diff -r 3e9ae9032273 -r b8cdd3d73022 src/HOL/Tools/Function/fundef_core.ML --- a/src/HOL/Tools/Function/fundef_core.ML Fri Oct 23 15:33:19 2009 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,956 +0,0 @@ -(* Title: HOL/Tools/Function/fundef_core.ML - Author: Alexander Krauss, TU Muenchen - -A package for general recursive function definitions: -Main functionality. -*) - -signature FUNDEF_CORE = -sig - val trace: bool Unsynchronized.ref - - 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 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 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. 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, 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 _ = trace_msg (K "Proving Replacement lemmas...") - val repLemmas = map (mk_replacement_lemma thy h ih_elim) clauses - - val _ = trace_msg (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 _ = trace_msg (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 (@{const_name 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 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 (More_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 |> 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 (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 (@{const_name 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 (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 (simpset_of ctxt) 1) - THEN (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 - - -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