src/HOL/simpdata.ML
author nipkow
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Added ex_imp
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(*  Title:      HOL/simpdata.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1991  University of Cambridge
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Instantiation of the generic simplifier
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*)
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open Simplifier;
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local
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fun prover s = prove_goal HOL.thy s (fn _ => [fast_tac HOL_cs 1]);
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val P_imp_P_iff_True = prover "P --> (P = True)" RS mp;
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val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection;
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val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp;
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val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection;
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fun atomize pairs =
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  let fun atoms th =
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        (case concl_of th of
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           Const("Trueprop",_) $ p =>
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             (case head_of p of
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                Const(a,_) =>
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                  (case assoc(pairs,a) of
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                     Some(rls) => flat (map atoms ([th] RL rls))
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                   | None => [th])
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              | _ => [th])
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         | _ => [th])
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  in atoms end;
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fun mk_meta_eq r = case concl_of r of
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        Const("==",_)$_$_ => r
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    |   _$(Const("op =",_)$_$_) => r RS eq_reflection
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    |   _$(Const("not",_)$_) => r RS not_P_imp_P_eq_False
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    |   _ => r RS P_imp_P_eq_True;
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(* last 2 lines requires all formulae to be of the from Trueprop(.) *)
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fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
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val imp_cong = impI RSN
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    (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
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        (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
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val o_apply = prove_goalw HOL.thy [o_def] "(f o g)(x) = f(g(x))"
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 (fn _ => [rtac refl 1]);
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val simp_thms = map prover
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 [ "(x=x) = True",
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   "(~True) = False", "(~False) = True", "(~ ~ P) = P",
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   "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))",
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   "(True=P) = P", "(P=True) = P",
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   "(True --> P) = P", "(False --> P) = True", 
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   "(P --> True) = True", "(P --> P) = True",
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   "(P --> False) = (~P)", "(P --> ~P) = (~P)",
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   "(P & True) = P", "(True & P) = P", 
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   "(P & False) = False", "(False & P) = False", "(P & P) = P",
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   "(P | True) = True", "(True | P) = True", 
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   "(P | False) = P", "(False | P) = P", "(P | P) = P",
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   "(!x.P) = P", "(? x.P) = P", "? x. x=t", "(? x. x=t & P(x)) = P(t)",
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   "(P|Q --> R) = ((P-->R)&(Q-->R))" ];
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in
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val meta_eq_to_obj_eq = prove_goal HOL.thy "x==y ==> x=y"
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  (fn [prem] => [rewtac prem, rtac refl 1]);
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val eq_sym_conv = prover "(x=y) = (y=x)";
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val conj_assoc = prover "((P&Q)&R) = (P&(Q&R))";
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val if_True = prove_goalw HOL.thy [if_def] "(if True then x else y) = x"
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 (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
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val if_False = prove_goalw HOL.thy [if_def] "(if False then x else y) = y"
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 (fn _=>[fast_tac (HOL_cs addIs [select_equality]) 1]);
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val if_P = prove_goal HOL.thy "P ==> (if P then x else y) = x"
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 (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]);
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val if_not_P = prove_goal HOL.thy "~P ==> (if P then x else y) = y"
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 (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]);
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val expand_if = prove_goal HOL.thy
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    "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))"
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 (fn _=> [ (res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1),
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         rtac (if_P RS ssubst) 2,
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         rtac (if_not_P RS ssubst) 1,
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         REPEAT(fast_tac HOL_cs 1) ]);
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val if_bool_eq = prove_goal HOL.thy
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                   "(if P then Q else R) = ((P-->Q) & (~P-->R))"
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                   (fn _ => [rtac expand_if 1]);
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(*Add congruence rules for = (instead of ==) *)
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infix 4 addcongs;
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fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection]);
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fun Addcongs congs = (simpset := !simpset addcongs congs);
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(*Add a simpset to a classical set!*)
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infix 4 addss;
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fun cs addss ss = cs addbefore asm_full_simp_tac ss 1;
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val mksimps_pairs =
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  [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
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   ("All", [spec]), ("True", []), ("False", []),
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   ("If", [if_bool_eq RS iffD1])];
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fun mksimps pairs = map mk_meta_eq o atomize pairs o gen_all;
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val HOL_ss = empty_ss
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      setmksimps (mksimps mksimps_pairs)
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      setsolver (fn prems => resolve_tac (TrueI::refl::prems) ORELSE' atac
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                             ORELSE' etac FalseE)
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      setsubgoaler asm_simp_tac
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      addsimps ([if_True, if_False, o_apply, conj_assoc] @ simp_thms)
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      addcongs [imp_cong];
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local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2)
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in
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fun split_tac splits = mktac (map mk_meta_eq splits)
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end;
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local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2)
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in
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fun split_inside_tac splits = mktac (map mk_meta_eq splits)
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end;
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(* eliminiation of existential quantifiers in assumptions *)
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val ex_all_equiv =
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  let val lemma1 = prove_goal HOL.thy
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        "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)"
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        (fn prems => [resolve_tac prems 1, etac exI 1]);
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      val lemma2 = prove_goalw HOL.thy [Ex_def]
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        "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)"
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        (fn prems => [REPEAT(resolve_tac prems 1)])
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  in equal_intr lemma1 lemma2 end;
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(* '&' congruence rule: not included by default!
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   May slow rewrite proofs down by as much as 50% *)
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val conj_cong = impI RSN
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    (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))"
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        (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
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val rev_conj_cong = impI RSN
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    (2, prove_goal HOL.thy "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))"
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        (fn _=> [fast_tac HOL_cs 1]) RS mp RS mp);
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(** 'if' congruence rules: neither included by default! *)
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(*Simplifies x assuming c and y assuming ~c*)
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val if_cong = prove_goal HOL.thy
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  "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\
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\  (if b then x else y) = (if c then u else v)"
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  (fn rew::prems =>
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   [stac rew 1, stac expand_if 1, stac expand_if 1,
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    fast_tac (HOL_cs addDs prems) 1]);
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(*Prevents simplification of x and y: much faster*)
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val if_weak_cong = prove_goal HOL.thy
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  "b=c ==> (if b then x else y) = (if c then x else y)"
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  (fn [prem] => [rtac (prem RS arg_cong) 1]);
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(*Prevents simplification of t: much faster*)
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val let_weak_cong = prove_goal HOL.thy
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  "a = b ==> (let x=a in t(x)) = (let x=b in t(x))"
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  (fn [prem] => [rtac (prem RS arg_cong) 1]);
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end;
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fun prove nm thm  = qed_goal nm HOL.thy thm (fn _ => [fast_tac HOL_cs 1]);
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prove "conj_commute" "(P&Q) = (Q&P)";
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prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))";
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val conj_comms = [conj_commute, conj_left_commute];
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prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)";
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prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)";
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prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)";
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prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)";
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prove "not_all" "(~ (! x.P(x))) = (? x.~P(x))";
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prove "not_ex"  "(~ (? x.P(x))) = (! x.~P(x))";
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prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))";
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prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))";
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prove "ex_imp" "((? x. P x) --> Q) = (!x. P x --> Q)";
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qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
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  (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
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qed_goal "if_distrib" HOL.thy
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  "f(if c then x else y) = (if c then f x else f y)" 
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  (fn _ => [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]);
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qed_goalw "o_assoc" HOL.thy [o_def] "(f o g) o h = (f o g o h)"
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  (fn _=>[rtac ext 1, rtac refl 1]);