(* Title: FOLP/ifol.ML
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Tactics and lemmas for ifol.thy (intuitionistic first-order logic)
*)
open IFOLP;
signature IFOLP_LEMMAS =
sig
val allE: thm
val all_cong: thm
val all_dupE: thm
val all_impE: thm
val box_equals: thm
val conjE: thm
val conj_cong: thm
val conj_impE: thm
val contrapos: thm
val disj_cong: thm
val disj_impE: thm
val eq_cong: thm
val ex1I: thm
val ex1E: thm
val ex1_equalsE: thm
(* val ex1_cong: thm****)
val ex_cong: thm
val ex_impE: thm
val iffD1: thm
val iffD2: thm
val iffE: thm
val iffI: thm
val iff_cong: thm
val iff_impE: thm
val iff_refl: thm
val iff_sym: thm
val iff_trans: thm
val impE: thm
val imp_cong: thm
val imp_impE: thm
val mp_tac: int -> tactic
val notE: thm
val notI: thm
val not_cong: thm
val not_impE: thm
val not_sym: thm
val not_to_imp: thm
val pred1_cong: thm
val pred2_cong: thm
val pred3_cong: thm
val pred_congs: thm list
val refl: thm
val rev_mp: thm
val simp_equals: thm
val subst: thm
val ssubst: thm
val subst_context: thm
val subst_context2: thm
val subst_context3: thm
val sym: thm
val trans: thm
val TrueI: thm
val uniq_assume_tac: int -> tactic
val uniq_mp_tac: int -> tactic
end;
structure IFOLP_Lemmas : IFOLP_LEMMAS =
struct
val TrueI = TrueI;
(*** Sequent-style elimination rules for & --> and ALL ***)
val conjE = prove_goal IFOLP.thy
"[| p:P&Q; !!x y.[| x:P; y:Q |] ==> f(x,y):R |] ==> ?a:R"
(fn prems=>
[ (REPEAT (resolve_tac prems 1
ORELSE (resolve_tac [conjunct1, conjunct2] 1 THEN
resolve_tac prems 1))) ]);
val impE = prove_goal IFOLP.thy
"[| p:P-->Q; q:P; !!x.x:Q ==> r(x):R |] ==> ?p:R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
val allE = prove_goal IFOLP.thy
"[| p:ALL x.P(x); !!y.y:P(x) ==> q(y):R |] ==> ?p:R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
(*Duplicates the quantifier; for use with eresolve_tac*)
val all_dupE = prove_goal IFOLP.thy
"[| p:ALL x.P(x); !!y z.[| y:P(x); z:ALL x.P(x) |] ==> q(y,z):R \
\ |] ==> ?p:R"
(fn prems=> [ (REPEAT (resolve_tac (prems@[spec]) 1)) ]);
(*** Negation rules, which translate between ~P and P-->False ***)
val notI = prove_goalw IFOLP.thy [not_def] "(!!x.x:P ==> q(x):False) ==> ?p:~P"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI]) 1)) ]);
val notE = prove_goalw IFOLP.thy [not_def] "[| p:~P; q:P |] ==> ?p:R"
(fn prems=>
[ (resolve_tac [mp RS FalseE] 1),
(REPEAT (resolve_tac prems 1)) ]);
(*This is useful with the special implication rules for each kind of P. *)
val not_to_imp = prove_goal IFOLP.thy
"[| p:~P; !!x.x:(P-->False) ==> q(x):Q |] ==> ?p:Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[impI,notE]) 1)) ]);
(* For substitution int an assumption P, reduce Q to P-->Q, substitute into
this implication, then apply impI to move P back into the assumptions.
To specify P use something like
eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1 *)
val rev_mp = prove_goal IFOLP.thy "[| p:P; q:P --> Q |] ==> ?p:Q"
(fn prems=> [ (REPEAT (resolve_tac (prems@[mp]) 1)) ]);
(*Contrapositive of an inference rule*)
val contrapos = prove_goal IFOLP.thy "[| p:~Q; !!y.y:P==>q(y):Q |] ==> ?a:~P"
(fn [major,minor]=>
[ (rtac (major RS notE RS notI) 1),
(etac minor 1) ]);
(** Unique assumption tactic.
Ignores proof objects.
