author | wenzelm |
Sun, 05 Jul 2015 15:02:30 +0200 | |
changeset 60642 | 48dd1cefb4ae |
parent 59936 | b8ffc3dc9e24 |
child 60695 | 757549b4bbe6 |
permissions | -rw-r--r-- |
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(* Title: HOL/Tools/Function/partial_function.ML |
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Author: Alexander Krauss, TU Muenchen |
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Partial function definitions based on least fixed points in ccpos. |
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*) |
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signature PARTIAL_FUNCTION = |
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sig |
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val init: string -> term -> term -> thm -> thm -> thm option -> declaration |
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val mono_tac: Proof.context -> int -> tactic |
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val add_partial_function: string -> (binding * typ option * mixfix) list -> |
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Attrib.binding * term -> local_theory -> local_theory |
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val add_partial_function_cmd: string -> (binding * string option * mixfix) list -> |
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Attrib.binding * string -> local_theory -> local_theory |
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end; |
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structure Partial_Function: PARTIAL_FUNCTION = |
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struct |
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(*** Context Data ***) |
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datatype setup_data = Setup_Data of |
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{fixp: term, |
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mono: term, |
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fixp_eq: thm, |
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fixp_induct: thm, |
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fixp_induct_user: thm option}; |
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structure Modes = Generic_Data |
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( |
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type T = setup_data Symtab.table; |
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val empty = Symtab.empty; |
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val extend = I; |
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fun merge data = Symtab.merge (K true) data; |
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) |
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fun init mode fixp mono fixp_eq fixp_induct fixp_induct_user phi = |
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let |
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val term = Morphism.term phi; |
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val thm = Morphism.thm phi; |
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val data' = Setup_Data |
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{fixp=term fixp, mono=term mono, fixp_eq=thm fixp_eq, |
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fixp_induct=thm fixp_induct, fixp_induct_user=Option.map thm fixp_induct_user}; |
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in |
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Modes.map (Symtab.update (mode, data')) |
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end |
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val known_modes = Symtab.keys o Modes.get o Context.Proof; |
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val lookup_mode = Symtab.lookup o Modes.get o Context.Proof; |
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(*** Automated monotonicity proofs ***) |
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fun strip_cases ctac = ctac #> Seq.map snd; |
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(*rewrite conclusion with k-th assumtion*) |
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fun rewrite_with_asm_tac ctxt k = |
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Subgoal.FOCUS (fn {context = ctxt', prems, ...} => |
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Local_Defs.unfold_tac ctxt' [nth prems k]) ctxt; |
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fun dest_case ctxt t = |
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case strip_comb t of |
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(Const (case_comb, _), args) => |
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(case Ctr_Sugar.ctr_sugar_of_case ctxt case_comb of |
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NONE => NONE |
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| SOME {case_thms, ...} => |
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let |
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val lhs = Thm.prop_of (hd case_thms) |
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|> HOLogic.dest_Trueprop |> HOLogic.dest_eq |> fst; |
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val arity = length (snd (strip_comb lhs)); |
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val conv = funpow (length args - arity) Conv.fun_conv |
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(Conv.rewrs_conv (map mk_meta_eq case_thms)); |
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in |
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SOME (nth args (arity - 1), conv) |
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end) |
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| _ => NONE; |
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(*split on case expressions*) |
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val split_cases_tac = Subgoal.FOCUS_PARAMS (fn {context = ctxt, ...} => |
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SUBGOAL (fn (t, i) => case t of |
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_ $ (_ $ Abs (_, _, body)) => |
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(case dest_case ctxt body of |
