# File ‹simpdata.ML›

```(*  Title:      FOL/simpdata.ML
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1994  University of Cambridge

Simplification data for FOL.
*)

(*Make meta-equalities.  The operator below is Trueprop*)

fun mk_meta_eq th =
(case Thm.concl_of th of
_ \$ \<^Const_>‹eq _ for _ _› => th RS @{thm eq_reflection}
| _ \$ \<^Const_>‹iff for _ _› => th RS @{thm iff_reflection}
| _ => error "conclusion must be a =-equality or <->");

fun mk_eq th =
(case Thm.concl_of th of
\<^Const_>‹Pure.eq _ for _ _› => th
| _ \$ \<^Const_>‹eq _ for _ _› => mk_meta_eq th
| _ \$ \<^Const_>‹iff for _ _› => mk_meta_eq th
| _ \$ \<^Const_>‹Not for _› => th RS @{thm iff_reflection_F}
| _  => th RS @{thm iff_reflection_T});

(*Replace premises x=y, X<->Y by X==Y*)
fun mk_meta_prems ctxt =
rule_by_tactic ctxt
(REPEAT_FIRST (resolve_tac ctxt [@{thm meta_eq_to_obj_eq}, @{thm def_imp_iff}]));

(*Congruence rules for = or <-> (instead of ==)*)
fun mk_meta_cong ctxt rl =
Drule.zero_var_indexes (mk_meta_eq (mk_meta_prems ctxt rl))
handle THM _ =>
error("Premises and conclusion of congruence rules must use =-equality or <->");

val mksimps_pairs =
[(\<^const_name>‹imp›, [@{thm mp}]), (\<^const_name>‹conj›, [@{thm conjunct1}, @{thm conjunct2}]),
(\<^const_name>‹All›, [@{thm spec}]), (\<^const_name>‹True›, []), (\<^const_name>‹False›, [])];

fun mk_atomize pairs =
let
fun atoms th =
(case Thm.concl_of th of
\<^Const_>‹Trueprop for p› =>
(case head_of p of
Const(a,_) =>
(case AList.lookup (op =) pairs a of
SOME(rls) => maps atoms ([th] RL rls)
| NONE => [th])
| _ => [th])
| _ => [th])
in atoms end;

fun mksimps pairs ctxt = map mk_eq o mk_atomize pairs o Variable.gen_all ctxt;

(** make simplification procedures for quantifier elimination **)
structure Quantifier1 = Quantifier1
(
(*abstract syntax*)
fun dest_eq \<^Const_>‹eq _ for s t› = SOME (s, t)
| dest_eq _ = NONE
fun dest_conj \<^Const_>‹conj for s t› = SOME (s, t)
| dest_conj _ = NONE
fun dest_imp \<^Const_>‹imp for s t› = SOME (s, t)
| dest_imp _ = NONE
val conj = \<^Const>‹conj›
val imp  = \<^Const>‹imp›
(*rules*)
val iff_reflection = @{thm iff_reflection}
val iffI = @{thm iffI}
val iff_trans = @{thm iff_trans}
val conjI= @{thm conjI}
val conjE= @{thm conjE}
val impI = @{thm impI}
val mp   = @{thm mp}
val uncurry = @{thm uncurry}
val exI  = @{thm exI}
val exE  = @{thm exE}
val iff_allI = @{thm iff_allI}
val iff_exI = @{thm iff_exI}
val all_comm = @{thm all_comm}
val ex_comm = @{thm ex_comm}
val atomize =
let val rules = @{thms atomize_all atomize_imp atomize_eq atomize_iff atomize_conj}
in fn ctxt => Raw_Simplifier.rewrite ctxt true rules end
);

(*** Case splitting ***)

structure Splitter = Splitter
(
val context = \<^context>
val mk_eq = mk_eq
val meta_eq_to_iff = @{thm meta_eq_to_iff}
val iffD = @{thm iffD2}
val disjE = @{thm disjE}
val conjE = @{thm conjE}
val exE = @{thm exE}
val contrapos = @{thm contrapos}
val contrapos2 = @{thm contrapos2}
val notnotD = @{thm notnotD}
val safe_tac = Cla.safe_tac
);

val split_tac = Splitter.split_tac;
val split_inside_tac = Splitter.split_inside_tac;
val split_asm_tac = Splitter.split_asm_tac;

(*** Standard simpsets ***)

val triv_rls = [@{thm TrueI}, @{thm refl}, reflexive_thm, @{thm iff_refl}, @{thm notFalseI}];

fun unsafe_solver ctxt =
FIRST' [resolve_tac ctxt (triv_rls @ Simplifier.prems_of ctxt),
assume_tac ctxt,
eresolve_tac ctxt @{thms FalseE}];

(*No premature instantiation of variables during simplification*)
fun safe_solver ctxt =
FIRST' [match_tac ctxt (triv_rls @ Simplifier.prems_of ctxt),
eq_assume_tac, ematch_tac ctxt @{thms FalseE}];

(*No simprules, but basic infastructure for simplification*)
val FOL_basic_ss =
empty_simpset \<^context>
setSSolver (mk_solver "FOL safe" safe_solver)
setSolver (mk_solver "FOL unsafe" unsafe_solver)
|> Simplifier.set_subgoaler asm_simp_tac
|> Simplifier.set_mksimps (mksimps mksimps_pairs)
|> Simplifier.set_mkcong mk_meta_cong
|> simpset_of;

fun unfold_tac ctxt ths =
ALLGOALS (full_simp_tac (clear_simpset (put_simpset FOL_basic_ss ctxt) addsimps ths));

(*** integration of simplifier with classical reasoner ***)

structure Clasimp = Clasimp
(
structure Simplifier = Simplifier
and Splitter = Splitter
and Classical = Cla
and Blast = Blast
val iffD1 = @{thm iffD1}
val iffD2 = @{thm iffD2}
val notE = @{thm notE}
);
open Clasimp;

```