--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/simpdata.ML Fri Mar 03 12:02:25 1995 +0100
@@ -0,0 +1,163 @@
+(* 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 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 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 --> 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)",
+ "(P|Q --> R) = ((P-->R)&(Q-->R))" ];
+
+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 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) ]);
+
+val if_bool_eq = prove_goal HOL.thy "if P Q R = ((P-->Q) & (~P-->R))"
+ (fn _ => [rtac expand_if 1]);
+
+infix addcongs;
+fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection]);
+
+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 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, conj_assoc] @ simp_thms)
+ addcongs [imp_cong];
+
+fun split_tac splits =
+ mk_case_split_tac (meta_eq_to_obj_eq RS iffD2) (map mk_meta_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!
+ 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);
+
+(** '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;
+
+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 "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
+prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";