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NONE => no_tac |
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| SOME (arg, conv) => |
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let open Conv in |
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if Term.is_open arg then no_tac |
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else ((DETERM o strip_cases o Induct.cases_tac ctxt false [[SOME arg]] NONE []) |
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THEN_ALL_NEW (rewrite_with_asm_tac ctxt 0) |
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THEN_ALL_NEW eresolve_tac ctxt @{thms thin_rl} |
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THEN_ALL_NEW (CONVERSION |
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(params_conv ~1 (fn ctxt' => |
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arg_conv (arg_conv (abs_conv (K conv) ctxt'))) ctxt))) i |
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end) |
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| _ => no_tac) 1); |
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(*monotonicity proof: apply rules + split case expressions*) |
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fun mono_tac ctxt = |
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K (Local_Defs.unfold_tac ctxt [@{thm curry_def}]) |
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THEN' (TRY o REPEAT_ALL_NEW |
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(resolve_tac ctxt (rev (Named_Theorems.get ctxt @{named_theorems partial_function_mono})) |
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ORELSE' split_cases_tac ctxt)); |
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(*** Auxiliary functions ***) |
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(*positional instantiation with computed type substitution. |
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internal version of attribute "[of s t u]".*) |
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fun cterm_instantiate' cts thm = |
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let |
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val thy = Thm.theory_of_thm thm; |
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val vs = rev (Term.add_vars (Thm.prop_of thm) []) |
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|> map (Thm.global_cterm_of thy o Var); |
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in |
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cterm_instantiate (zip_options vs cts) thm |
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end; |
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(*Returns t $ u, but instantiates the type of t to make the |
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application type correct*) |
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fun apply_inst ctxt t u = |
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let |
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val thy = Proof_Context.theory_of ctxt; |
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val T = domain_type (fastype_of t); |
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val T' = fastype_of u; |
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val subst = Sign.typ_match thy (T, T') Vartab.empty |
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handle Type.TYPE_MATCH => raise TYPE ("apply_inst", [T, T'], [t, u]) |
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in |
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map_types (Envir.norm_type subst) t $ u |
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end; |
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fun head_conv cv ct = |
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if can Thm.dest_comb ct then Conv.fun_conv (head_conv cv) ct else cv ct; |
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(*** currying transformation ***) |
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fun curry_const (A, B, C) = |
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Const (@{const_name Product_Type.curry}, |
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[HOLogic.mk_prodT (A, B) --> C, A, B] ---> C); |
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fun mk_curry f = |
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case fastype_of f of |
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Type ("fun", [Type (_, [S, T]), U]) => |
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curry_const (S, T, U) $ f |
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| T => raise TYPE ("mk_curry", [T], [f]); |
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(* iterated versions. Nonstandard left-nested tuples arise naturally |
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from "split o split o split"*) |
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fun curry_n arity = funpow (arity - 1) mk_curry; |
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fun uncurry_n arity = funpow (arity - 1) HOLogic.mk_split; |
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val curry_uncurry_ss = |
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simpset_of (put_simpset HOL_basic_ss @{context} |
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addsimps [@{thm Product_Type.curry_split}, @{thm Product_Type.split_curry}]) |
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val split_conv_ss = |
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simpset_of (put_simpset HOL_basic_ss @{context} |
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addsimps [@{thm Product_Type.split_conv}]); |
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val curry_K_ss = |
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simpset_of (put_simpset HOL_basic_ss @{context} |
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addsimps [@{thm Product_Type.curry_K}]); |
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(* instantiate generic fixpoint induction and eliminate the canonical assumptions; |
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curry induction predicate *) |
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fun specialize_fixp_induct ctxt args fT fT_uc F curry uncurry mono_thm f_def rule = |
