(* Title: HOL/simpdata.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Instantiation of the generic simplifier
*)
open Simplifier;
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 r =
case concl_of r of
Const("Trueprop",_) $ p =>
(case p of
Const("op -->",_)$_$_ => atomize(r RS mp)
| Const("op &",_)$_$_ => atomize(r RS conjunct1) @
atomize(r RS conjunct2)
| Const("All",_)$_ => atomize(r RS spec)
| Const("True",_) => []
| Const("False",_) => []
| _ => [r])
| _ => [r];
fun mk_eq r = case concl_of r of
Const("==",_)$_$_ => r
| _$(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;
fun mk_rews thm = map mk_eq (atomize(gen_all thm));
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 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 & 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",
"(P|Q --> R) = ((P-->R)&(Q-->R))" ];
val meta_obj_reflection = prove_goal HOL.thy "x==y ==> x=y"
(fn [prem] => [rewtac prem, rtac refl 1]);
in
val conj_assoc = prover "((P&Q)&R) = (P&(Q&R))";
val conj_commute = prover "(P&Q) = (Q&P)";
val conj_left_commute = prover "(P&(Q&R)) = (Q&(P&R))";
val conj_comms = [conj_commute, conj_left_commute];
val if_True = prove_goalw HOL.thy [if_def] "if(True,x,y) = x"
(fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
val if_False = prove_goalw HOL.thy [if_def] "if(False,x,y) = y"
(fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
val if_P = prove_goal HOL.thy "P ==> if(P,x,y) = x"
(fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
val if_not_P = prove_goal HOL.thy "~P ==> if(P,x,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,x,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) ]);
infix addcongs;
fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection]);
val HOL_ss = empty_ss
setmksimps mk_rews
setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac
ORELSE' etac FalseE)
setsubgoaler asm_simp_tac
addsimps ([if_True, if_False, o_apply, conj_assoc] @ simp_thms)
addcongs [imp_cong];
fun split_tac splits =
mk_case_split_tac (meta_obj_reflection RS iffD2) (map mk_eq splits);
(* 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! *)
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);
(** '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,x,y) = if(c,u,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,x,y) = if(c,x,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;