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folderived.ml
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(* ========================================================================= *)
(* First-order derived rules in the LCF setup. *)
(* *)
(* Copyright (c) 2003-2007, John Harrison. (See "LICENSE.txt" for details.) *)
(* ========================================================================= *)
(* ------------------------------------------------------------------------- *)
(* ****** *)
(* ------------------------------------------ eq_sym *)
(* |- s = t ==> t = s *)
(* ------------------------------------------------------------------------- *)
let eq_sym s t =
let rth = axiom_eqrefl s in
funpow 2 (fun th -> modusponens (imp_swap th) rth)
(axiom_predcong "=" [s; s] [t; s]);;
(* ------------------------------------------------------------------------- *)
(* |- s = t ==> t = u ==> s = u. *)
(* ------------------------------------------------------------------------- *)
let eq_trans s t u =
let th1 = axiom_predcong "=" [t; u] [s; u] in
let th2 = modusponens (imp_swap th1) (axiom_eqrefl u) in
imp_trans (eq_sym s t) th2;;
(* ------------------------------------------------------------------------- *)
(* ---------------------------- icongruence *)
(* |- s = t ==> tm[s] = tm[t] *)
(* ------------------------------------------------------------------------- *)
let rec icongruence s t stm ttm =
if stm = ttm then add_assum (mk_eq s t) (axiom_eqrefl stm)
else if stm = s & ttm = t then imp_refl (mk_eq s t) else
match (stm,ttm) with
(Fn(fs,sa),Fn(ft,ta)) when fs = ft & length sa = length ta ->
let ths = map2 (icongruence s t) sa ta in
let ts = map (consequent ** concl) ths in
imp_trans_chain ths (axiom_funcong fs (map lhs ts) (map rhs ts))
| _ -> failwith "icongruence: not congruent";;
(* ------------------------------------------------------------------------- *)
(* Example. *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
icongruence <<|s|>> <<|t|>> <<|f(s,g(s,t,s),u,h(h(s)))|>>
<<|f(s,g(t,t,s),u,h(h(t)))|>>;;
END_INTERACTIVE;;
(* ------------------------------------------------------------------------- *)
(* |- (forall x. p ==> q(x)) ==> p ==> (forall x. q(x)) *)
(* ------------------------------------------------------------------------- *)
let gen_right_th x p q =
imp_swap(imp_trans (axiom_impall x p) (imp_swap(axiom_allimp x p q)));;
(* ------------------------------------------------------------------------- *)
(* |- p ==> q *)
(* ------------------------------------- genimp "x" *)
(* |- (forall x. p) ==> (forall x. q) *)
(* ------------------------------------------------------------------------- *)
let genimp x th =
let p,q = dest_imp(concl th) in
modusponens (axiom_allimp x p q) (gen x th);;
(* ------------------------------------------------------------------------- *)
(* If |- p ==> q[x] then |- p ==> forall x. q[x] *)
(* ------------------------------------------------------------------------- *)
let gen_right x th =
let p,q = dest_imp(concl th) in
modusponens (gen_right_th x p q) (gen x th);;
(* ------------------------------------------------------------------------- *)
(* |- (forall x. p(x) ==> q) ==> (exists x. p(x)) ==> q *)
(* ------------------------------------------------------------------------- *)
let exists_left_th x p q =
let p' = Imp(p,False) and q' = Imp(q,False) in
let th1 = genimp x (imp_swap(imp_trans_th p q False)) in
let th2 = imp_trans th1 (gen_right_th x q' p') in
let th3 = imp_swap(imp_trans_th q' (Forall(x,p')) False) in
let th4 = imp_trans2 (imp_trans th2 th3) (axiom_doubleneg q) in
let th5 = imp_add_concl False (genimp x (iff_imp2 (axiom_not p))) in
let th6 = imp_trans (iff_imp1 (axiom_not (Forall(x,Not p)))) th5 in
let th7 = imp_trans (iff_imp1(axiom_exists x p)) th6 in
imp_swap(imp_trans th7 (imp_swap th4));;
(* ------------------------------------------------------------------------- *)
(* If |- p(x) ==> q then |- (exists x. p(x)) ==> q *)
(* ------------------------------------------------------------------------- *)
let exists_left x th =
let p,q = dest_imp(concl th) in
modusponens (exists_left_th x p q) (gen x th);;
(* ------------------------------------------------------------------------- *)
(* |- x = t ==> p ==> q [x not in t and not free in q] *)
(* --------------------------------------------------------------- subspec *)
(* |- (forall x. p) ==> q *)
(* ------------------------------------------------------------------------- *)
let subspec th =
match concl th with
Imp(Atom(R("=",[Var x;t])) as e,Imp(p,q)) ->
let th1 = imp_trans (genimp x (imp_swap th))
(exists_left_th x e q) in
modusponens (imp_swap th1) (axiom_existseq x t)
| _ -> failwith "subspec: wrong sort of theorem";;
(* ------------------------------------------------------------------------- *)
(* |- x = y ==> p[x] ==> q[y] [x not in FV(q); y not in FV(p) or x == y] *)
(* --------------------------------------------------------- subalpha *)
