(* 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;
signature HOL_LEMMAS =
sig
val allE: thm
val all_dupE: thm
val allI: thm
val arg_cong: thm
val fun_cong: thm
val box_equals: thm
val case_tac: string -> int -> tactic
val ccontr: thm
val classical: thm
val cong: thm
val conjunct1: thm
val conjunct2: thm
val conjE: thm
val conjI: thm
val contrapos: thm
val disjCI: thm
val disjE: thm
val disjI1: thm
val disjI2: thm
val eqTrueI: thm
val eqTrueE: thm
val ex1E: thm
val ex1I: thm
val exCI: thm
val exI: thm
val exE: thm
val excluded_middle: thm
val FalseE: thm
val False_neq_True: thm
val iffCE : thm
val iffD1: thm
val iffD2: thm
val iffE: thm
val iffI: thm
val impCE: thm
val impE: thm
val not_sym: thm
val notE: thm
val notI: thm
val notnotD : thm
val rev_mp: thm
val select_equality: thm
val spec: thm
val sstac: thm list -> int -> tactic
val ssubst: thm
val stac: thm -> int -> tactic
val strip_tac: int -> tactic
val swap: thm
val sym: thm
val trans: thm
val TrueI: thm
end;
structure HOL_Lemmas : HOL_LEMMAS =
struct
(** Equality **)
val sym = prove_goal 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*)
val ssubst = standard (sym RS subst);
val trans = prove_goal 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 *)
val box_equals = prove_goal 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 **)
(*similar to AP_THM in Gordon's HOL*)
val fun_cong = prove_goal 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*)
val arg_cong = prove_goal HOL.thy "x=y ==> f(x)=f(y)"
(fn [prem] => [rtac (prem RS subst) 1, rtac refl 1]);
val cong = prove_goal 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 **)
val iffI = prove_goal HOL.thy
"[| P ==> Q; Q ==> P |] ==> P=Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI, iff RS mp RS mp]) 1)) ]);
val iffD2 = prove_goal HOL.thy "[| P=Q; Q |] ==> P"
(fn prems =>
[rtac ssubst 1, resolve_tac prems 1, resolve_tac prems 1]);
val iffD1 = sym RS iffD2;
val iffE = prove_goal 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 **)
val TrueI = prove_goalw HOL.thy [True_def] "True"
(fn _ => [rtac refl 1]);
val eqTrueI = prove_goal HOL.thy "P ==> P=True"
(fn prems => [REPEAT(resolve_tac ([iffI,TrueI]@prems) 1)]);
val eqTrueE = prove_goal HOL.thy "P=True ==> P"
(fn prems => [REPEAT(resolve_tac (prems@[TrueI,iffD2]) 1)]);
(** Universal quantifier **)
val allI = prove_goalw HOL.thy [All_def] "(!!x::'a. P(x)) ==> !x. P(x)"
(fn prems => [resolve_tac (prems RL [eqTrueI RS ext]) 1]);
val spec = prove_goalw HOL.thy [All_def] "! x::'a.P(x) ==> P(x)"
(fn prems => [rtac eqTrueE 1, resolve_tac (prems RL [fun_cong]) 1]);
val allE = prove_goal HOL.thy "[| !x.P(x); P(x) ==> R |] ==> R"
(fn major::prems=>
[ (REPEAT (resolve_tac (prems @ [major RS spec]) 1)) ]);
val all_dupE = prove_goal 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! **)
val FalseE = prove_goal HOL.thy "False ==> P"
(fn prems => [rtac spec 1, fold_tac [False_def], resolve_tac prems 1]);
val False_neq_True = prove_goal HOL.thy "False=True ==> P"
(fn [prem] => [rtac (prem RS eqTrueE RS FalseE) 1]);
(** Negation **)
val notI = prove_goalw HOL.thy [not_def] "(P ==> False) ==> ~P"
(fn prems=> [rtac impI 1, eresolve_tac prems 1]);
val notE = prove_goalw HOL.thy [not_def] "[| ~P; P |] ==> R"
(fn prems => [rtac (mp RS FalseE) 1, REPEAT(resolve_tac prems 1)]);
(** Implication **)
val impE = prove_goal 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. *)
val rev_mp = prove_goal HOL.thy "[| P; P --> Q |] ==> Q"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
val contrapos = prove_goal HOL.thy "[| ~Q; P==>Q |] ==> ~P"