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let |
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val cert = Thm.cterm_of ctxt |
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val ([P], ctxt') = Variable.variant_fixes ["P"] ctxt |
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val P_inst = Abs ("f", fT_uc, Free (P, fT --> HOLogic.boolT) $ (curry $ Bound 0)) |
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in |
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(* FIXME ctxt vs. ctxt' (!?) *) |
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rule |
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|> cterm_instantiate' [SOME (cert uncurry), NONE, SOME (cert curry), NONE, SOME (cert P_inst)] |
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|> Tactic.rule_by_tactic ctxt |
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(Simplifier.simp_tac (put_simpset curry_uncurry_ss ctxt) 3 (* discharge U (C f) = f *) |
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THEN Simplifier.simp_tac (put_simpset curry_K_ss ctxt) 4 (* simplify bot case *) |
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THEN Simplifier.full_simp_tac (put_simpset curry_uncurry_ss ctxt) 5) (* simplify induction step *) |
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|> (fn thm => thm OF [mono_thm, f_def]) |
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|> Conv.fconv_rule (Conv.concl_conv ~1 (* simplify conclusion *) |
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(Raw_Simplifier.rewrite ctxt false [mk_meta_eq @{thm Product_Type.curry_split}])) |
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|> singleton (Variable.export ctxt' ctxt) |
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end |
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fun mk_curried_induct args ctxt inst_rule = |
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let |
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val cert = Thm.cterm_of ctxt |
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val ([P], ctxt') = Variable.variant_fixes ["P"] ctxt |
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val split_paired_all_conv = |
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Conv.every_conv (replicate (length args - 1) (Conv.rewr_conv @{thm split_paired_all})) |
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val split_params_conv = |
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Conv.params_conv ~1 (fn ctxt' => |
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Conv.implies_conv split_paired_all_conv Conv.all_conv) |
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val (P_var, x_var) = |
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Thm.prop_of inst_rule |> Logic.strip_imp_concl |> HOLogic.dest_Trueprop |
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|> strip_comb |> apsnd hd |
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|> apply2 dest_Var |
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val P_rangeT = range_type (snd P_var) |
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val PT = map (snd o dest_Free) args ---> P_rangeT |
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val x_inst = cert (foldl1 HOLogic.mk_prod args) |
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val P_inst = cert (uncurry_n (length args) (Free (P, PT))) |
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val inst_rule' = inst_rule |
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|> Tactic.rule_by_tactic ctxt |
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(Simplifier.simp_tac (put_simpset curry_uncurry_ss ctxt) 4 |
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THEN Simplifier.simp_tac (put_simpset curry_uncurry_ss ctxt) 3 |
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THEN CONVERSION (split_params_conv ctxt |
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then_conv (Conv.forall_conv (K split_paired_all_conv) ctxt)) 3) |
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|> Thm.instantiate ([], [(P_var, P_inst), (x_var, x_inst)]) |
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|> Simplifier.full_simplify (put_simpset split_conv_ss ctxt) |
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|> singleton (Variable.export ctxt' ctxt) |
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in |
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inst_rule' |
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end; |
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(*** partial_function definition ***) |
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fun gen_add_partial_function prep mode fixes_raw eqn_raw lthy = |
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let |
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val setup_data = the (lookup_mode lthy mode) |
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handle Option.Option => error (cat_lines ["Unknown mode " ^ quote mode ^ ".", |
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"Known modes are " ^ commas_quote (known_modes lthy) ^ "."]); |
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val Setup_Data {fixp, mono, fixp_eq, fixp_induct, fixp_induct_user} = setup_data; |
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val ((fixes, [(eq_abinding, eqn)]), _) = prep fixes_raw [eqn_raw] lthy; |
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val ((_, plain_eqn), args_ctxt) = Variable.focus eqn lthy; |
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val ((f_binding, fT), mixfix) = the_single fixes; |
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val fname = Binding.name_of f_binding; |
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val cert = Thm.cterm_of lthy; |
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val (lhs, rhs) = HOLogic.dest_eq (HOLogic.dest_Trueprop plain_eqn); |
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val (head, args) = strip_comb lhs; |
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val argnames = map (fst o dest_Free) args; |
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val F = fold_rev lambda (head :: args) rhs; |
