src/HOL/simpdata.ML
 author nipkow Thu Apr 11 08:30:25 1996 +0200 (1996-04-11) changeset 1655 5be64540f275 parent 1548 afe750876848 child 1660 8cb42cd97579 permissions -rw-r--r--
1 (*  Title:      HOL/simpdata.ML
2     ID:         \$Id\$
3     Author:     Tobias Nipkow
4     Copyright   1991  University of Cambridge
6 Instantiation of the generic simplifier
7 *)
9 open Simplifier;
11 local
13 fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
15 val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
16 val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
18 val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
19 val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
21 fun atomize pairs =
22   let fun atoms th =
23         (case concl_of th of
24            Const("Trueprop",_) \$ p =>
26                 Const(a,_) =>
27                   (case assoc(pairs,a) of
28                      Some(rls) => flat (map atoms ([th] RL rls))
29                    | None => [th])
30               | _ => [th])
31          | _ => [th])
32   in atoms end;
34 fun mk_meta_eq r = case concl_of r of
35         Const("==",_)\$_\$_ => r
36     |   _\$(Const("op =",_)\$_\$_) => r RS eq_reflection
37     |   _\$(Const("not",_)\$_) => r RS not_P_imp_P_eq_False
38     |   _ => r RS P_imp_P_eq_True;
39 (* last 2 lines requires all formulae to be of the from Trueprop(.) *)
41 fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
43 val imp_cong = impI RSN
44     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
45         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
47 val o_apply = prove_goalw HOL.thy [o_def] "(f o g)(x) = f(g(x))"
48  (fn _ => [rtac refl 1]);
50 val simp_thms = map prover
51  [ "(x=x) = True",
52    "(~True) = False", "(~False) = True", "(~ ~ P) = P",
53    "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
54    "(True=P) = P", "(P=True) = P",
55    "(True --> P) = P", "(False --> P) = True",
56    "(P --> True) = True", "(P --> P) = True",
57    "(P --> False) = (~P)", "(P --> ~P) = (~P)",
58    "(P & True) = P", "(True & P) = P",
59    "(P & False) = False", "(False & P) = False", "(P & P) = P",
60    "(P | True) = True", "(True | P) = True",
61    "(P | False) = P", "(False | P) = P", "(P | P) = P",
62    "(!x.P) = P", "(? x.P) = P", "? x. x=t", "(? x. x=t & P(x)) = P(t)",
63    "(P|Q --> R) = ((P-->R)&(Q-->R))" ];
65 in
67 val meta_eq_to_obj_eq = prove_goal HOL.thy "x==y ==> x=y"
68   (fn [prem] => [rewtac prem, rtac refl 1]);
70 val eq_sym_conv = prover "(x=y) = (y=x)";
72 val conj_assoc = prover "((P&Q)&R) = (P&(Q&R))";
74 val if_True = prove_goalw HOL.thy [if_def] "(if True then x else y) = x"
75  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
77 val if_False = prove_goalw HOL.thy [if_def] "(if False then x else y) = y"
78  (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
80 val if_P = prove_goal HOL.thy "P ==> (if P then x else y) = x"
81  (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
83 val if_not_P = prove_goal HOL.thy "~P ==> (if P then x else y) = y"
84  (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
86 val expand_if = prove_goal HOL.thy
87     "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
88  (fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
89          rtac (if_P RS ssubst) 2,
90          rtac (if_not_P RS ssubst) 1,
91          REPEAT(fast_tac HOL_cs 1) ]);
93 val if_bool_eq = prove_goal HOL.thy
94                    "(if P then Q else R) = ((P-->Q) & (~P-->R))"
95                    (fn _ => [rtac expand_if 1]);
103 (*Add a simpset to a classical set!*)
107 val mksimps_pairs =
108   [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
109    ("All", [spec]), ("True", []), ("False", []),
110    ("If", [if_bool_eq RS iffD1])];
112 fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
114 val HOL_ss = empty_ss
115       setmksimps (mksimps mksimps_pairs)
116       setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac
117                              ORELSE' etac FalseE)
118       setsubgoaler asm_simp_tac
119       addsimps ([if_True, if_False, o_apply, conj_assoc] @ simp_thms)
122 local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
123 in
124 fun split_tac splits = mktac (map mk_meta_eq splits)
125 end;
128 (* eliminiation of existential quantifiers in assumptions *)
130 val ex_all_equiv =
131   let val lemma1 = prove_goal HOL.thy
132         "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
133         (fn prems => [resolve_tac prems 1, etac exI 1]);
134       val lemma2 = prove_goalw HOL.thy [Ex_def]
135         "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
136         (fn prems => [REPEAT(resolve_tac prems 1)])
137   in equal_intr lemma1 lemma2 end;
139 (* '&' congruence rule: not included by default!
140    May slow rewrite proofs down by as much as 50% *)
142 val conj_cong = impI RSN
143     (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
144         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
146 val rev_conj_cong = impI RSN
147     (2, prove_goal HOL.thy "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
148         (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
150 (** 'if' congruence rules: neither included by default! *)
152 (*Simplifies x assuming c and y assuming ~c*)
153 val if_cong = prove_goal HOL.thy
154   "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
155 \  (if b then x else y) = (if c then u else v)"
156   (fn rew::prems =>
157    [stac rew 1, stac expand_if 1, stac expand_if 1,
158     fast_tac (HOL_cs addDs prems) 1]);
160 (*Prevents simplification of x and y: much faster*)
161 val if_weak_cong = prove_goal HOL.thy
162   "b=c ==> (if b then x else y) = (if c then x else y)"
163   (fn [prem] => [rtac (prem RS arg_cong) 1]);
165 (*Prevents simplification of t: much faster*)
166 val let_weak_cong = prove_goal HOL.thy
167   "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
168   (fn [prem] => [rtac (prem RS arg_cong) 1]);
170 end;
172 fun prove nm thm  = qed_goal nm HOL.thy thm (fn _ => [fast_tac HOL_cs 1]);
174 prove "conj_commute" "(P&Q) = (Q&P)";
175 prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
176 val conj_comms = [conj_commute, conj_left_commute];
178 prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
179 prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
181 prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
182 prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
184 prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
185 prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
187 qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
188   (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
190 qed_goal "if_distrib" HOL.thy
191   "f(if c then x else y) = (if c then f x else f y)"
192   (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
194 qed_goalw "o_assoc" HOL.thy [o_def] "(f o g) o h = (f o g o h)"
195   (fn _=>[rtac ext 1, rtac refl 1]);