(* Title: HOL/simpdata.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Instantiation of the generic simplifier
*)
section "Simplifier";
open Simplifier;
(*** Addition of rules to simpsets and clasets simultaneously ***)
(*Takes UNCONDITIONAL theorems of the form A<->B to
the Safe Intr rule B==>A and
the Safe Destruct rule A==>B.
Also ~A goes to the Safe Elim rule A ==> ?R
Failing other cases, A is added as a Safe Intr rule*)
local
val iff_const = HOLogic.eq_const HOLogic.boolT;
fun addIff th =
(case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
(Const("Not",_) $ A) =>
AddSEs [zero_var_indexes (th RS notE)]
| (con $ _ $ _) =>
if con=iff_const
then (AddSIs [zero_var_indexes (th RS iffD2)];
AddSDs [zero_var_indexes (th RS iffD1)])
else AddSIs [th]
| _ => AddSIs [th];
Addsimps [th])
handle _ => error ("AddIffs: theorem must be unconditional\n" ^
string_of_thm th)
fun delIff th =
(case HOLogic.dest_Trueprop (#prop(rep_thm th)) of
(Const("Not",_) $ A) =>
Delrules [zero_var_indexes (th RS notE)]
| (con $ _ $ _) =>
if con=iff_const
then Delrules [zero_var_indexes (th RS iffD2),
make_elim (zero_var_indexes (th RS iffD1))]
else Delrules [th]
| _ => Delrules [th];
Delsimps [th])
handle _ => warning("DelIffs: ignoring conditional theorem\n" ^
string_of_thm th)
in
val AddIffs = seq addIff
val DelIffs = seq delIff
end;
local
fun prover s = prove_goal HOL.thy s (fn _ => [blast_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 gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
in
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(.) *)
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 & (P & Q)) = (P & Q)",
"(P | True) = True", "(True | P) = True",
"(P | False) = P", "(False | P) = P",
"(P | P) = P", "(P | (P | Q)) = (P | Q)",
"((~P) = (~Q)) = (P=Q)",
"(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x",
"(? x. x=t & P(x)) = P(t)",
"(! x. t=x --> P(x)) = P(t)" ];
(*Add congruence rules for = (instead of ==) *)
infix 4 addcongs delcongs;
fun ss addcongs congs = ss addeqcongs (map standard (congs RL [eq_reflection]));
fun ss delcongs congs = ss deleqcongs (map standard (congs RL [eq_reflection]));
fun Addcongs congs = (simpset := !simpset addcongs congs);
fun Delcongs congs = (simpset := !simpset delcongs congs);
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 _=> [blast_tac HOL_cs 1]) RS mp RS mp);
(*Miniscoping: pushing in existential quantifiers*)
val ex_simps = map prover
["(EX x. P x & Q) = ((EX x. P x) & Q)",
"(EX x. P & Q x) = (P & (EX x. Q x))",
"(EX x. P x | Q) = ((EX x. P x) | Q)",
"(EX x. P | Q x) = (P | (EX x. Q x))",
"(EX x. P x --> Q) = ((ALL x. P x) --> Q)",
"(EX x. P --> Q x) = (P --> (EX x. Q x))"];
(*Miniscoping: pushing in universal quantifiers*)
val all_simps = map prover
["(ALL x. P x & Q) = ((ALL x. P x) & Q)",
"(ALL x. P & Q x) = (P & (ALL x. Q x))",
"(ALL x. P x | Q) = ((ALL x. P x) | Q)",
"(ALL x. P | Q x) = (P | (ALL x. Q x))",
"(ALL x. P x --> Q) = ((EX x. P x) --> Q)",
"(ALL x. P --> Q x) = (P --> (ALL x. Q x))"];
(*** Simplification procedure for turning ? x. ... & x = t & ...
into ? x. x = t & ... & ...
