst fixes)
in
(info,
lthy |> Local_Theory.declaration false (add_function_data o morph_function_data info))
end
in
((goal_state, afterqed), lthy')
end
*}
ML {*
fun gen_function is_external prep default_constraint fixspec eqns config lthy =
let
val ((goal_state, afterqed), lthy') =
prepare_function is_external prep default_constraint fixspec eqns config lthy
in
lthy'
|> Proof.theorem NONE (snd oo afterqed) [[(Logic.unprotect (concl_of goal_state), [])]]
|> Proof.refine (Method.primitive_text (K goal_state))
|> Seq.hd
end
*}
ML {*
val function = gen_function false Specification.check_spec (Type_Infer.anyT HOLogic.typeS)
val function_cmd = gen_function true Specification.read_spec "_::type"
fun add_case_cong n thy =
let
val cong = #case_cong (Datatype.the_info thy n)
|> safe_mk_meta_eq
in
Context.theory_map
(Function_Ctx_Tree.map_function_congs (Thm.add_thm cong)) thy
end
val setup_case_cong = Datatype.interpretation (K (fold add_case_cong))
(* setup *)
val setup =
Attrib.setup @{binding fundef_cong}
(Attrib.add_del Function_Ctx_Tree.cong_add Function_Ctx_Tree.cong_del)
"declaration of congruence rule for function definitions"
#> setup_case_cong
#> Function_Relation.setup
#> Function_Common.Termination_Simps.setup
val get_congs = Function_Ctx_Tree.get_function_congs
fun get_info ctxt t = Item_Net.retrieve (get_function ctxt) t
|> the_single |> snd
(* outer syntax *)
val _ =
Outer_Syntax.local_theory_to_proof "nominal_primrec" "define recursive functions for nominal types"
Keyword.thy_goal
(function_parser default_config
>> (fn ((config, fixes), statements) => function_cmd fixes statements config))
*}
ML {* trace := true *}
lemma test:
assumes a: "[[x]]lst. t = [[x]]lst. t'"
shows "t = t'"
using a
apply(simp add: Abs_eq_iff)
apply(erule exE)
apply(simp only: alphas)
apply(erule conjE)+
apply(rule sym)
apply(clarify)
apply(rule supp_perm_eq)
apply(subgoal_tac "set [x] \<sharp>* p")
apply(auto simp add: fresh_star_def)[1]
apply(simp)
apply(simp add: fresh_star_def)
apply(simp add: fresh_perm)
done
lemma test2:
assumes "a \<sharp> x" "c \<sharp> x" "b \<sharp> y" "c \<sharp> y"
and "(a \<rightleftharpoons> c) \<bullet> x = (b \<rightleftharpoons> c) \<bullet> y"
shows "x = y"
using assms by (simp add: swap_fresh_fresh)
lemma test3:
assumes "x = y"
and "a \<sharp> x" "c \<sharp> x" "b \<sharp> y" "c \<sharp> y"
shows "(a \<rightleftharpoons> c) \<bullet> x = (b \<rightleftharpoons> c) \<bullet> y"
using assms by (simp add: swap_fresh_fresh)
nominal_primrec
depth :: "lam \<Rightarrow> nat"
where
"depth (Var x) = 1"
| "depth (App t1 t2) = (max (depth t1) (depth t2)) + 1"
| "depth (Lam x t) = (depth t) + 1"
apply(rule_tac y="x" in lam.exhaust)
apply(simp_all)[3]
apply(simp_all only: lam.distinct)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(subst (asm) Abs_eq_iff)
apply(erule exE)
apply(simp add: alphas)
apply(clarify)
oops
lemma removeAll_eqvt[eqvt]:
shows "p \<bullet> (removeAll x xs) = removeAll (p \<bullet> x) (p \<bullet> xs)"
by (induct xs) (auto)
nominal_primrec
frees_lst :: "lam \<Rightarrow> atom list"
where
"frees_lst (Var x) = [atom x]"
| "frees_lst (App t1 t2) = (frees_lst t1) @ (frees_lst t2)"
| "frees_lst (Lam x t) = removeAll (atom x) (frees_lst t)"
apply(rule_tac y="x" in lam.exhaust)
apply(simp_all)[3]
apply(simp_all only: lam.distinct)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: Abs_eq_iff)
apply(erule exE)
apply(simp add: alphas)
apply(simp add: atom_eqvt)
apply(clarify)
oops
nominal_primrec
subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_ [_ ::= _]" [100,100,100] 100)
where
"(Var x)[y ::= s] = (if x=y then s else (Var x))"
| "(App t\<^isub>1 t\<^isub>2)[y ::= s] = App (t\<^isub>1[y ::= s]) (t\<^isub>2[y ::= s])"
| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam x t)[y ::= s] = Lam x (t[y ::= s])"
apply(case_tac x)
apply(simp)
apply(rule_tac y="a" and c="(b, c)" in lam.strong_exhaust)
apply(simp add: lam.eq_iff lam.distinct)
