(* Title: HOL/HOL.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68
*)
(** Equality **)
section "=";
qed_goal "sym" HOL.thy "s=t ==> t=s"
(fn prems => [cut_facts_tac prems 1, etac subst 1, rtac refl 1]);
(*calling "standard" reduces maxidx to 0*)
bind_thm ("ssubst", (sym RS subst));
qed_goal "trans" HOL.thy "[| r=s; s=t |] ==> r=t"
(fn prems =>
[rtac subst 1, resolve_tac prems 1, resolve_tac prems 1]);
val prems = goal thy "(A == B) ==> A = B";
by (rewrite_goals_tac prems);
by (rtac refl 1);
qed "def_imp_eq";
(*Useful with eresolve_tac for proving equalties from known equalities.
a = b
| |
c = d *)
qed_goal "box_equals" HOL.thy
"[| a=b; a=c; b=d |] ==> c=d"
(fn prems=>
[ (rtac trans 1),
(rtac trans 1),
(rtac sym 1),
(REPEAT (resolve_tac prems 1)) ]);
(** Congruence rules for meta-application **)
section "Congruence";
(*similar to AP_THM in Gordon's HOL*)
qed_goal "fun_cong" HOL.thy "(f::'a=>'b) = g ==> f(x)=g(x)"
(fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
(*similar to AP_TERM in Gordon's HOL and FOL's subst_context*)
qed_goal "arg_cong" HOL.thy "x=y ==> f(x)=f(y)"
(fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
qed_goal "cong" HOL.thy
"[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
(fn [prem1,prem2] =>
[rtac (prem1 RS subst) 1, rtac (prem2 RS subst) 1, rtac refl 1]);
(** Equality of booleans -- iff **)
section "iff";
qed_goal "iffI" HOL.thy
"[| P ==> Q; Q ==> P |] ==> P=Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);
qed_goal "iffD2" HOL.thy "[| P=Q; Q |] ==> P"
(fn prems =>
[rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
qed_goal "rev_iffD2" HOL.thy "!!P. [| Q; P=Q |] ==> P"
(fn _ => [etac iffD2 1, assume_tac 1]);
bind_thm ("iffD1", sym RS iffD2);
bind_thm ("rev_iffD1", sym RSN (2, rev_iffD2));
qed_goal "iffE" HOL.thy
"[| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R"
(fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2, p1 RS iffD1, p2, impI])1)]);
(** True **)
section "True";
qed_goalw "TrueI" HOL.thy [True_def] "True"
(fn _ => [rtac refl 1]);
qed_goal "eqTrueI" HOL.thy "P ==> P=True"
(fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
qed_goal "eqTrueE" HOL.thy "P=True ==> P"
(fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
(** Universal quantifier **)
section "!";
qed_goalw "allI" HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
(fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
qed_goalw "spec" HOL.thy [All_def] "! x::'a. P(x) ==> P(x)"
(fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
qed_goal "allE" HOL.thy "[| !x. P(x); P(x) ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
qed_goal "all_dupE" HOL.thy
"[| ! x. P(x); [| P(x); ! x. P(x) |] ==> R |] ==> R"
(fn prems =>
[ (REPEAT (resolve_tac (prems @ (prems RL [spec])) 1)) ]);
(** False ** Depends upon spec; it is impossible to do propositional logic
before quantifiers! **)
section "False";
qed_goalw "FalseE" HOL.thy [False_def] "False ==> P"
(fn [major] => [rtac (major RS spec) 1]);
qed_goal "False_neq_True" HOL.thy "False=True ==> P"
(fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
(** Negation **)
section "~";
qed_goalw "notI" HOL.thy [not_def] "(P ==> False) ==> ~P"
(fn prems=> [rtac impI 1, eresolve_tac prems 1]);
qed_goal "False_not_True" HOL.thy "False ~= True"
(K [rtac notI 1, etac False_neq_True 1]);
qed_goal "True_not_False" HOL.thy "True ~= False"
(K [rtac notI 1, dtac sym 1, etac False_neq_True 1]);
qed_goalw "notE" HOL.thy [not_def] "[| ~P; P |] ==> R"
(fn prems => [rtac (prems MRS mp RS FalseE) 1]);
bind_thm ("classical2", notE RS notI);
qed_goal "rev_notE" HOL.thy "!!P R. [| P; ~P |] ==> R"
(fn _ => [REPEAT (ares_tac [notE] 1)]);
(** Implication **)
section "-->";
qed_goal "impE" HOL.thy "[| P-->Q; P; Q ==> R |] ==> R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
(* Reduces Q to P-->Q, allowing substitution in P. *)
qed_goal "rev_mp" HOL.thy "[| P; P --> Q |] ==> Q"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