Fails unless one assumption is equal and exactly one is unifiable
**)
local
fun discard_proof (Const("Proof",_) $ P $ _) = P;
in
val uniq_assume_tac =
SUBGOAL
(fn (prem,i) =>
let val hyps = map discard_proof (Logic.strip_assums_hyp prem)
and concl = discard_proof (Logic.strip_assums_concl prem)
in
if exists (fn hyp => hyp aconv concl) hyps
then case distinct (filter (fn hyp=> could_unify(hyp,concl)) hyps) of
[_] => assume_tac i
| _ => no_tac
else no_tac
end);
end;
(*** Modus Ponens Tactics ***)
(*Finds P-->Q and P in the assumptions, replaces implication by Q *)
fun mp_tac i = eresolve_tac [notE,make_elim mp] i THEN assume_tac i;
(*Like mp_tac but instantiates no variables*)
fun uniq_mp_tac i = eresolve_tac [notE,impE] i THEN uniq_assume_tac i;
(*** If-and-only-if ***)
val iffI = prove_goalw IFOLP.thy [iff_def]
"[| !!x.x:P ==> q(x):Q; !!x.x:Q ==> r(x):P |] ==> ?p:P<->Q"
(fn prems=> [ (REPEAT (ares_tac (prems@[conjI, impI]) 1)) ]);
(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *)
val iffE = prove_goalw IFOLP.thy [iff_def]
"[| p:P <-> Q; !!x y.[| x:P-->Q; y:Q-->P |] ==> q(x,y):R |] ==> ?p:R"
(fn prems => [ (resolve_tac [conjE] 1), (REPEAT (ares_tac prems 1)) ]);
(* Destruct rules for <-> similar to Modus Ponens *)
val iffD1 = prove_goalw IFOLP.thy [iff_def] "[| p:P <-> Q; q:P |] ==> ?p:Q"
(fn prems => [ (rtac (conjunct1 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
val iffD2 = prove_goalw IFOLP.thy [iff_def] "[| p:P <-> Q; q:Q |] ==> ?p:P"
(fn prems => [ (rtac (conjunct2 RS mp) 1), (REPEAT (ares_tac prems 1)) ]);
val iff_refl = prove_goal IFOLP.thy "?p:P <-> P"
(fn _ => [ (REPEAT (ares_tac [iffI] 1)) ]);
val iff_sym = prove_goal IFOLP.thy "p:Q <-> P ==> ?p:P <-> Q"
(fn [major] =>
[ (rtac (major RS iffE) 1),
(rtac iffI 1),
(REPEAT (eresolve_tac [asm_rl,mp] 1)) ]);
val iff_trans = prove_goal IFOLP.thy "[| p:P <-> Q; q:Q<-> R |] ==> ?p:P <-> R"
(fn prems =>
[ (cut_facts_tac prems 1),
(rtac iffI 1),
(REPEAT (eresolve_tac [asm_rl,iffE] 1 ORELSE mp_tac 1)) ]);
(*** Unique existence. NOTE THAT the following 2 quantifications
EX!x such that [EX!y such that P(x,y)] (sequential)
EX!x,y such that P(x,y) (simultaneous)
do NOT mean the same thing. The parser treats EX!x y.P(x,y) as sequential.
***)
val ex1I = prove_goalw IFOLP.thy [ex1_def]
"[| p:P(a); !!x u.u:P(x) ==> f(u) : x=a |] ==> ?p:EX! x. P(x)"
(fn prems => [ (REPEAT (ares_tac (prems@[exI,conjI,allI,impI]) 1)) ]);
val ex1E = prove_goalw IFOLP.thy [ex1_def]
"[| p:EX! x.P(x); \
\ !!x u v. [| u:P(x); v:ALL y. P(y) --> y=x |] ==> f(x,u,v):R |] ==>\
\ ?a : R"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (eresolve_tac [exE,conjE] 1 ORELSE ares_tac prems 1)) ]);
(*** <-> congruence rules for simplification ***)
(*Use iffE on a premise. For conj_cong, imp_cong, all_cong, ex_cong*)
fun iff_tac prems i =
resolve_tac (prems RL [iffE]) i THEN
REPEAT1 (eresolve_tac [asm_rl,mp] i);
val conj_cong = prove_goal IFOLP.thy
"[| p:P <-> P'; !!x.x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P&Q) <-> (P'&Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (ares_tac [iffI,conjI] 1
ORELSE eresolve_tac [iffE,conjE,mp] 1
ORELSE iff_tac prems 1)) ]);
val disj_cong = prove_goal IFOLP.thy