apply(auto)[1]
apply(simp add: lam.eq_iff lam.distinct)
apply(auto)[1]
apply(simp add: fresh_star_def lam.eq_iff lam.distinct)
apply(simp_all add: lam.distinct)[5]
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(erule conjE)+
oops
nominal_primrec
depth :: "lam \<Rightarrow> nat"
where
"depth (Var x) = 1"
| "depth (App t1 t2) = (max (depth t1) (depth t2)) + 1"
| "depth (Lam x t) = (depth t) + 1"
apply(rule_tac y="x" in lam.exhaust)
apply(simp_all)[3]
apply(simp_all only: lam.distinct)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
(*
apply(subst (asm) Abs_eq_iff)
apply(erule exE)
apply(simp add: alphas)
apply(clarify)
*)
apply(subgoal_tac "finite (supp (x, xa, t, ta, depth_sumC t, depth_sumC ta))")
apply(erule_tac ?'a="name" in obtain_at_base)
unfolding fresh_def[symmetric]
apply(drule_tac a="atom x" and b="atom xa" and c="atom a" in test3)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(rule_tac a="atom x" and b="atom xa" and c="atom a" in test2)
apply(simp add: pure_fresh)
apply(simp add: pure_fresh)
apply(simp add: pure_fresh)
apply(simp add: pure_fresh)
apply(simp add: eqvt_at_def)
apply(drule test)
apply(simp)
apply(simp add: finite_supp)
done
termination depth
apply(relation "measure size")
apply(auto simp add: lam.size)
done
thm depth.psimps
thm depth.simps
lemma swap_set_not_in_at:
fixes a b::"'a::at"
and S::"'a::at set"
assumes a: "a \<notin> S" "b \<notin> S"
shows "(a \<leftrightarrow> b) \<bullet> S = S"
unfolding permute_set_eq
using a by (auto simp add: permute_flip_at)
lemma removeAll_eqvt[eqvt]:
shows "p \<bullet> (removeAll x xs) = removeAll (p \<bullet> x) (p \<bullet> xs)"
by (induct xs) (auto)
nominal_primrec
frees_lst :: "lam \<Rightarrow> atom list"
where
"frees_lst (Var x) = [atom x]"
| "frees_lst (App t1 t2) = (frees_lst t1) @ (frees_lst t2)"
| "frees_lst (Lam x t) = removeAll (atom x) (frees_lst t)"
apply(rule_tac y="x" in lam.exhaust)
apply(simp_all)[3]
apply(simp_all only: lam.distinct)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: Abs_eq_iff)
apply(erule exE)
apply(simp add: alphas)
apply(simp add: atom_eqvt)
apply(clarify)
apply(rule trans)
apply(rule sym)
apply(rule_tac p="p" in supp_perm_eq)
oops
nominal_primrec
frees :: "lam \<Rightarrow> name set"
where
"frees (Var x) = {x}"
| "frees (App t1 t2) = (frees t1) \<union> (frees t2)"
| "frees (Lam x t) = (frees t) - {x}"
apply(rule_tac y="x" in lam.exhaust)
apply(simp_all)[3]
apply(simp_all only: lam.distinct)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(subgoal_tac "finite (supp (x, xa, t, ta, frees_sumC t, frees_sumC ta))")
apply(erule_tac ?'a="name" in obtain_at_base)
unfolding fresh_def[symmetric]
apply(drule_tac a="atom x" and b="atom xa" and c="atom a" in test3)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp)
apply(drule test)
apply(rule_tac t="frees_sumC t - {x}" and s="(x \<leftrightarrow> a) \<bullet> (frees_sumC t - {x})" in subst)
oops
thm Abs_eq_iff[simplified alphas]
lemma Abs_set_eq_iff2:
fixes x y::"'a::fs"
shows "[bs]set. x = [cs]set. y \<longleftrightarrow>
(\<exists>p. supp ([bs]set. x) = supp ([cs]set. y) \<and>
supp ([bs]set. x) \<sharp>* p \<and>
p \<bullet> x = y \<and> p \<bullet> bs = cs)"
unfolding Abs_eq_iff
unfolding alphas
unfolding supp_Abs
by simp
lemma Abs_set_eq_iff3:
fixes x y::"'a::fs"
assumes a: "finite bs" "finite cs"
shows "[bs]set. x = [cs]set. y \<longleftrightarrow>
(\<exists>p. supp ([bs]set. x) = supp ([cs]set. y) \<and>
supp ([bs]set. x) \<sharp>* p \<and>
p \<bullet> x = y \<and> p \<bullet> bs = cs \<and>
supp p \<subseteq> bs \<union> cs)"
unfolding Abs_eq_iff
unfolding alphas
unfolding supp_Abs
apply(auto)
using set_renaming_perm
sorry
lemma Abs_eq_iff2:
fixes x y::"'a::fs"
shows "[bs]lst. x = [cs]lst. y \<longleftrightarrow>
(\<exists>p. supp ([bs]lst. x) = supp ([cs]lst. y) \<and>
supp ([bs]lst. x) \<sharp>* p \<and>
p \<bullet> x = y \<and> p \<bullet> bs = cs)"
unfolding Abs_eq_iff
