src/HOL/Tools/TFL/casesplit.ML

changeset 23150 | 073a65f0bc40 |

child 29265 | 5b4247055bd7 |

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Tools/TFL/casesplit.ML Thu May 31 13:18:52 2007 +0200 @@ -0,0 +1,331 @@ +(* Title: HOL/Tools/TFL/casesplit.ML + ID: $Id$ + Author: Lucas Dixon, University of Edinburgh + +A structure that defines a tactic to program case splits. + + casesplit_free : + string * typ -> int -> thm -> thm Seq.seq + + casesplit_name : + string -> int -> thm -> thm Seq.seq + +These use the induction theorem associated with the recursive data +type to be split. + +The structure includes a function to try and recursively split a +conjecture into a list sub-theorems: + + splitto : thm list -> thm -> thm +*) + +(* logic-specific *) +signature CASE_SPLIT_DATA = +sig + val dest_Trueprop : term -> term + val mk_Trueprop : term -> term + val atomize : thm list + val rulify : thm list +end; + +structure CaseSplitData_HOL : CASE_SPLIT_DATA = +struct +val dest_Trueprop = HOLogic.dest_Trueprop; +val mk_Trueprop = HOLogic.mk_Trueprop; + +val atomize = thms "induct_atomize"; +val rulify = thms "induct_rulify"; +val rulify_fallback = thms "induct_rulify_fallback"; + +end; + + +signature CASE_SPLIT = +sig + (* failure to find a free to split on *) + exception find_split_exp of string + + (* getting a case split thm from the induction thm *) + val case_thm_of_ty : theory -> typ -> thm + val cases_thm_of_induct_thm : thm -> thm + + (* case split tactics *) + val casesplit_free : + string * typ -> int -> thm -> thm Seq.seq + val casesplit_name : string -> int -> thm -> thm Seq.seq + + (* finding a free var to split *) + val find_term_split : + term * term -> (string * typ) option + val find_thm_split : + thm -> int -> thm -> (string * typ) option + val find_thms_split : + thm list -> int -> thm -> (string * typ) option + + (* try to recursively split conjectured thm to given list of thms *) + val splitto : thm list -> thm -> thm + + (* for use with the recdef package *) + val derive_init_eqs : + theory -> + (thm * int) list -> term list -> (thm * int) list +end; + +functor CaseSplitFUN(Data : CASE_SPLIT_DATA) = +struct + +val rulify_goals = MetaSimplifier.rewrite_goals_rule Data.rulify; +val atomize_goals = MetaSimplifier.rewrite_goals_rule Data.atomize; + +(* beta-eta contract the theorem *) +fun beta_eta_contract thm = + let + val thm2 = equal_elim (Thm.beta_conversion true (Thm.cprop_of thm)) thm + val thm3 = equal_elim (Thm.eta_conversion (Thm.cprop_of thm2)) thm2 + in thm3 end; + +(* make a casethm from an induction thm *) +val cases_thm_of_induct_thm = + Seq.hd o (ALLGOALS (fn i => REPEAT (etac Drule.thin_rl i))); + +(* get the case_thm (my version) from a type *) +fun case_thm_of_ty sgn ty = + let + val dtypestab = DatatypePackage.get_datatypes sgn; + val ty_str = case ty of + Type(ty_str, _) => ty_str + | TFree(s,_) => error ("Free type: " ^ s) + | TVar((s,i),_) => error ("Free variable: " ^ s) + val dt = case Symtab.lookup dtypestab ty_str + of SOME dt => dt + | NONE => error ("Not a Datatype: " ^ ty_str) + in + cases_thm_of_induct_thm (#induction dt) + end; + +(* + val ty = (snd o hd o map Term.dest_Free o Term.term_frees) t; +*) + + +(* for use when there are no prems to the subgoal *) +(* does a case split on the given variable *) +fun mk_casesplit_goal_thm sgn (vstr,ty) gt = + let + val x = Free(vstr,ty) + val abst = Abs(vstr, ty, Term.abstract_over (x, gt)); + + val ctermify = Thm.cterm_of sgn; + val ctypify = Thm.ctyp_of sgn; + val case_thm = case_thm_of_ty sgn ty; + + val abs_ct = ctermify abst; + val free_ct = ctermify x; + + val casethm_vars = rev (Term.term_vars (Thm.concl_of case_thm)); + + val casethm_tvars = Term.term_tvars (Thm.concl_of case_thm); + val (Pv, Dv, type_insts) = + case (Thm.concl_of case_thm) of + (_ $ ((Pv as Var(P,Pty)) $ (Dv as Var(D, Dty)))) => + (Pv, Dv, + Sign.typ_match sgn (Dty, ty) Vartab.empty) + | _ => error "not a valid case thm"; + val type_cinsts = map (fn (ixn, (S, T)) => (ctypify (TVar (ixn, S)), ctypify T)) + (Vartab.dest type_insts); + val cPv = ctermify (Envir.subst_TVars type_insts Pv); + val cDv = ctermify (Envir.subst_TVars type_insts Dv); + in + (beta_eta_contract + (case_thm + |> Thm.instantiate (type_cinsts, []) + |> Thm.instantiate ([], [(cPv, abs_ct), (cDv, free_ct)]))) + end; + + +(* for use when there are no prems to the subgoal *) +(* does a case split on the given variable (Free fv) *) +fun casesplit_free fv i th = + let + val (subgoalth, exp) = IsaND.fix_alls i th; + val subgoalth' = atomize_goals subgoalth; + val gt = Data.dest_Trueprop (Logic.get_goal (Thm.prop_of subgoalth') 1); + val sgn = Thm.theory_of_thm th; + + val splitter_thm = mk_casesplit_goal_thm sgn fv gt; + val nsplits = Thm.nprems_of splitter_thm; + + val split_goal_th = splitter_thm RS subgoalth'; + val rulified_split_goal_th = rulify_goals split_goal_th; + in + IsaND.export_back exp rulified_split_goal_th + end; + + +(* for use when there are no prems to the subgoal *) +(* does a case split on the given variable *) +fun casesplit_name vstr i th = + let + val (subgoalth, exp) = IsaND.fix_alls i th; + val subgoalth' = atomize_goals subgoalth; + val gt = Data.dest_Trueprop (Logic.get_goal (Thm.prop_of subgoalth') 1); + + val freets = Term.term_frees gt; + fun getter x = + let val (n,ty) = Term.dest_Free x in + (if vstr = n orelse vstr = Name.dest_skolem n + then SOME (n,ty) else NONE ) + handle Fail _ => NONE (* dest_skolem *) + end; + val (n,ty) = case Library.get_first getter freets + of SOME (n, ty) => (n, ty) + | _ => error ("no such variable " ^ vstr); + val sgn = Thm.theory_of_thm th; + + val splitter_thm = mk_casesplit_goal_thm sgn (n,ty) gt; + val nsplits = Thm.nprems_of splitter_thm; + + val split_goal_th = splitter_thm RS subgoalth'; + + val rulified_split_goal_th = rulify_goals split_goal_th; + in + IsaND.export_back exp rulified_split_goal_th + end; + + +(* small example: +Goal "P (x :: nat) & (C y --> Q (y :: nat))"; +by (rtac (thm "conjI") 1); +val th = topthm(); +val i = 2; +val vstr = "y"; + +by (casesplit_name "y" 2); + +val th = topthm(); +val i = 1; +val th' = casesplit_name "x" i th; +*) + + +(* the find_XXX_split functions are simply doing a lightwieght (I +think) term matching equivalent to find where to do the next split *) + +(* assuming two twems are identical except for a free in one at a +subterm, or constant in another, ie assume that one term is a plit of +another, then gives back the free variable that has been split. *) +exception find_split_exp of string +fun find_term_split (Free v, _ $ _) = SOME v + | find_term_split (Free v, Const _) = SOME v + | find_term_split (Free v, Abs _) = SOME v (* do we really want this case? *) + | find_term_split (Free v, Var _) = NONE (* keep searching *) + | find_term_split (a $ b, a2 $ b2) = + (case find_term_split (a, a2) of + NONE => find_term_split (b,b2) + | vopt => vopt) + | find_term_split (Abs(_,ty,t1), Abs(_,ty2,t2)) = + find_term_split (t1, t2) + | find_term_split (Const (x,ty), Const(x2,ty2)) = + if x = x2 then NONE else (* keep searching *) + raise find_split_exp (* stop now *) + "Terms are not identical upto a free varaible! (Consts)" + | find_term_split (Bound i, Bound j) = + if i = j then NONE else (* keep searching *) + raise find_split_exp (* stop now *) + "Terms are not identical upto a free varaible! (Bound)" + | find_term_split (a, b) = + raise find_split_exp (* stop now *) + "Terms are not identical upto a free varaible! (Other)"; + +(* assume that "splitth" is a case split form of subgoal i of "genth", +then look for a free variable to split, breaking the subgoal closer to +splitth. *) +fun find_thm_split splitth i genth = + find_term_split (Logic.get_goal (Thm.prop_of genth) i, + Thm.concl_of splitth) handle find_split_exp _ => NONE; + +(* as above but searches "splitths" for a theorem that suggest a case split *) +fun find_thms_split splitths i genth = + Library.get_first (fn sth => find_thm_split sth i genth) splitths; + + +(* split the subgoal i of "genth" until we get to a member of +splitths. Assumes that genth will be a general form of splitths, that +can be case-split, as needed. Otherwise fails. Note: We assume that +all of "splitths" are split to the same level, and thus it doesn't +matter which one we choose to look for the next split. Simply add +search on splitthms and split variable, to change this. *) +(* Note: possible efficiency measure: when a case theorem is no longer +useful, drop it? *) +(* Note: This should not be a separate tactic but integrated into the +case split done during recdef's case analysis, this would avoid us +having to (re)search for variables to split. *) +fun splitto splitths genth = + let + val _ = not (null splitths) orelse error "splitto: no given splitths"; + val sgn = Thm.theory_of_thm genth; + + (* check if we are a member of splitths - FIXME: quicker and + more flexible with discrim net. *) + fun solve_by_splitth th split = + Thm.biresolution false [(false,split)] 1 th; + + fun split th = + (case find_thms_split splitths 1 th of + NONE => + (writeln "th:"; + Display.print_thm th; writeln "split ths:"; + Display.print_thms splitths; writeln "\n--"; + error "splitto: cannot find variable to split on") + | SOME v => + let + val gt = Data.dest_Trueprop (List.nth(Thm.prems_of th, 0)); + val split_thm = mk_casesplit_goal_thm sgn v gt; + val (subthms, expf) = IsaND.fixed_subgoal_thms split_thm; + in + expf (map recsplitf subthms) + end) + + and recsplitf th = + (* note: multiple unifiers! we only take the first element, + probably fine -- there is probably only one anyway. *) + (case Library.get_first (Seq.pull o solve_by_splitth th) splitths of + NONE => split th + | SOME (solved_th, more) => solved_th) + in + recsplitf genth + end; + + +(* Note: We dont do this if wf conditions fail to be solved, as each +case may have a different wf condition - we could group the conditions +togeather and say that they must be true to solve the general case, +but that would hide from the user which sub-case they were related +to. Probably this is not important, and it would work fine, but I +prefer leaving more fine grain control to the user. *) + +(* derive eqs, assuming strict, ie the rules have no assumptions = all + the well-foundness conditions have been solved. *) +fun derive_init_eqs sgn rules eqs = + let + fun get_related_thms i = + List.mapPartial ((fn (r, x) => if x = i then SOME r else NONE)); + fun add_eq (i, e) xs = + (e, (get_related_thms i rules), i) :: xs + fun solve_eq (th, [], i) = + error "derive_init_eqs: missing rules" + | solve_eq (th, [a], i) = (a, i) + | solve_eq (th, splitths as (_ :: _), i) = (splitto splitths th, i); + val eqths = + map (Thm.trivial o Thm.cterm_of sgn o Data.mk_Trueprop) eqs; + in + [] + |> fold_index add_eq eqths + |> map solve_eq + |> rev + end; + +end; + + +structure CaseSplit = CaseSplitFUN(CaseSplitData_HOL);