(* Title: HOL/intr_elim.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
Introduction/elimination rule module -- for Inductive/Coinductive Definitions
*)
signature INDUCTIVE_ARG = (** Description of a (co)inductive def **)
sig
val thy : theory (*new theory with inductive defs*)
val monos : thm list (*monotonicity of each M operator*)
val con_defs : thm list (*definitions of the constructors*)
end;
(*internal items*)
signature INDUCTIVE_I =
sig
val rec_tms : term list (*the recursive sets*)
val intr_tms : term list (*terms for the introduction rules*)
end;
signature INTR_ELIM =
sig
val thy : theory (*copy of input theory*)
val defs : thm list (*definitions made in thy*)
val mono : thm (*monotonicity for the lfp definition*)
val unfold : thm (*fixed-point equation*)
val intrs : thm list (*introduction rules*)
val elim : thm (*case analysis theorem*)
val raw_induct : thm (*raw induction rule from Fp.induct*)
val mk_cases : thm list -> string -> thm (*generates case theorems*)
val rec_names : string list (*names of recursive sets*)
end;
(*prove intr/elim rules for a fixedpoint definition*)
functor Intr_elim_Fun
(structure Inductive: sig include INDUCTIVE_ARG INDUCTIVE_I end
and Fp: FP) : INTR_ELIM =
struct
open Logic Inductive Ind_Syntax;
val rec_names = map (#1 o dest_Const o head_of) rec_tms;
val big_rec_name = space_implode "_" rec_names;
val _ = deny (big_rec_name mem map ! (stamps_of_thy thy))
("Definition " ^ big_rec_name ^
" would clash with the theory of the same name!");
(*fetch fp definitions from the theory*)
val big_rec_def::part_rec_defs =
map (get_def thy)
(case rec_names of [_] => rec_names | _ => big_rec_name::rec_names);
val sign = sign_of thy;
(********)
val _ = writeln " Proving monotonicity...";
val Const("==",_) $ _ $ (Const(_,fpT) $ fp_abs) =
big_rec_def |> rep_thm |> #prop |> unvarify;
(*For the type of the argument of mono*)
val [monoT] = binder_types fpT;
val mono =
prove_goalw_cterm []
(cterm_of sign (mk_Trueprop (Const("mono", monoT-->boolT) $ fp_abs)))
(fn _ =>
[rtac monoI 1,
REPEAT (ares_tac (basic_monos @ monos) 1)]);
val unfold = standard (mono RS (big_rec_def RS Fp.Tarski));
(********)
val _ = writeln " Proving the introduction rules...";
fun intro_tacsf disjIn prems =
[(*insert prems and underlying sets*)
cut_facts_tac prems 1,
rtac (unfold RS ssubst) 1,
REPEAT (resolve_tac [Part_eqI,CollectI] 1),
(*Now 1-2 subgoals: the disjunction, perhaps equality.*)
rtac disjIn 1,
(*Not ares_tac, since refl must be tried before any equality assumptions;
backtracking may occur if the premises have extra variables!*)
DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 ORELSE assume_tac 1)];
(*combines disjI1 and disjI2 to access the corresponding nested disjunct...*)
val mk_disj_rls =
let fun f rl = rl RS disjI1
and g rl = rl RS disjI2
in accesses_bal(f, g, asm_rl) end;
val intrs = map (uncurry (prove_goalw_cterm part_rec_defs))
(map (cterm_of sign) intr_tms ~~
map intro_tacsf (mk_disj_rls(length intr_tms)));
(********)
val _ = writeln " Proving the elimination rule...";
(*Includes rules for Suc and Pair since they are common constructions*)
val elim_rls = [asm_rl, FalseE, Suc_neq_Zero, Zero_neq_Suc,
make_elim Suc_inject,
refl_thin, conjE, exE, disjE];
(*Breaks down logical connectives in the monotonic function*)
val basic_elim_tac =
REPEAT (SOMEGOAL (eresolve_tac (elim_rls@sumprod_free_SEs)
ORELSE' bound_hyp_subst_tac))
THEN prune_params_tac;
val elim = rule_by_tactic basic_elim_tac (unfold RS equals_CollectD);
(*Applies freeness of the given constructors, which *must* be unfolded by
the given defs. Cannot simply use the local con_defs because con_defs=[]
for inference systems.
fun con_elim_tac defs =
rewrite_goals_tac defs THEN basic_elim_tac THEN fold_tac defs;
*)
fun con_elim_tac simps =
let val elim_tac = REPEAT o (eresolve_tac (elim_rls@sumprod_free_SEs))
in ALLGOALS(EVERY'[elim_tac,
asm_full_simp_tac (nat_ss addsimps simps),
elim_tac,
REPEAT o bound_hyp_subst_tac])
THEN prune_params_tac
end;
(*String s should have the form t:Si where Si is an inductive set*)
fun mk_cases defs s =
rule_by_tactic (con_elim_tac defs)
(assume_read thy s RS elim);
val defs = big_rec_def::part_rec_defs;
val raw_induct = standard ([big_rec_def, mono] MRS Fp.induct);
end;