where the latter can be rewritten via (? x. x = t & P(x)) = P(t)
***)
local
fun def(eq as (c as Const("op =",_)) $ s $ t) =
if s = Bound 0 andalso not(loose_bvar1(t,0)) then Some eq else
if t = Bound 0 andalso not(loose_bvar1(s,0)) then Some(c$t$s)
else None
| def _ = None;
fun extract(Const("op &",_) $ P $ Q) =
(case def P of
Some eq => Some(eq,Q)
| None => (case def Q of
Some eq => Some(eq,P)
| None =>
(case extract P of
Some(eq,P') => Some(eq, HOLogic.conj $ P' $ Q)
| None => (case extract Q of
Some(eq,Q') => Some(eq,HOLogic.conj $ P $ Q')
| None => None))))
| extract _ = None;
fun prove_eq(ceqt) =
let val tac = rtac eq_reflection 1 THEN rtac iffI 1 THEN
ALLGOALS(EVERY'[etac exE, REPEAT o (etac conjE),
rtac exI, REPEAT o (ares_tac [conjI] ORELSE' etac sym)])
in rule_by_tactic tac (trivial ceqt) end;
fun rearrange sg _ (F as ex $ Abs(x,T,P)) =
(case extract P of
None => None
| Some(eq,Q) =>
let val ceqt = cterm_of sg
(Logic.mk_equals(F,ex $ Abs(x,T,HOLogic.conj$eq$Q)))
in Some(prove_eq ceqt) end)
| rearrange _ _ _ = None;
val pattern = read_cterm (sign_of HOL.thy) ("? x. P(x) & Q(x)",HOLogic.boolT)
in
val defEX_regroup = mk_simproc "defined EX" [pattern] rearrange;
end;
(* elimination 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;
end;
(* Elimination of True from asumptions: *)
val True_implies_equals = prove_goal HOL.thy
"(True ==> PROP P) == PROP P"
(fn _ => [rtac equal_intr_rule 1, atac 2,
METAHYPS (fn prems => resolve_tac prems 1) 1,
rtac TrueI 1]);
fun prove nm thm = qed_goal nm HOL.thy thm (fn _ => [blast_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 "conj_assoc" "((P&Q)&R) = (P&(Q&R))";
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 "disj_assoc" "((P|Q)|R) = (P|(Q|R))";
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_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))";
prove "imp_conjL" "((P&Q) -->R) = (P --> (Q --> R))";
prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))";
(*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*)
prove "imp_disj_not1" "((P --> Q | R)) = (~Q --> P --> R)";
prove "imp_disj_not2" "((P --> Q | R)) = (~R --> P --> Q)";
prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
prove "not_imp" "(~(P --> Q)) = (P & ~Q)";
prove "not_iff" "(P~=Q) = (P = (~Q))";
(*Avoids duplication of subgoals after expand_if, when the true and false
cases boil down to the same thing.*)
prove "cases_simp" "((P --> Q) & (~P --> Q)) = 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)))";
(* '&' congruence rule: not included by default!
May slow rewrite proofs down by as much as 50% *)
let val th = prove_goal HOL.thy
"(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
(fn _=> [blast_tac HOL_cs 1])
in bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp))) end;
let val th = prove_goal HOL.thy
"(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
(fn _=> [blast_tac HOL_cs 1])
in bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp))) end;
(* '|' congruence rule: not included by default! *)
let val th = prove_goal HOL.thy
"(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))"
(fn _=> [blast_tac HOL_cs 1])
in bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp))) end;
prove "eq_sym_conv" "(x=y) = (y=x)";
qed_goalw "o_apply" HOL.thy [o_def] "(f o g) x = f (g x)"
(fn _ => [rtac refl 1]);
qed_goal "meta_eq_to_obj_eq" HOL.thy "x==y ==> x=y"
(fn [prem] => [rewtac prem, rtac refl 1]);
qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x"
(fn _=>[blast_tac (HOL_cs addIs [select_equality]) 1]);
qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y"
(fn _=>[blast_tac (HOL_cs addIs [select_equality]) 1]);
qed_goal "if_P" HOL.thy "P ==> (if P then x else y) = x"
(fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
(*
qed_goal "if_not_P" 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 ]);
*)
qed_goalw "if_not_P" HOL.thy [if_def] "!!P. ~P ==> (if P then x else y) = y"
(fn _ => [blast_tac (HOL_cs addIs [select_equality]) 1]);
qed_goal "expand_if" 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),
stac if_P 2,
stac if_not_P 1,
REPEAT(blast_tac HOL_cs 1) ]);
qed_goal "if_bool_eq" HOL.thy
"(if P then Q else R) = ((P-->Q) & (~P-->R))"
(fn _ => [rtac expand_if 1]);
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;
qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
(fn _ => [split_tac [expand_if] 1, blast_tac HOL_cs 1]);
(** 'if' congruence rules: neither included by default! *)
(*Simplifies x assuming c and y assuming ~c*)
qed_goal "if_cong" 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,
blast_tac (HOL_cs addDs prems) 1]);
(*Prevents simplification of x and y: much faster*)
qed_goal "if_weak_cong" 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*)
qed_goal "let_weak_cong" HOL.thy
