(* Title: HOL/simpdata.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Instantiation of the generic simplifier
*)
open Simplifier;
(*** Integration of simplifier with classical reasoner ***)
(*Add a simpset to a classical set!*)
infix 4 addss;
fun cs addss ss = cs addbefore asm_full_simp_tac ss 1;
fun Addss ss = (claset := !claset addbefore asm_full_simp_tac ss 1);
(*Maybe swap the safe_tac and simp_tac lines?**)
fun auto_tac (cs,ss) =
TRY (safe_tac cs) THEN
ALLGOALS (asm_full_simp_tac ss) THEN
REPEAT (FIRSTGOAL (best_tac (cs addss ss)));
fun Auto_tac() = auto_tac (!claset, !simpset);
fun auto() = by (Auto_tac());
local
fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
fun atomize pairs =
let fun atoms th =
(case concl_of th of
Const("Trueprop",_) $ p =>
(case head_of p of
Const(a,_) =>
(case assoc(pairs,a) of
Some(rls) => flat (map atoms ([th] RL rls))
| None => [th])
| _ => [th])
| _ => [th])
in atoms end;
fun mk_meta_eq r = case concl_of r of
Const("==",_)$_$_ => r
| _$(Const("op =",_)$_$_) => r RS eq_reflection
| _$(Const("not",_)$_) => r RS not_P_imp_P_eq_False
| _ => r RS P_imp_P_eq_True;
(* last 2 lines requires all formulae to be of the from Trueprop(.) *)
fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
val simp_thms = map prover
[ "(x=x) = True",
"(~True) = False", "(~False) = True", "(~ ~ P) = P",
"(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
"(True=P) = P", "(P=True) = P",
"(True --> P) = P", "(False --> P) = True",
"(P --> True) = True", "(P --> P) = True",
"(P --> False) = (~P)", "(P --> ~P) = (~P)",
"(P & True) = P", "(True & P) = P",
"(P & False) = False", "(False & P) = False", "(P & P) = P",
"(P | True) = True", "(True | P) = True",
"(P | False) = P", "(False | P) = P", "(P | P) = P",
"(!x.P) = P", "(? x.P) = P", "? x. x=t",
"(? x. x=t & P(x)) = P(t)", "(! x. x=t --> P(x)) = P(t)" ];
in
val meta_eq_to_obj_eq = prove_goal HOL.thy "x==y ==> x=y"
(fn [prem] => [rewtac prem, rtac refl 1]);
val eq_sym_conv = prover "(x=y) = (y=x)";
val conj_assoc = prover "((P&Q)&R) = (P&(Q&R))";
val disj_assoc = prover "((P|Q)|R) = (P|(Q|R))";
val imp_disj = prover "(P|Q --> R) = ((P-->R)&(Q-->R))";
val if_True = prove_goalw HOL.thy [if_def] "(if True then x else y) = x"
(fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
val if_False = prove_goalw HOL.thy [if_def] "(if False then x else y) = y"
(fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
val if_P = prove_goal HOL.thy "P ==> (if P then x else y) = x"
(fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
val if_not_P = prove_goal HOL.thy "~P ==> (if P then x else y) = y"
(fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
val expand_if = prove_goal HOL.thy
"P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
(fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
rtac (if_P RS ssubst) 2,
rtac (if_not_P RS ssubst) 1,
REPEAT(fast_tac HOL_cs 1) ]);
val if_bool_eq = prove_goal HOL.thy
"(if P then Q else R) = ((P-->Q) & (~P-->R))"
(fn _ => [rtac expand_if 1]);
(*Add congruence rules for = (instead of ==) *)
infix 4 addcongs;
fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection]);
fun Addcongs congs = (simpset := !simpset addcongs congs);
val mksimps_pairs =
[("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
("All", [spec]), ("True", []), ("False", []),
("If", [if_bool_eq RS iffD1])];
fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
val imp_cong = impI RSN
(2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
(fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
val o_apply = prove_goalw HOL.thy [o_def] "(f o g)(x) = f(g(x))"
(fn _ => [rtac refl 1]);
val HOL_ss = empty_ss
setmksimps (mksimps mksimps_pairs)
setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac
ORELSE' etac FalseE)
setsubgoaler asm_simp_tac
addsimps ([if_True, if_False, o_apply, imp_disj, conj_assoc, disj_assoc]
@ simp_thms)
addcongs [imp_cong];
(*In general it seems wrong to add distributive laws by default: they
might cause exponential blow-up. This one has been added for a while
and cannot be removed without affecting existing proofs. Moreover,
rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
grounds that it allows simplification of R in the two cases.*)
local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
in
fun split_tac splits = mktac (map mk_meta_eq splits)
end;
local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2)
in
fun split_inside_tac splits = mktac (map mk_meta_eq splits)
end;
(* eliminiation of existential quantifiers in assumptions *)
val ex_all_equiv =
let val lemma1 = prove_goal HOL.thy
"(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
(fn prems => [resolve_tac prems 1, etac exI 1]);
val lemma2 = prove_goalw HOL.thy [Ex_def]
"(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
(fn prems => [REPEAT(resolve_tac prems 1)])
in equal_intr lemma1 lemma2 end;
(* '&' congruence rule: not included by default!
May slow rewrite proofs down by as much as 50% *)
val conj_cong = impI RSN
(2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
(fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
val rev_conj_cong = impI RSN
(2, prove_goal HOL.thy "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
(fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
(** 'if' congruence rules: neither included by default! *)
(*Simplifies x assuming c and y assuming ~c*)
val if_cong = prove_goal HOL.thy
"[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
\ (if b then x else y) = (if c then u else v)"
(fn rew::prems =>
[stac rew 1, stac expand_if 1, stac expand_if 1,
fast_tac (HOL_cs addDs prems) 1]);
(*Prevents simplification of x and y: much faster*)
val if_weak_cong = prove_goal HOL.thy
"b=c ==> (if b then x else y) = (if c then x else y)"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
(*Prevents simplification of t: much faster*)
val let_weak_cong = prove_goal HOL.thy
"a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
end;
fun prove nm thm = qed_goal nm HOL.thy thm (fn _ => [fast_tac HOL_cs 1]);
prove "conj_commute" "(P&Q) = (Q&P)";
prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
val conj_comms = [conj_commute, conj_left_commute];
prove "disj_commute" "(P|Q) = (Q|P)";
prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))";
val disj_comms = [disj_commute, disj_left_commute];
prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))";
prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))";
prove "imp_conj_distrib" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
prove "imp_conj" "((P&Q)-->R) = (P --> (Q --> R))";
prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
prove "not_iff" "(P~=Q) = (P = (~Q))";
prove "not_all" "(~ (! x.P(x))) = (? x.~P(x))";
prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)";
prove "not_ex" "(~ (? x.P(x))) = (! x.~P(x))";
prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)";
prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
(fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
qed_goal "if_distrib" HOL.thy
"f(if c then x else y) = (if c then f x else f y)"
(fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
qed_goalw "o_assoc" HOL.thy [o_def] "f o (g o h) = (f o g o h)"
(fn _=>[rtac ext 1, rtac refl 1]);