(fn [major,minor]=>
[ (rtac (major RS notE RS notI) 1),
(etac minor 1) ]);
(* ~(?t = ?s) ==> ~(?s = ?t) *)
val [not_sym] = compose(sym,2,contrapos);
(** Existential quantifier **)
val exI = prove_goalw HOL.thy [Ex_def] "P(x) ==> ? x::'a.P(x)"
(fn prems => [rtac selectI 1, resolve_tac prems 1]);
val exE = prove_goalw HOL.thy [Ex_def]
"[| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q"
(fn prems => [REPEAT(resolve_tac prems 1)]);
(** Conjunction **)
val conjI = prove_goalw HOL.thy [and_def] "[| P; Q |] ==> P&Q"
(fn prems =>
[REPEAT (resolve_tac (prems@[allI,impI]) 1 ORELSE etac (mp RS mp) 1)]);
val conjunct1 = prove_goalw HOL.thy [and_def] "[| P & Q |] ==> P"
(fn prems =>
[resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
val conjunct2 = prove_goalw HOL.thy [and_def] "[| P & Q |] ==> Q"
(fn prems =>
[resolve_tac (prems RL [spec] RL [mp]) 1, REPEAT(ares_tac [impI] 1)]);
val conjE = prove_goal 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 *)
val disjI1 = prove_goalw HOL.thy [or_def] "P ==> P|Q"
(fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
val disjI2 = prove_goalw HOL.thy [or_def] "Q ==> P|Q"
(fn [prem] => [REPEAT(ares_tac [allI,impI, prem RSN (2,mp)] 1)]);
val disjE = prove_goalw 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, atac 1, rtac (a3 RS impI) 1, atac 1]);
(** CCONTR -- classical logic **)
val ccontr = prove_goal HOL.thy "(~P ==> False) ==> P"
(fn prems =>
[rtac (True_or_False RS (disjE RS eqTrueE)) 1, atac 1,
rtac spec 1, fold_tac [False_def], resolve_tac prems 1,
rtac ssubst 1, atac 1, rewtac not_def,
REPEAT (ares_tac [impI] 1) ]);
val ccontr = prove_goalw HOL.thy [not_def] "(~P ==> False) ==> P"
(fn prems =>
[rtac (True_or_False RS (disjE RS eqTrueE)) 1, atac 1,
rtac spec 1, fold_tac [False_def], resolve_tac prems 1,
rtac ssubst 1, atac 1, REPEAT (ares_tac [impI] 1) ]);
val classical = prove_goal HOL.thy "(~P ==> P) ==> P"
(fn prems => [rtac ccontr 1, REPEAT (ares_tac (prems@[notE]) 1)]);
(*Double negation law*)
val notnotD = prove_goal HOL.thy "~~P ==> P"
(fn [major]=>
[ (rtac classical 1), (eresolve_tac [major RS notE] 1) ]);
(** Unique existence **)
val ex1I = prove_goalw 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)]);
val ex1E = prove_goalw 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 **)
val select_equality = prove_goal HOL.thy
"[| P(a); !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a"
(fn prems => [ resolve_tac prems 1,
rtac selectI 1,
resolve_tac prems 1 ]);
(** Classical intro rules for disjunction and existential quantifiers *)
val disjCI = prove_goal 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)) ]);
val excluded_middle = prove_goal HOL.thy "~P | P"
(fn _ => [ (REPEAT (ares_tac [disjCI] 1)) ]);
(*Classical implies (-->) elimination. *)
val impCE = prove_goal 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. *)
val iffCE = prove_goal 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))) ]);
val exCI = prove_goal HOL.thy "(! x. ~P(x) ==> P(a)) ==> ? x.P(x)"
(fn prems=>
[ (rtac ccontr 1),
(REPEAT (ares_tac (prems@[exI,allI,notI,notE]) 1)) ]);
(*Required by the "classical" module*)
val swap = prove_goal HOL.thy "~P ==> (~Q ==> P) ==> Q"
(fn major::prems=>
[ rtac ccontr 1, rtac (major RS notE) 1, REPEAT (ares_tac prems 1)]);
(* case distinction *)
val case_split_thm = prove_goal 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 = rtac(th RS ssubst);
fun sstac ths = EVERY' (map stac ths);
fun strip_tac i = REPEAT(resolve_tac [impI,allI] i);
end;
open HOL_Lemmas;