(* Title: HOL/Tools/function_package/fundef_core.ML
Author: Alexander Krauss, TU Muenchen
A package for general recursive function definitions:
Main functionality.
*)
signature FUNDEF_CORE =
sig
val prepare_fundef : FundefCommon.fundef_config
-> string (* defname *)
-> ((bstring * typ) * mixfix) list (* defined symbol *)
-> ((bstring * typ) list * term list * term * term) list (* specification *)
-> local_theory
-> (term (* f *)
* thm (* goalstate *)
* (thm -> FundefCommon.fundef_result) (* continuation *)
) * local_theory
end
structure FundefCore : FUNDEF_CORE =
struct
val boolT = HOLogic.boolT
val mk_eq = HOLogic.mk_eq
open FundefLib
open FundefCommon
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: FundefCtxTree.ctx_tree,
lGI: thm,
RCs: rec_call_info list
}
(* Theory dependencies. *)
val Pair_inject = @{thm Product_Type.Pair_inject};
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};
fun find_calls tree =
let
fun add_Ri (fixes,assumes) (_ $ arg) _ (_, xs) = ([], (fixes, assumes, arg) :: xs)
| add_Ri _ _ _ _ = raise Match
in
rev (FundefCtxTree.traverse_tree add_Ri tree [])
end
(** 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')
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 (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
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 **)
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 (assume o cterm_of thy) gs
val case_hyp = 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
(* lowlevel term function *)
fun abstract_over_list vs body =
let
exception SAME;
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 handle SAME => u) handle SAME => t $ abs lev v u)
| _ => raise SAME);
in
fold_index (fn (i,v) => fn t => abs i v t handle SAME => t) vs body
end
fun mk_clause_info globals G f no cdata qglr tree RCs GIntro_thm RIntro_thms =
let
val Globals {h, fvar, x, ...} = 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
|> forall_elim (cert f)
|> fold forall_elim cqs
|> fold Thm.elim_implies ags
fun mk_call_info (rcfix, rcassm, rcarg) RI =
let
val llRI = RI
|> fold forall_elim cqs
|> fold (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
(* replace this by a table later*)
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
(* 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 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 forall_elim (cqsj @ cqsi) (* "Gsj => Gsi => lhsj = lhsi ==> rhsj = rhsi" *)
|> fold Thm.elim_implies agsj
|> fold Thm.elim_implies agsi
|> Thm.elim_implies ((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 forall_elim (cqsi @ cqsj) (* "Gsi => Gsj => lhsi = lhsj ==> rhsi = rhsj" *)
|> fold Thm.elim_implies agsi
|> fold Thm.elim_implies agsj
|> Thm.elim_implies (assume lhsi_eq_lhsj)
|> (fn thm => thm RS sym) (* "Gsi, Gsj, lhsi = lhsj |-- rhsj = rhsi" *)
end
end
(* Generates the replacement lemma in fully quantified form. *)
fun mk_replacement_lemma thy h ih_elim clause =
let
val ClauseInfo {cdata=ClauseContext {qs, lhs, rhs, 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, ...} => assume (cterm_of thy (subst_bounds (rev qs, h_assum)))) RCs
val (eql, _) = FundefCtxTree.rewrite_by_tree thy h ih_elim_case (Ris ~~ h_assums) tree
val replace_lemma = (eql RS meta_eq_to_obj_eq)
|> implies_intr (cprop_of case_hyp)
|> fold_rev (implies_intr o cprop_of) h_assums
|> fold_rev (implies_intr o cprop_of) ags
|> fold_rev forall_intr cqs
|> Thm.close_derivation
in
replace_lemma
end
fun mk_uniqueness_clause thy globals f compat_store 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, ...}, ...} = 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 compat = get_compat_thm thy compat_store i j cctxi cctxj
val Ghsj' = map (fn RCInfo {h_assum, ...} => assume (cterm_of thy (subst_bounds (rev qsj', h_assum)))) RCsj
val RLj_import =
RLj |> fold forall_elim cqsj'
|> fold Thm.elim_implies agsj'
|> fold Thm.elim_implies Ghsj'
val y_eq_rhsj'h = assume (cterm_of thy (HOLogic.mk_Trueprop (mk_eq (y, rhsj'h))))
val lhsi_eq_lhsj' = assume (cterm_of thy (HOLogic.mk_Trueprop (mk_eq (lhsi, lhsj')))) (* lhs_i = lhs_j' |-- lhs_i = lhs_j' *)
in
(trans OF [case_hyp, lhsi_eq_lhsj']) (* lhs_i = lhs_j' |-- x = lhs_j' *)
|> 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 (implies_intr o cprop_of) Ghsj'
|> fold_rev (implies_intr o cprop_of) agsj' (* lhs_i = lhs_j' , y = rhs_j_h' |-- Gj', Rj1'...Rjk' ==> y = rhs_i_f *)
|> implies_intr (cprop_of y_eq_rhsj'h)
|> implies_intr (cprop_of lhsi_eq_lhsj')
|> fold_rev forall_intr (cterm_of thy h :: cqsj')
end
fun mk_uniqueness_case ctxt thy globals G f ihyp ih_intro G_cases compat_store clauses rep_lemmas clausei =
let