qed_goal "contrapos" HOL.thy "[| ~Q; P==>Q |] ==> ~P"
(fn [major,minor]=>
[ (rtac (major RS notE RS notI) 1),
(etac minor 1) ]);
qed_goal "rev_contrapos" HOL.thy "[| P==>Q; ~Q |] ==> ~P"
(fn [major,minor]=>
[ (rtac (minor RS contrapos) 1), (etac major 1) ]);
(* ~(?t = ?s) ==> ~(?s = ?t) *)
bind_thm("not_sym", sym COMP rev_contrapos);
(** Existential quantifier **)
section "?";
qed_goalw "exI" HOL.thy [Ex_def] "P x ==> ? x::'a. P x"
(fn prems => [rtac selectI 1, resolve_tac prems 1]);
qed_goalw "exE" HOL.thy [Ex_def]
"[| ? x::'a. P(x); !!x. P(x) ==> Q |] ==> Q"
(fn prems => [REPEAT(resolve_tac prems 1)]);
(** Conjunction **)
section "&";
qed_goalw "conjI" HOL.thy [and_def] "[| P; Q |] ==> P&Q"
(fn prems =>
[REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
qed_goalw "conjunct1" HOL.thy [and_def] "[| P & Q |] ==> P"
(fn prems =>
[resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
qed_goalw "conjunct2" HOL.thy [and_def] "[| P & Q |] ==> Q"
(fn prems =>
[resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
qed_goal "conjE" HOL.thy "[| P&Q; [| P; Q |] ==> R |] ==> R"
(fn prems =>
[cut_facts_tac prems 1, resolve_tac prems 1,
etac conjunct1 1, etac conjunct2 1]);
qed_goal "context_conjI" HOL.thy "[| P; P ==> Q |] ==> P & Q"
(fn prems => [REPEAT(resolve_tac (conjI::prems) 1)]);
(** Disjunction *)
section "|";
qed_goalw "disjI1" HOL.thy [or_def] "P ==> P|Q"
(fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
qed_goalw "disjI2" HOL.thy [or_def] "Q ==> P|Q"
(fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
qed_goalw "disjE" HOL.thy [or_def] "[| P | Q; P ==> R; Q ==> R |] ==> R"
(fn [a1,a2,a3] =>
[rtac (mp RS mp) 1, rtac spec 1, rtac a1 1,
rtac (a2 RS impI) 1, assume_tac 1, rtac (a3 RS impI) 1, assume_tac 1]);
(** CCONTR -- classical logic **)
section "classical logic";
qed_goalw "classical" HOL.thy [not_def] "(~P ==> P) ==> P"
(fn [prem] =>
[rtac (True_or_False RS (disjE RS eqTrueE)) 1, assume_tac 1,
rtac (impI RS prem RS eqTrueI) 1,
etac subst 1, assume_tac 1]);
val ccontr = FalseE RS classical;
(*Double negation law*)
qed_goal "notnotD" HOL.thy "~~P ==> P"
(fn [major]=>
[ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
qed_goal "contrapos2" HOL.thy "[| Q; ~ P ==> ~ Q |] ==> P" (fn [p1,p2] => [
rtac classical 1,
dtac p2 1,
etac notE 1,
rtac p1 1]);
qed_goal "swap2" HOL.thy "[| P; Q ==> ~ P |] ==> ~ Q" (fn [p1,p2] => [
rtac notI 1,
dtac p2 1,
etac notE 1,
rtac p1 1]);
(** Unique existence **)
section "?!";
qed_goalw "ex1I" HOL.thy [Ex1_def]
"[| P(a); !!x. P(x) ==> x=a |] ==> ?! x. P(x)"
(fn prems =>
[REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)]);
(*Sometimes easier to use: the premises have no shared variables. Safe!*)
qed_goal "ex_ex1I" HOL.thy
"[| ? x. P(x); !!x y. [| P(x); P(y) |] ==> x=y |] ==> ?! x. P(x)"
(fn [ex,eq] => [ (rtac (ex RS exE) 1),
(REPEAT (ares_tac [ex1I,eq] 1)) ]);
qed_goalw "ex1E" HOL.thy [Ex1_def]
"[| ?! x. P(x); !!x. [| P(x); ! y. P(y) --> y=x |] ==> R |] ==> R"
(fn major::prems =>
[rtac (major RS exE) 1, REPEAT (etac conjE 1 ORELSE ares_tac prems 1)]);
Goal "?! x. P x ==> ? x. P x";
by (etac ex1E 1);
by (rtac exI 1);
by (assume_tac 1);
qed "ex1_implies_ex";
(** Select: Hilbert's Epsilon-operator **)
section "@";
(*Easier to apply than selectI: conclusion has only one occurrence of P*)
qed_goal "selectI2" HOL.thy
"[| P a; !!x. P x ==> Q x |] ==> Q (@x. P x)"
(fn prems => [ resolve_tac prems 1,
rtac selectI 1,
resolve_tac prems 1 ]);
(*Easier to apply than selectI2 if witness ?a comes from an EX-formula*)
qed_goal "selectI2EX" HOL.thy
"[| ? a. P a; !!x. P x ==> Q x |] ==> Q (Eps P)"
(fn [major,minor] => [rtac (major RS exE) 1, etac selectI2 1, etac minor 1]);
qed_goal "select_equality" HOL.thy
"[| P a; !!x. P x ==> x=a |] ==> (@x. P x) = a"
(fn prems => [ rtac selectI2 1,
REPEAT (ares_tac prems 1) ]);
qed_goalw "select1_equality" HOL.thy [Ex1_def]