"a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
(*In general it seems wrong to add distributive laws by default: they
might cause exponential blow-up. But imp_disjL has been in 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.*)
val mksimps_pairs =
[("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
("All", [spec]), ("True", []), ("False", []),
("If", [if_bool_eq RS iffD1])];
fun unsafe_solver prems = FIRST'[resolve_tac (TrueI::refl::prems),
atac, etac FalseE];
(*No premature instantiation of variables during simplification*)
fun safe_solver prems = FIRST'[match_tac (TrueI::refl::prems),
eq_assume_tac, ematch_tac [FalseE]];
val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac
setSSolver safe_solver
setSolver unsafe_solver
setmksimps (mksimps mksimps_pairs);
val HOL_ss =
HOL_basic_ss addsimps
([triv_forall_equality, (* prunes params *)
True_implies_equals, (* prune asms `True' *)
if_True, if_False, if_cancel,
o_apply, imp_disjL, conj_assoc, disj_assoc,
de_Morgan_conj, de_Morgan_disj, not_imp,
not_all, not_ex, cases_simp]
@ ex_simps @ all_simps @ simp_thms)
addsimprocs [defEX_regroup]
addcongs [imp_cong];
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]);
val prems = goal HOL.thy "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
by (case_tac "P" 1);
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
val expand_case = result();
fun expand_case_tac P i =
res_inst_tac [("P",P)] expand_case i THEN
Simp_tac (i+1) THEN
Simp_tac i;
(*** Install simpsets and datatypes in theory structure ***)
simpset := HOL_ss;
exception SS_DATA of simpset;
let fun merge [] = SS_DATA empty_ss
| merge ss = let val ss = map (fn SS_DATA x => x) ss;
in SS_DATA (foldl merge_ss (hd ss, tl ss)) end;
fun put (SS_DATA ss) = simpset := ss;
fun get () = SS_DATA (!simpset);
in add_thydata "HOL"
("simpset", ThyMethods {merge = merge, put = put, get = get})
end;
fun simpset_of tname =
case get_thydata tname "simpset" of
None => empty_ss
| Some (SS_DATA ss) => ss;
type dtype_info = {case_const:term,
case_rewrites:thm list,
constructors:term list,
induct_tac: string -> int -> tactic,
nchotomy: thm,
exhaustion: thm,
exhaust_tac: string -> int -> tactic,
case_cong:thm};
exception DT_DATA of (string * dtype_info) list;
val datatypes = ref [] : (string * dtype_info) list ref;
let fun merge [] = DT_DATA []
| merge ds =
let val ds = map (fn DT_DATA x => x) ds;
in DT_DATA (foldl (gen_union eq_fst) (hd ds, tl ds)) end;
fun put (DT_DATA ds) = datatypes := ds;
fun get () = DT_DATA (!datatypes);
in add_thydata "HOL"
("datatypes", ThyMethods {merge = merge, put = put, get = get})
end;
(*** Integration of simplifier with classical reasoner ***)
(* rot_eq_tac rotates the first equality premise of subgoal i to the front,
fails if there is no equaliy or if an equality is already at the front *)
local
fun is_eq (Const ("Trueprop", _) $ (Const("op =" ,_) $ _ $ _)) = true
| is_eq _ = false;
fun find_eq n [] = None
| find_eq n (t :: ts) = if (is_eq t) then Some n
else find_eq (n + 1) ts;
in
val rot_eq_tac =
SUBGOAL (fn (Bi,i) =>
case find_eq 0 (Logic.strip_assums_hyp Bi) of
None => no_tac
| Some 0 => no_tac
| Some n => rotate_tac n i)
end;
(*an unsatisfactory fix for the incomplete asm_full_simp_tac!
better: asm_really_full_simp_tac, a yet to be implemented version of
asm_full_simp_tac that applies all equalities in the
premises to all the premises *)
fun safe_asm_more_full_simp_tac ss = TRY o rot_eq_tac THEN'
safe_asm_full_simp_tac ss;
(*Add a simpset to a classical set!*)
infix 4 addSss addss;
fun cs addSss ss = cs addSaltern (CHANGED o (safe_asm_more_full_simp_tac ss));
fun cs addss ss = cs addbefore asm_full_simp_tac ss;
fun Addss ss = (claset := !claset addss ss);
(*Designed to be idempotent, except if best_tac instantiates variables
in some of the subgoals*)
type clasimpset = (claset * simpset);
val HOL_css = (HOL_cs, HOL_ss);
fun pair_upd1 f ((a,b),x) = (f(a,x), b);
fun pair_upd2 f ((a,b),x) = (a, f(b,x));
infix 4 addSIs2 addSEs2 addSDs2 addIs2 addEs2 addDs2
addsimps2 delsimps2 addcongs2 delcongs2;
fun op addSIs2 arg = pair_upd1 (op addSIs) arg;
fun op addSEs2 arg = pair_upd1 (op addSEs) arg;
fun op addSDs2 arg = pair_upd1 (op addSDs) arg;
fun op addIs2 arg = pair_upd1 (op addIs ) arg;
fun op addEs2 arg = pair_upd1 (op addEs ) arg;
fun op addDs2 arg = pair_upd1 (op addDs ) arg;
fun op addsimps2 arg = pair_upd2 (op addsimps) arg;
fun op delsimps2 arg = pair_upd2 (op delsimps) arg;
fun op addcongs2 arg = pair_upd2 (op addcongs) arg;
fun op delcongs2 arg = pair_upd2 (op delcongs) arg;
fun auto_tac (cs,ss) =
let val cs' = cs addss ss
in EVERY [TRY (safe_tac cs'),
REPEAT (FIRSTGOAL (fast_tac cs')),
TRY (safe_tac (cs addSss ss)),
prune_params_tac]
end;
fun Auto_tac () = auto_tac (!claset, !simpset);
fun auto () = by (Auto_tac ());