val Globals {x, y, ranT, fvar, ...} = globals
val ClauseInfo {cdata = ClauseContext {lhs, rhs, qs, 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 (implies_intr o cprop_of) CCas
|> fold_rev (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 = assume (cterm_of thy (HOLogic.mk_Trueprop (G $ lhs $ y)))
val unique_clauses = map2 (mk_uniqueness_clause thy globals f compat_store clausei) clauses rep_lemmas
val uniqueness = G_cases
|> forall_elim (cterm_of thy lhs)
|> forall_elim (cterm_of thy y)
|> forall_elim P
|> Thm.elim_implies G_lhs_y
|> fold Thm.elim_implies unique_clauses
|> implies_intr (cprop_of G_lhs_y)
|> 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])
|> implies_intr (cprop_of case_hyp)
|> fold_rev (implies_intr o cprop_of) ags
|> fold_rev forall_intr cqs
val function_value =
existence
|> implies_intr ihyp
|> implies_intr (cprop_of case_hyp)
|> forall_intr (cterm_of thy x)
|> forall_elim (cterm_of thy lhs)
|> curry (op RS) refl
in
(exactly_one, function_value)
end
fun prove_stuff ctxt globals G f R clauses complete compat compat_store G_elim 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 ("Ex1", (ranT --> boolT) --> boolT) $
Abs ("y", ranT, G $ Bound 1 $ Bound 0))))
|> cterm_of thy
val ihyp_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 _ = Output.debug (K "Proving Replacement lemmas...")
val repLemmas = map (mk_replacement_lemma thy h ih_elim) clauses
val _ = Output.debug (K "Proving cases for unique existence...")
val (ex1s, values) =
split_list (map (mk_uniqueness_case ctxt thy globals G f ihyp ih_intro G_elim compat_store clauses repLemmas) clauses)
val _ = Output.debug (K "Proving: Graph is a function")
val graph_is_function = complete
|> Thm.forall_elim_vars 0
|> fold (curry op COMP) ex1s
|> implies_intr (ihyp)
|> implies_intr (cterm_of thy (HOLogic.mk_Trueprop (mk_acc domT R $ x)))
|> forall_intr (cterm_of thy x)
|> (fn it => Drule.compose_single (it, 2, acc_induct_rule)) (* "EX! y. (?x,y):G" *)
|> (fn it => fold (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 (implies_intr o cprop_of) compat
|> implies_intr (cprop_of complete)
in
(goalstate, values)
end
fun define_graph Gname fvar domT ranT clauses RCss lthy =
let
val GT = domT --> ranT --> boolT
val Gvar = Free (the_single (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 (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 (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
val (GIntro_thms, (G, G_elim, G_induct, lthy)) =
FundefInductiveWrap.inductive_def G_intros ((dest_Free Gvar, NoSyn), lthy)
in
((G, GIntro_thms, G_elim, G_induct), lthy)
end
fun define_function fdefname (fname, mixfix) domT ranT G default lthy =
let
val f_def =
Abs ("x", domT, Const ("FunDef.THE_default", ranT --> (ranT --> boolT) --> ranT) $ (default $ Bound 0) $
Abs ("y", ranT, G $ Bound 1 $ Bound 0))
|> Syntax.check_term lthy
val ((f, (_, f_defthm)), lthy) =
LocalTheory.define Thm.internalK ((Binding.name (function_name fname), mixfix), ((Binding.name fdefname, []), f_def)) lthy
in
((f, f_defthm), lthy)
end
fun define_recursion_relation Rname domT ranT fvar f qglrs clauses RCss lthy =
let
val RT = domT --> domT --> boolT
val Rvar = Free (the_single (Variable.variant_frees lthy [] [(Rname, RT)]))
fun mk_RIntro (ClauseContext {qs, gs, lhs, ...}, (oqs, _, _, _)) (rcfix, rcassm, rcarg) =
HOLogic.mk_Trueprop (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 (RIntro_thmss, (R, R_elim, _, lthy)) =
fold_burrow FundefInductiveWrap.inductive_def R_intross ((dest_Free Rvar, NoSyn), lthy)
in
((R, RIntro_thmss, 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 (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
((assume z_smaller) RS ((assume lhs_acc) RS acc_downward))
|> (fn it => it COMP graph_is_function)
|> implies_intr z_smaller
|> forall_intr (cterm_of thy z)
|> (fn it => it COMP valthm)
|> implies_intr lhs_acc
|> asm_simplify (HOL_basic_ss addsimps [f_iff])
|> fold_rev (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 Orderings.predicate1I} RS @{thm accp_subset_induct}
fun binder_conv cv ctxt = Conv.arg_conv (Conv.abs_conv (K cv) ctxt);
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 = 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 = 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 (binder_conv (arg1_conv (arg_conv (arg_conv case_hyp_conv))) ctxt) aihyp
end
fun mk_Prec (RCInfo {llRI, RIvs, CCas, rcarg, ...}) =
sih |> forall_elim (cterm_of thy rcarg)
|> Thm.elim_implies llRI
|> fold_rev (implies_intr o cprop_of) CCas
|> fold_rev (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 = assume step
|> fold 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)