(* |- (forall x. p) ==> (forall y. q) *)
(* ------------------------------------------------------------------------- *)
let subalpha th =
match concl th with
Imp(Atom(R("=",[Var x;Var y])),Imp(p,q)) ->
if x = y then genimp x (modusponens th (axiom_eqrefl(Var x)))
else gen_right y (subspec th)
| _ -> failwith "subalpha: wrong sort of theorem";;
(* ------------------------------------------------------------------------- *)
(* ---------------------------------- isubst *)
(* |- s = t ==> p[s] ==> p[t] *)
(* ------------------------------------------------------------------------- *)
let rec isubst s t sfm tfm =
if sfm = tfm then add_assum (mk_eq s t) (imp_refl tfm) else
match (sfm,tfm) with
Atom(R(p,sa)),Atom(R(p',ta)) when p = p' & length sa = length ta ->
let ths = map2 (icongruence s t) sa ta in
let ls,rs = unzip (map (dest_eq ** consequent ** concl) ths) in
imp_trans_chain ths (axiom_predcong p ls rs)
| Imp(sp,sq),Imp(tp,tq) ->
let th1 = imp_trans (eq_sym s t) (isubst t s tp sp)
and th2 = isubst s t sq tq in
imp_trans_chain [th1; th2] (imp_mono_th sp tp sq tq)
| Forall(x,p),Forall(y,q) ->
if x = y then
imp_trans (gen_right x (isubst s t p q)) (axiom_allimp x p q)
else
let z = Var(variant x (unions [fv p; fv q; fvt s; fvt t])) in
let th1 = isubst (Var x) z p (subst (x |=> z) p)
and th2 = isubst z (Var y) (subst (y |=> z) q) q in
let th3 = subalpha th1 and th4 = subalpha th2 in
let th5 = isubst s t (consequent(concl th3))
(antecedent(concl th4)) in
imp_swap (imp_trans2 (imp_trans th3 (imp_swap th5)) th4)
| _ ->
let sth = iff_imp1(expand_connective sfm)
and tth = iff_imp2(expand_connective tfm) in
let th1 = isubst s t (consequent(concl sth))
(antecedent(concl tth)) in
imp_swap(imp_trans sth (imp_swap(imp_trans2 th1 tth)));;
(* ------------------------------------------------------------------------- *)
(* *)
(* -------------------------------------------- alpha "z" <<forall x. p[x]>> *)
(* |- (forall x. p[x]) ==> (forall z. p'[z]) *)
(* *)
(* [Restriction that z is not free in the initial p[x].] *)
(* ------------------------------------------------------------------------- *)
let alpha z fm =
match fm with
Forall(x,p) -> let p' = subst (x |=> Var z) p in
subalpha(isubst (Var x) (Var z) p p')
| _ -> failwith "alpha: not a universal formula";;
(* ------------------------------------------------------------------------- *)
(* *)
(* -------------------------------- ispec t <<forall x. p[x]>> *)
(* |- (forall x. p[x]) ==> p'[t] *)
(* ------------------------------------------------------------------------- *)
let rec ispec t fm =
match fm with
Forall(x,p) ->
if mem x (fvt t) then
let th = alpha (variant x (union (fvt t) (var p))) fm in
imp_trans th (ispec t (consequent(concl th)))
else subspec(isubst (Var x) t p (subst (x |=> t) p))
| _ -> failwith "ispec: non-universal formula";;
(* ------------------------------------------------------------------------- *)
(* Specialization rule. *)
(* ------------------------------------------------------------------------- *)
let spec t th = modusponens (ispec t (concl th)) th;;
(* ------------------------------------------------------------------------- *)
(* An example. *)
(* ------------------------------------------------------------------------- *)
START_INTERACTIVE;;
ispec <<|y|>> <<forall x y z. x + y + z = z + y + x>>;;
(* ------------------------------------------------------------------------- *)
(* Additional tests not in main text. *)
(* ------------------------------------------------------------------------- *)
isubst <<|x + x|>> <<|2 * x|>>
<<x + x = x ==> x = 0>> <<2 * x = x ==> x = 0>>;;
isubst <<|x + x|>> <<|2 * x|>>
<<(x + x = y + y) ==> (y + y + y = x + x + x)>>
<<2 * x = y + y ==> y + y + y = x + 2 * x>>;;
ispec <<|x|>> <<forall x y z. x + y + z = y + z + z>> ;;
ispec <<|x|>> <<forall x. x = x>> ;;
ispec <<|w + y + z|>> <<forall x y z. x + y + z = y + z + z>> ;;
ispec <<|x + y + z|>> <<forall x y z. x + y + z = y + z + z>> ;;
ispec <<|x + y + z|>> <<forall x y z. nothing_much>> ;;
isubst <<|x + x|>> <<|2 * x|>>
<<(x + x = y + y) <=> (something \/ y + y + y = x + x + x)>> ;;
isubst <<|x + x|>> <<|2 * x|>>
<<(exists x. x = 2) <=> exists y. y + x + x = y + y + y>>
<<(exists x. x = 2) <=> (exists y. y + 2 * x = y + y + y)>>;;
isubst <<|x|>> <<|y|>>
<<(forall z. x = z) <=> (exists x. y < z) /\ (forall y. y < x)>>
<<(forall z. y = z) <=> (exists x. y < z) /\ (forall y'. y' < y)>>;;
(* ------------------------------------------------------------------------- *)
(* The bug is now fixed. *)
(* ------------------------------------------------------------------------- *)
ispec <<|x'|>> <<forall x x' x''. x + x' + x'' = 0>>;;
ispec <<|x''|>> <<forall x x' x''. x + x' + x'' = 0>>;;
ispec <<|x' + x''|>> <<forall x x' x''. x + x' + x'' = 0>>;;
ispec <<|x + x' + x''|>> <<forall x x' x''. x + x' + x'' = 0>>;;
ispec <<|2 * x|>> <<forall x x'. x + x' = x' + x>>;;
END_INTERACTIVE;;