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val arity = length args; |
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val (aTs, bTs) = chop arity (binder_types fT); |
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val tupleT = foldl1 HOLogic.mk_prodT aTs; |
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val fT_uc = tupleT :: bTs ---> body_type fT; |
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val f_uc = Var ((fname, 0), fT_uc); |
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val x_uc = Var (("x", 0), tupleT); |
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val uncurry = lambda head (uncurry_n arity head); |
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val curry = lambda f_uc (curry_n arity f_uc); |
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val F_uc = |
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lambda f_uc (uncurry_n arity (F $ curry_n arity f_uc)); |
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val mono_goal = apply_inst lthy mono (lambda f_uc (F_uc $ f_uc $ x_uc)) |
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|> HOLogic.mk_Trueprop |
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|> Logic.all x_uc; |
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val mono_thm = Goal.prove_internal lthy [] (cert mono_goal) |
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(K (mono_tac lthy 1)) |
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val inst_mono_thm = Thm.forall_elim (cert x_uc) mono_thm |
40107 | 264 |
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val f_def_rhs = curry_n arity (apply_inst lthy fixp F_uc); |
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val f_def_binding = Binding.concealed (Binding.name (Thm.def_name fname)); |
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val ((f, (_, f_def)), lthy') = Local_Theory.define |
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((f_binding, mixfix), ((f_def_binding, []), f_def_rhs)) lthy; |
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val eqn = HOLogic.mk_eq (list_comb (f, args), |
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Term.betapplys (F, f :: args)) |
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|> HOLogic.mk_Trueprop; |
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val unfold = |
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(cterm_instantiate' (map (SOME o cert) [uncurry, F, curry]) fixp_eq |
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OF [inst_mono_thm, f_def]) |
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|> Tactic.rule_by_tactic lthy' (Simplifier.simp_tac (put_simpset curry_uncurry_ss lthy') 1); |
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val specialized_fixp_induct = |
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specialize_fixp_induct lthy' args fT fT_uc F curry uncurry inst_mono_thm f_def fixp_induct |
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|> Drule.rename_bvars' (map SOME (fname :: fname :: argnames)); |
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282 |
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val mk_raw_induct = |
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cterm_instantiate' [SOME (cert uncurry), NONE, SOME (cert curry)] |
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#> mk_curried_induct args args_ctxt |
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#> singleton (Variable.export args_ctxt lthy') |
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#> (fn thm => cterm_instantiate' [SOME (cert F)] thm OF [inst_mono_thm, f_def]) |
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#> Drule.rename_bvars' (map SOME (fname :: argnames @ argnames)) |
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289 |
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val raw_induct = Option.map mk_raw_induct fixp_induct_user |
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val rec_rule = let open Conv in |
292 |
Goal.prove lthy' (map (fst o dest_Free) args) [] eqn (fn _ => |
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CONVERSION ((arg_conv o arg1_conv o head_conv o rewr_conv) (mk_meta_eq unfold)) 1 |
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THEN resolve_tac lthy' @{thms refl} 1) end; |
40107 | 295 |
in |
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lthy' |
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|> Local_Theory.note (eq_abinding, [rec_rule]) |
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|-> (fn (_, rec') => |
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Spec_Rules.add Spec_Rules.Equational ([f], rec') |
300 |
#> Local_Theory.note ((Binding.qualify true fname (Binding.name "simps"), []), rec') #> snd) |
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|> (Local_Theory.note ((Binding.qualify true fname (Binding.name "mono"), []), [mono_thm]) #> snd) |
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|> (case raw_induct of NONE => I | SOME thm => |
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Local_Theory.note ((Binding.qualify true fname (Binding.name "raw_induct"), []), [thm]) #> snd) |
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|> (Local_Theory.note ((Binding.qualify true fname (Binding.name "fixp_induct"), []), [specialized_fixp_induct]) #> snd) |
40107 | 305 |
end; |
306 |
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val add_partial_function = gen_add_partial_function Specification.check_spec; |
|
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val add_partial_function_cmd = gen_add_partial_function Specification.read_spec; |
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val mode = @{keyword "("} |-- Parse.xname --| @{keyword ")"}; |
40107 | 311 |
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val _ = |
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Outer_Syntax.local_theory @{command_keyword partial_function} "define partial function" |
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((mode -- (Parse.fixes -- (Parse.where_ |-- Parse_Spec.spec))) |
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>> (fn (mode, (fixes, spec)) => add_partial_function_cmd mode fixes spec)); |
40107 | 316 |
|
317 |
end |