|> symmetric (* P lhs == P x *)
|> (fn eql => equal_elim eql P_lhs) (* "P x" *)
|> implies_intr (cprop_of case_hyp)
|> fold_rev (implies_intr o cprop_of) ags
|> fold_rev 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 *)
|> implies_intr ihyp
|> implies_intr (cprop_of x_D)
|> forall_intr (cterm_of thy x)
val subset_induct_rule =
acc_subset_induct
|> (curry op COMP) (assume D_subset)
|> (curry op COMP) (assume D_dcl)
|> (curry op COMP) (assume a_D)
|> (curry op COMP) istep
|> fold_rev implies_intr steps
|> implies_intr a_D
|> implies_intr D_dcl
|> implies_intr D_subset
val subset_induct_all = fold_rev (forall_intr o cterm_of thy) [P, a, D] subset_induct_rule
val simple_induct_rule =
subset_induct_rule
|> forall_intr (cterm_of thy D)
|> 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]))
|> forall_intr (cterm_of thy z)
|> forall_intr (cterm_of thy x))
|> forall_intr (cterm_of thy a)
|> forall_intr (cterm_of thy P)
in
simple_induct_rule
end
(* FIXME: This should probably use fixed goals, to be more reliable and faster *)
fun mk_domain_intro ctxt (Globals {domT, ...}) R R_cases clause =
let
val thy = ProofContext.theory_of ctxt
val ClauseInfo {cdata = ClauseContext {qs, gs, lhs, rhs, 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 (local_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 {x, z, ...} = globals
val ClauseInfo {cdata = ClauseContext {qs,cqs,ags,lhs,rhs,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 = map (fn (ctx,thm) => FundefCtxTree.export_thm thy ctx thm) (u @ sub)
val hyp = HOLogic.mk_Trueprop (R' $ arg $ lhs)
|> fold_rev (curry Logic.mk_implies o prop_of) used (* additional hyps *)
|> FundefCtxTree.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 = assume hyp
|> fold forall_elim cqs
|> fold Thm.elim_implies ags
|> FundefCtxTree.import_thm thy (fixes, assumes)
|> fold Thm.elim_implies used (* "(arg, lhs) : R'" *)
val z_eq_arg = assume (cterm_of thy (HOLogic.mk_Trueprop (mk_eq (z, arg))))
val acc = thm COMP ih_case
val z_acc_local = acc
|> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (K (symmetric (z_eq_arg RS eq_reflection)))))
val ethm = z_acc_local
|> FundefCtxTree.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
FundefCtxTree.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 ("FunDef.in_rel", HOLogic.mk_setT (HOLogic.mk_prodT (domT, domT)) --> fastype_of R) $ Rrel
val wfR' = cterm_of thy (HOLogic.mk_Trueprop (Const (@{const_name "Wellfounded.wfP"}, (domT --> domT --> boolT) --> boolT) $ R')) (* "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 = 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
|> forall_elim (cterm_of thy z)
|> forall_elim (cterm_of thy x)
|> forall_elim (cterm_of thy (acc_R $ z))
|> curry op COMP (assume R_z_x)
|> fold_rev (curry op COMP) cases
|> implies_intr R_z_x
|> forall_intr (cterm_of thy z)
|> (fn it => it COMP accI)
|> implies_intr ihyp
|> forall_intr (cterm_of thy x)
|> (fn it => Drule.compose_single(it,2,wf_induct_rule))
|> curry op RS (assume wfR')
|> forall_intr_vars
|> (fn it => it COMP allI)
|> fold implies_intr hyps
|> implies_intr wfR'
|> forall_intr (cterm_of thy R')
|> forall_elim (cterm_of thy (inrel_R))
|> curry op RS wf_in_rel
|> full_simplify (HOL_basic_ss addsimps [in_rel_def])
|> 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 (local_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, qs, 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 (local_simpset_of ctxt) 1)
THEN (etac not_acc_down 1)
THEN ((etac R_cases) THEN_ALL_NEW (simp_default_tac (local_simpset_of ctxt))) 1)
|> fold_rev forall_intr_rename (map fst oqs ~~ cqs)
end
in
map2 mk_trsimp clauses psimps
end
fun prepare_fundef config defname [((fname, fT), mixfix)] abstract_qglrs lthy =
let
val FundefConfig {domintros, tailrec, default=default_str, ...} = config
val fvar = Free (fname, fT)
val domT = domain_type fT
val ranT = range_type fT
val default = Syntax.parse_term lthy default_str
|> TypeInfer.constrain 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, ...}) =
FundefCtxTree.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), 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 (FundefCtxTree.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 ranT fvar f abstract_qglrs clauses RCss) lthy
val (_, lthy) =
LocalTheory.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 |> assume
val compat = mk_compat_proof_obligations domT ranT fvar f abstract_qglrs |> map (cert #> 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) 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
FundefResult {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
end