"[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P|Q) <-> (P'|Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (eresolve_tac [iffE,disjE,disjI1,disjI2] 1
ORELSE ares_tac [iffI] 1
ORELSE mp_tac 1)) ]);
val imp_cong = prove_goal IFOLP.thy
"[| p:P <-> P'; !!x.x:P' ==> q(x):Q <-> Q' |] ==> ?p:(P-->Q) <-> (P'-->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (ares_tac [iffI,impI] 1
ORELSE eresolve_tac [iffE] 1
ORELSE mp_tac 1 ORELSE iff_tac prems 1)) ]);
val iff_cong = prove_goal IFOLP.thy
"[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P<->Q) <-> (P'<->Q')"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (eresolve_tac [iffE] 1
ORELSE ares_tac [iffI] 1
ORELSE mp_tac 1)) ]);
val not_cong = prove_goal IFOLP.thy
"p:P <-> P' ==> ?p:~P <-> ~P'"
(fn prems =>
[ (cut_facts_tac prems 1),
(REPEAT (ares_tac [iffI,notI] 1
ORELSE mp_tac 1
ORELSE eresolve_tac [iffE,notE] 1)) ]);
val all_cong = prove_goal IFOLP.thy
"(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(ALL x.P(x)) <-> (ALL x.Q(x))"
(fn prems =>
[ (REPEAT (ares_tac [iffI,allI] 1
ORELSE mp_tac 1
ORELSE eresolve_tac [allE] 1 ORELSE iff_tac prems 1)) ]);
val ex_cong = prove_goal IFOLP.thy
"(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(EX x.P(x)) <-> (EX x.Q(x))"
(fn prems =>
[ (REPEAT (eresolve_tac [exE] 1 ORELSE ares_tac [iffI,exI] 1
ORELSE mp_tac 1
ORELSE iff_tac prems 1)) ]);
(*NOT PROVED
val ex1_cong = prove_goal IFOLP.thy
"(!!x.f(x):P(x) <-> Q(x)) ==> ?p:(EX! x.P(x)) <-> (EX! x.Q(x))"
(fn prems =>
[ (REPEAT (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1
ORELSE mp_tac 1
ORELSE iff_tac prems 1)) ]);
*)
(*** Equality rules ***)
val refl = ieqI;
val subst = prove_goal IFOLP.thy "[| p:a=b; q:P(a) |] ==> ?p : P(b)"
(fn [prem1,prem2] => [ rtac (prem2 RS rev_mp) 1, (rtac (prem1 RS ieqE) 1),
rtac impI 1, atac 1 ]);
val sym = prove_goal IFOLP.thy "q:a=b ==> ?c:b=a"
(fn [major] => [ (rtac (major RS subst) 1), (rtac refl 1) ]);
val trans = prove_goal IFOLP.thy "[| p:a=b; q:b=c |] ==> ?d:a=c"
(fn [prem1,prem2] => [ (rtac (prem2 RS subst) 1), (rtac prem1 1) ]);
(** ~ b=a ==> ~ a=b **)
val not_sym = prove_goal IFOLP.thy "p:~ b=a ==> ?q:~ a=b"
(fn [prem] => [ (rtac (prem RS contrapos) 1), (etac sym 1) ]);
(*calling "standard" reduces maxidx to 0*)
val ssubst = standard (sym RS subst);
(*A special case of ex1E that would otherwise need quantifier expansion*)
val ex1_equalsE = prove_goal IFOLP.thy
"[| p:EX! x.P(x); q:P(a); r:P(b) |] ==> ?d:a=b"
(fn prems =>
[ (cut_facts_tac prems 1),
(etac ex1E 1),
(rtac trans 1),
(rtac sym 2),
(REPEAT (eresolve_tac [asm_rl, spec RS mp] 1)) ]);
(** Polymorphic congruence rules **)
val subst_context = prove_goal IFOLP.thy
"[| p:a=b |] ==> ?d:t(a)=t(b)"
(fn prems=>
[ (resolve_tac (prems RL [ssubst]) 1),
(resolve_tac [refl] 1) ]);
val subst_context2 = prove_goal IFOLP.thy
"[| p:a=b; q:c=d |] ==> ?p:t(a,c)=t(b,d)"
(fn prems=>
[ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
val subst_context3 = prove_goal IFOLP.thy
"[| p:a=b; q:c=d; r:e=f |] ==> ?p:t(a,c,e)=t(b,d,f)"
(fn prems=>
[ (EVERY1 (map rtac ((prems RL [ssubst]) @ [refl]))) ]);
(*Useful with eresolve_tac for proving equalties from known equalities.