unfolding alphas
unfolding supp_Abs
by simp
lemma Abs_eq_iff3:
fixes x y::"'a::fs"
shows "[bs]lst. x = [cs]lst. y \<longleftrightarrow>
(\<exists>p. supp ([bs]lst. x) = supp ([cs]lst. y) \<and>
supp ([bs]lst. x) \<sharp>* p \<and>
p \<bullet> x = y \<and> p \<bullet> bs = cs \<and>
supp p \<subseteq> set bs \<union> set cs)"
unfolding Abs_eq_iff
unfolding alphas
unfolding supp_Abs
apply(auto)
using list_renaming_perm
apply -
apply(drule_tac x="bs" in meta_spec)
apply(drule_tac x="p" in meta_spec)
apply(erule exE)
apply(rule_tac x="q" in exI)
apply(simp add: set_eqvt)
sorry
nominal_primrec
subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_ [_ ::= _]" [100,100,100] 100)
where
"(Var x)[y ::= s] = (if x=y then s else (Var x))"
| "(App t\<^isub>1 t\<^isub>2)[y ::= s] = App (t\<^isub>1[y ::= s]) (t\<^isub>2[y ::= s])"
| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam x t)[y ::= s] = Lam x (t[y ::= s])"
apply(case_tac x)
apply(simp)
apply(rule_tac y="a" and c="(b, c)" in lam.strong_exhaust)
apply(simp add: lam.eq_iff lam.distinct)
apply(auto)[1]
apply(simp add: lam.eq_iff lam.distinct)
apply(auto)[1]
apply(simp add: fresh_star_def lam.eq_iff lam.distinct)
apply(simp_all add: lam.distinct)[5]
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(erule conjE)+
apply(subst (asm) Abs_eq_iff3)
apply(erule exE)
apply(erule conjE)+
apply(clarify)
apply(perm_simp)
apply(simp)
apply(rule trans)
apply(rule sym)
apply(rule_tac p="p" in supp_perm_eq)
apply(rule fresh_star_supp_conv)
apply(drule fresh_star_supp_conv)
apply(simp add: Abs_fresh_star_iff)
thm fresh_eqvt_at
apply(rule_tac f="subst_sumC" in fresh_eqvt_at)
apply(assumption)
apply(simp add: finite_supp)
prefer 2
apply(simp)
apply(simp add: eqvt_at_def)
apply(perm_simp)
apply(subgoal_tac "p \<bullet> ya = ya")
apply(subgoal_tac "p \<bullet> sa = sa")
apply(simp)
apply(rule supp_perm_eq)
apply(rule fresh_star_supp_conv)
apply(auto simp add: fresh_star_def fresh_Pair)[1]
apply(rule supp_perm_eq)
apply(rule fresh_star_supp_conv)
apply(auto simp add: fresh_star_def fresh_Pair)[1]
nominal_primrec
subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam" ("_ [_ ::= _]" [100,100,100] 100)
where
"(Var x)[y ::= s] = (if x=y then s else (Var x))"
| "(App t\<^isub>1 t\<^isub>2)[y ::= s] = App (t\<^isub>1[y ::= s]) (t\<^isub>2[y ::= s])"
| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam x t)[y ::= s] = Lam x (t[y ::= s])"
apply(case_tac x)
apply(simp)
apply(rule_tac y="a" and c="(b, c)" in lam.strong_exhaust)
apply(simp add: lam.eq_iff lam.distinct)
apply(auto)[1]
apply(simp add: lam.eq_iff lam.distinct)
apply(auto)[1]
apply(simp add: fresh_star_def lam.eq_iff lam.distinct)
apply(simp_all add: lam.distinct)[5]
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(subgoal_tac "finite (supp (ya, sa, x, xa, t, ta, subst_sumC (t, ya, sa), subst_sumC (ta, ya, sa)))")
apply(erule_tac ?'a="name" in obtain_at_base)
unfolding fresh_def[symmetric]
apply(rule_tac a="atom x" and b="atom xa" and c="atom a" in test2)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(erule conjE)+
apply(drule_tac a="atom x" and b="atom xa" and c="atom a" in test3)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: Abs_fresh_iff)
apply(simp add: eqvt_at_def)
apply(drule test)
apply(simp)
apply(subst (2) swap_fresh_fresh)
apply(simp)
apply(simp)
apply(subst (2) swap_fresh_fresh)
apply(simp)
apply(simp)
apply(subst (3) swap_fresh_fresh)
apply(simp)
apply(simp)
apply(subst (3) swap_fresh_fresh)
apply(simp)
apply(simp)
apply(simp)
apply(simp add: finite_supp)
done
(* this should not work *)
nominal_primrec
bnds :: "lam \<Rightarrow> name set"
where
"bnds (Var x) = {}"
| "bnds (App t1 t2) = (bnds t1) \<union> (bnds t2)"
| "bnds (Lam x t) = (bnds t) \<union> {x}"
apply(rule_tac y="x" in lam.exhaust)
apply(simp_all)[3]
apply(simp_all only: lam.distinct)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
apply(simp add: lam.eq_iff)
sorry
end