Nuprl Lemma : mFOL-abstract-rename
∀[Dom:Type]. ∀[S:FOStruct(Dom)].
∀x,y:ℤ. ∀fmla:mFOL().
((¬(x ∈ mFOL-boundvars(fmla)))
⇒ (∀a1,a2:FOAssignment(Dom).
((a2 = a1[y := a1 x] ∈ FOAssignment(Dom))
⇒ (Dom,S,a2 |= mFOL-abstract(fmla) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(fmla;y;x)) ∈ ℙ))))
Proof
Definitions occuring in Statement :
mFOL-abstract: mFOL-abstract(fmla),
mFOL-rename: mFOL-rename(fmla;old;new),
mFOL-boundvars: mFOL-boundvars(fmla),
mFOL: mFOL(),
FOSatWith: Dom,S,a |= fmla,
update-assignment: a[x := v],
FOStruct: FOStruct(Dom),
FOAssignment: FOAssignment(Dom),
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Lemmas :
mFOL-induction,
not_wf,
l_member_wf,
mFOL-boundvars_wf,
all_wf,
FOAssignment_wf,
update-assignment_wf,
FOSatWith_wf,
mFOL-abstract_wf,
mFOL-rename_wf,
mFOL_wf,
list_induction,
equal_wf,
list_wf,
map_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
map_nil_lemma,
nil_wf,
map_cons_lemma,
cons_wf,
squash_wf,
true_wf,
iff_weakening_equal,
member-union,
int-deq_wf,
assert_of_eq_atom,
and_wf,
neg_assert_of_eq_atom,
eq_atom_wf,
or_wf,
mFOconnect_wf,
member-insert,
decidable__equal_int,
false_wf,
not-equal-2,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
add-associates,
equal-wf-base,
int_subtype_base,
exists_wf,
mFOquant_wf,
FOStruct_wf
\mforall{}[Dom:Type]. \mforall{}[S:FOStruct(Dom)].
\mforall{}x,y:\mBbbZ{}. \mforall{}fmla:mFOL().
((\mneg{}(x \mmember{} mFOL-boundvars(fmla)))
{}\mRightarrow{} (\mforall{}a1,a2:FOAssignment(Dom).
((a2 = a1[y := a1 x])
{}\mRightarrow{} (Dom,S,a2 |= mFOL-abstract(fmla) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(fmla;y;x))))))
Date html generated:
2015_07_17-AM-07_54_28
Last ObjectModification:
2015_02_03-PM-09_39_04
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