a = b
| |
c = d *)
val box_equals = prove_goal IFOLP.thy
"[| p:a=b; q:a=c; r:b=d |] ==> ?p:c=d"
(fn prems=>
[ (resolve_tac [trans] 1),
(resolve_tac [trans] 1),
(resolve_tac [sym] 1),
(REPEAT (resolve_tac prems 1)) ]);
(*Dual of box_equals: for proving equalities backwards*)
val simp_equals = prove_goal IFOLP.thy
"[| p:a=c; q:b=d; r:c=d |] ==> ?p:a=b"
(fn prems=>
[ (resolve_tac [trans] 1),
(resolve_tac [trans] 1),
(REPEAT (resolve_tac (prems @ (prems RL [sym])) 1)) ]);
(** Congruence rules for predicate letters **)
val pred1_cong = prove_goal IFOLP.thy
"p:a=a' ==> ?p:P(a) <-> P(a')"
(fn prems =>
[ (cut_facts_tac prems 1),
(rtac iffI 1),
(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
val pred2_cong = prove_goal IFOLP.thy
"[| p:a=a'; q:b=b' |] ==> ?p:P(a,b) <-> P(a',b')"
(fn prems =>
[ (cut_facts_tac prems 1),
(rtac iffI 1),
(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
val pred3_cong = prove_goal IFOLP.thy
"[| p:a=a'; q:b=b'; r:c=c' |] ==> ?p:P(a,b,c) <-> P(a',b',c')"
(fn prems =>
[ (cut_facts_tac prems 1),
(rtac iffI 1),
(DEPTH_SOLVE (eresolve_tac [asm_rl, subst, ssubst] 1)) ]);
(*special cases for free variables P, Q, R, S -- up to 3 arguments*)
val pred_congs =
flat (map (fn c =>
map (fn th => read_instantiate [("P",c)] th)
[pred1_cong,pred2_cong,pred3_cong])
(explode"PQRS"));
(*special case for the equality predicate!*)
val eq_cong = read_instantiate [("P","op =")] pred2_cong;
(*** Simplifications of assumed implications.
Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE
used with mp_tac (restricted to atomic formulae) is COMPLETE for
intuitionistic propositional logic. See
R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
(preprint, University of St Andrews, 1991) ***)
val conj_impE = prove_goal IFOLP.thy
"[| p:(P&Q)-->S; !!x.x:P-->(Q-->S) ==> q(x):R |] ==> ?p:R"
(fn major::prems=>
[ (REPEAT (ares_tac ([conjI, impI, major RS mp]@prems) 1)) ]);
val disj_impE = prove_goal IFOLP.thy
"[| p:(P|Q)-->S; !!x y.[| x:P-->S; y:Q-->S |] ==> q(x,y):R |] ==> ?p:R"
(fn major::prems=>
[ (DEPTH_SOLVE (ares_tac ([disjI1, disjI2, impI, major RS mp]@prems) 1)) ]);
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since Q must be provable -- backtracking needed. *)
val imp_impE = prove_goal IFOLP.thy
"[| p:(P-->Q)-->S; !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q; !!x.x:S ==> r(x):R |] ==> \
\ ?p:R"
(fn major::prems=>
[ (REPEAT (ares_tac ([impI, major RS mp]@prems) 1)) ]);
(*Simplifies the implication. Classical version is stronger.
Still UNSAFE since ~P must be provable -- backtracking needed. *)
val not_impE = prove_goal IFOLP.thy
"[| p:~P --> S; !!y.y:P ==> q(y):False; !!y.y:S ==> r(y):R |] ==> ?p:R"
(fn major::prems=>
[ (REPEAT (ares_tac ([notI, impI, major RS mp]@prems) 1)) ]);
(*Simplifies the implication. UNSAFE. *)
val iff_impE = prove_goal IFOLP.thy
"[| p:(P<->Q)-->S; !!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q; \
\ !!x y.[| x:Q; y:P-->S |] ==> r(x,y):P; !!x.x:S ==> s(x):R |] ==> ?p:R"
(fn major::prems=>
[ (REPEAT (ares_tac ([iffI, impI, major RS mp]@prems) 1)) ]);
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*)
val all_impE = prove_goal IFOLP.thy
"[| p:(ALL x.P(x))-->S; !!x.q:P(x); !!y.y:S ==> r(y):R |] ==> ?p:R"
(fn major::prems=>
[ (REPEAT (ares_tac ([allI, impI, major RS mp]@prems) 1)) ]);
(*Unsafe: (EX x.P(x))-->S is equivalent to ALL x.P(x)-->S. *)
val ex_impE = prove_goal IFOLP.thy
"[| p:(EX x.P(x))-->S; !!y.y:P(a)-->S ==> q(y):R |] ==> ?p:R"
(fn major::prems=>
[ (REPEAT (ares_tac ([exI, impI, major RS mp]@prems) 1)) ]);
end;
open IFOLP_Lemmas;