# File ‹Tools/simpdata.ML›

```(*  Title:      HOL/Tools/simpdata.ML
Author:     Tobias Nipkow

Instantiation of the generic simplifier for HOL.
*)

(** tools setup **)

structure Quantifier1 = Quantifier1
(
(*abstract syntax*)
fun dest_eq (Const(\<^const_name>‹HOL.eq›,_) \$ s \$ t) = SOME (s, t)
| dest_eq _ = NONE;
fun dest_conj (Const(\<^const_name>‹HOL.conj›,_) \$ s \$ t) = SOME (s, t)
| dest_conj _ = NONE;
fun dest_imp (Const(\<^const_name>‹HOL.implies›,_) \$ s \$ t) = SOME (s, t)
| dest_imp _ = NONE;
val conj = HOLogic.conj
val imp  = HOLogic.imp
(*rules*)
val iff_reflection = @{thm eq_reflection}
val iffI = @{thm iffI}
val iff_trans = @{thm 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_conj}
in fn ctxt => Raw_Simplifier.rewrite ctxt true rules end
);

structure Simpdata =
struct

fun mk_meta_eq r = r RS @{thm eq_reflection};
fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;

fun mk_eq th =
(case Thm.concl_of th of
(*expects Trueprop if not == *)
Const (\<^const_name>‹Pure.eq›,_) \$ _ \$ _ => th
| _ \$ (Const (\<^const_name>‹HOL.eq›, _) \$ _ \$ _) => mk_meta_eq th
| _ \$ (Const (\<^const_name>‹Not›, _) \$ _) => th RS @{thm Eq_FalseI}
| _ => th RS @{thm Eq_TrueI})

fun mk_eq_True (_: Proof.context) r =
SOME (HOLogic.mk_obj_eq r RS @{thm Eq_TrueI}) handle Thm.THM _ => NONE;

(* Produce theorems of the form
(P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
*)

fun lift_meta_eq_to_obj_eq ctxt i st =
let
fun count_imp (Const (\<^const_name>‹HOL.simp_implies›, _) \$ _ \$ P) = 1 + count_imp P
| count_imp _ = 0;
val j = count_imp (Logic.strip_assums_concl (Thm.term_of (Thm.cprem_of st i)))
in
if j = 0 then @{thm meta_eq_to_obj_eq}
else
let
val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
val mk_simp_implies = fold_rev (fn R => fn S =>
Const (\<^const_name>‹HOL.simp_implies›, propT --> propT --> propT) \$ R \$ S) Ps;
in
Goal.prove_global (Proof_Context.theory_of ctxt) []
[mk_simp_implies \<^prop>‹(x::'a) == y›]
(mk_simp_implies \<^prop>‹(x::'a) = y›)
(fn {context = ctxt, prems} => EVERY
[rewrite_goals_tac ctxt @{thms simp_implies_def},
REPEAT (assume_tac ctxt 1 ORELSE
resolve_tac ctxt
(@{thm meta_eq_to_obj_eq} ::
map (rewrite_rule ctxt @{thms simp_implies_def}) prems) 1)])
end
end;

(*Congruence rules for = (instead of ==)*)
fun mk_meta_cong ctxt rl =
let
val rl' = Seq.hd (TRYALL (fn i => fn st =>
resolve_tac ctxt [lift_meta_eq_to_obj_eq ctxt i st] i st) rl)
in
mk_meta_eq rl' handle THM _ =>
if can Logic.dest_equals (Thm.concl_of rl') then rl'
else error "Conclusion of congruence rules must be =-equality"
end |> zero_var_indexes;

fun mk_atomize ctxt pairs =
let
fun atoms thm =
let
fun res th = map (fn rl => th RS rl);   (*exception THM*)
val thm_ctxt = Variable.declare_thm thm ctxt;
fun res_fixed rls =
if Thm.maxidx_of (Thm.adjust_maxidx_thm ~1 thm) = ~1 then res thm rls
else Variable.trade (K (fn [thm'] => res thm' rls)) thm_ctxt [thm];
in
case Thm.concl_of thm
of Const (\<^const_name>‹Trueprop›, _) \$ p => (case head_of p
of Const (a, _) => (case AList.lookup (op =) pairs a
of SOME rls => (maps atoms (res_fixed rls) handle THM _ => [thm])
| NONE => [thm])
| _ => [thm])
| _ => [thm]
end;
in atoms end;

fun mksimps pairs ctxt = map_filter (try mk_eq) o mk_atomize ctxt pairs o Variable.gen_all ctxt;

fun unsafe_solver_tac ctxt =
let
val sol_thms =
reflexive_thm :: @{thm TrueI} :: @{thm refl} :: Simplifier.prems_of ctxt;
fun sol_tac i =
FIRST
[resolve_tac ctxt sol_thms i,
assume_tac ctxt i,
eresolve_tac ctxt @{thms FalseE} i] ORELSE
(match_tac ctxt [@{thm conjI}]
THEN_ALL_NEW sol_tac) i
in
(fn i => REPEAT_DETERM (match_tac ctxt @{thms simp_impliesI} i)) THEN' sol_tac
end;

val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;

(*No premature instantiation of variables during simplification*)
fun safe_solver_tac ctxt =
(fn i => REPEAT_DETERM (match_tac ctxt @{thms simp_impliesI} i)) THEN'
FIRST' [match_tac ctxt (reflexive_thm :: @{thm TrueI} :: @{thm refl} :: Simplifier.prems_of ctxt),
eq_assume_tac, ematch_tac ctxt @{thms FalseE}];

val safe_solver = mk_solver "HOL safe" safe_solver_tac;

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

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

(* integration of simplifier with classical reasoner *)

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

val mksimps_pairs =
[(\<^const_name>‹HOL.implies›, [@{thm mp}]),
(\<^const_name>‹HOL.conj›, [@{thm conjunct1}, @{thm conjunct2}]),
(\<^const_name>‹All›, [@{thm spec}]),
(\<^const_name>‹True›, []),
(\<^const_name>‹False›, []),
(\<^const_name>‹If›, [@{thm if_bool_eq_conj} RS @{thm iffD1}])];

val HOL_basic_ss =
empty_simpset \<^context>
setSSolver safe_solver
setSolver unsafe_solver
|> Simplifier.set_subgoaler asm_simp_tac
|> Simplifier.set_mksimps (mksimps mksimps_pairs)
|> Simplifier.set_mkeqTrue mk_eq_True
|> Simplifier.set_mkcong mk_meta_cong
|> simpset_of;

fun hol_simplify ctxt rews =
Simplifier.full_simplify (put_simpset HOL_basic_ss ctxt addsimps rews);

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

end;

structure Splitter = Simpdata.Splitter;
structure Clasimp = Simpdata.Clasimp;
```