(* Title: HOL/HOL.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
For HOL.thy
Derived rules from Appendix of Mike Gordons HOL Report, Cambridge TR 68
*)
open HOL;
(** 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]);
(*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]);
val iffD1 = sym RS 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_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]);
(** 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)]);
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)]);
(** 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 ]);
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_goal "select_eq_Ex" HOL.thy "P (@ x. P x) = (? x. P x)" (fn prems => [
rtac iffI 1,
etac exI 1,
etac exE 1,
etac selectI 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))]);
(*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] => [cut_facts_tac [excluded_middle] 1, etac disjE 1,
etac p2 1, etac p1 1]);
fun case_tac a = res_inst_tac [("P",a)] case_split_thm;
(** Standard abbreviations **)
fun stac th = CHANGED o rtac (th RS ssubst);
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
(** strip 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 trans funs end;
fun qed_spec_mp name =
let val thm = normalize_thm [RSspec,RSmp] (result())
in bind_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;
(*Thus, assignments to the references claset and simpset are recorded
with theory "HOL". These files cannot be loaded directly in ROOT.ML.*)
use "hologic.ML";
use "cladata.ML";
use "simpdata.ML";