"!!P. [| ?!x. P x; P a |] ==> (@x. P x) = a" (K [
rtac select_equality 1, atac 1,
etac exE 1, etac conjE 1,
rtac allE 1, atac 1,
etac impE 1, atac 1, etac ssubst 1,
etac allE 1, etac impE 1, atac 1, etac ssubst 1,
rtac refl 1]);
qed_goal "select_eq_Ex" HOL.thy "P (@ x. P x) = (? x. P x)" (K [
rtac iffI 1,
etac exI 1,
etac exE 1,
etac selectI 1]);
qed_goal "Eps_eq" HOL.thy "(@y. y=x) = x" (K [
rtac select_equality 1,
rtac refl 1,
atac 1]);
qed_goal "Eps_sym_eq" HOL.thy "(Eps (op = x)) = x" (K [
rtac select_equality 1,
rtac refl 1,
etac sym 1]);
(** Classical intro rules for disjunction and existential quantifiers *)
section "classical intro rules";
qed_goal "disjCI" HOL.thy "(~Q ==> P) ==> P|Q"
(fn prems=>
[ (rtac classical 1),
(REPEAT (ares_tac (prems@[disjI1,notI]) 1)),
(REPEAT (ares_tac (prems@[disjI2,notE]) 1)) ]);
qed_goal "excluded_middle" HOL.thy "~P | P"
(fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
(*For disjunctive case analysis*)
fun excluded_middle_tac sP =
res_inst_tac [("Q",sP)] (excluded_middle RS disjE);
(*Classical implies (-->) elimination. *)
qed_goal "impCE" HOL.thy "[| P-->Q; ~P ==> R; Q ==> R |] ==> R"
(fn major::prems=>
[ rtac (excluded_middle RS disjE) 1,
REPEAT (DEPTH_SOLVE_1 (ares_tac (prems @ [major RS mp]) 1))]);
(*This version of --> elimination works on Q before P. It works best for
those cases in which P holds "almost everywhere". Can't install as
default: would break old proofs.*)
qed_goal "impCE'" thy
"[| P-->Q; Q ==> R; ~P ==> R |] ==> R"
(fn major::prems=>
[ (resolve_tac [excluded_middle RS disjE] 1),
(DEPTH_SOLVE (ares_tac (prems@[major RS mp]) 1)) ]);
(*Classical <-> elimination. *)
qed_goal "iffCE" HOL.thy
"[| P=Q; [| P; Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R"
(fn major::prems =>
[ (rtac (major RS iffE) 1),
(REPEAT (DEPTH_SOLVE_1
(eresolve_tac ([asm_rl,impCE,notE]@prems) 1))) ]);
qed_goal "exCI" HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x. P(x)"
(fn prems=>
[ (rtac ccontr 1),
(REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ]);
(* case distinction *)
qed_goal "case_split_thm" HOL.thy "[| P ==> Q; ~P ==> Q |] ==> Q"
(fn [p1,p2] => [rtac (excluded_middle RS disjE) 1,
etac p2 1, etac p1 1]);
fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
(** Standard abbreviations **)
(*Apply an equality or definition ONCE.
Fails unless the substitution has an effect*)
fun stac th =
let val th' = th RS def_imp_eq handle _ => th
in CHANGED_GOAL (rtac (th' RS ssubst))
end;
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
(** strip ! and --> from proved goal while preserving !-bound var names **)
local
(* Use XXX to avoid forall_intr failing because of duplicate variable name *)
val myspec = read_instantiate [("P","?XXX")] spec;
val _ $ (_ $ (vx as Var(_,vxT))) = concl_of myspec;
val cvx = cterm_of (#sign(rep_thm myspec)) vx;
val aspec = forall_intr cvx myspec;
in
fun RSspec th =
(case concl_of th of
_ $ (Const("All",_) $ Abs(a,_,_)) =>
let val ca = cterm_of (#sign(rep_thm th)) (Var((a,0),vxT))
in th RS forall_elim ca aspec end
| _ => raise THM("RSspec",0,[th]));
fun RSmp th =
(case concl_of th of
_ $ (Const("op -->",_)$_$_) => th RS mp
| _ => raise THM("RSmp",0,[th]));
fun normalize_thm funs =
let fun trans [] th = th
| trans (f::fs) th = (trans funs (f th)) handle THM _ => trans fs th
in zero_var_indexes o trans funs end;
fun qed_spec_mp name =
let val thm = normalize_thm [RSspec,RSmp] (result())
in ThmDatabase.ml_store_thm(name, thm) end;
fun qed_goal_spec_mp name thy s p =
bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goal thy s p));
fun qed_goalw_spec_mp name thy defs s p =
bind_thm (name, normalize_thm [RSspec,RSmp] (prove_goalw thy defs s p));
end;
(* attributes *)
local
fun gen_rulify x = Attrib.no_args (Drule.rule_attribute (K (normalize_thm [RSspec, RSmp]))) x;
in
val hol_setup =
[Attrib.add_attributes
[("rulify", (gen_rulify, gen_rulify), "put theorem into standard rule form")]];
end;