Nuprl Lemma : FOL-bound-rename
This lemma shows that there is a formula
with the same abstract meaning whose bound variables are disjoint from
a given list of variables.⋅
∀fmla:mFOL(). ∀L:ℤ List.
(∃fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(fmla) ∈ (ℤ List))
∧ (FOL-abstract(fmla') = FOL-abstract(fmla) ∈ AbstractFOFormula+(mFOL-freevars(fmla)))
∧ l_disjoint(ℤ;L;mFOL-boundvars(fmla')))])
Proof
Definitions occuring in Statement :
FOL-abstract: FOL-abstract(fmla),
mFOL-boundvars: mFOL-boundvars(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
AbstractFOFormula+: AbstractFOFormula+(vs),
l_disjoint: l_disjoint(T;l1;l2),
list: T List,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
guard: {T},
cand: A c∧ B,
mFOL-boundvars: mFOL-boundvars(fmla),
mFOatomic: name(vars),
mFOL_ind: mFOL_ind,
squash: ↓T,
true: True,
mFOL-freevars: mFOL-freevars(fmla),
mFOconnect: mFOconnect(knd;left;right),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
FOL-abstract: FOL-abstract(fmla),
l_disjoint: l_disjoint(T;l1;l2),
not: ¬A,
or: P ∨ Q,
false: False,
decidable: Dec(P),
exists: ∃x:A. B[x],
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
top: Top,
bfalse: ff,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nat: ℕ,
subtract: n - m,
less_than': less_than'(a;b),
l_member: (x ∈ l),
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
mFOquant: mFOquant(isall;var;body),
FOQuantifier+: FOQuantifier+(isall),
AbstractFOFormula+: AbstractFOFormula+(vs),
FOStruct+: FOStruct+{i:l}(Dom),
FOStruct: FOStruct(Dom),
l_contains: A ⊆ B,
FOAssignment: FOAssignment(vs,Dom),
update-assignment: a[x := v],
FOAssigment-rename: FOAssigment-rename(a;fmla;x;y),
nequal: a ≠ b ∈ T
Lemmas referenced :
mFOL-induction,
all_wf,
list_wf,
sq_exists_wf,
mFOL_wf,
equal-wf-base,
mFOL-freevars_wf,
list_subtype_base,
int_subtype_base,
length_wf,
equal_wf,
AbstractFOFormula+_wf,
FOL-abstract_wf,
l_disjoint_wf,
mFOL-boundvars_wf,
istype-atom,
istype-int,
bool_wf,
mFOatomic_wf,
l_disjoint_nil2,
atom_subtype_base,
subtype_rel-equal,
mFOconnect_wf,
squash_wf,
true_wf,
istype-universe,
val-union_wf,
valueall-type_wf,
deq_wf,
int-deq_wf,
int-valueall-type,
subtype_rel_self,
iff_weakening_equal,
FOConnective+_wf,
equal_functionality_wrt_subtype_rel2,
member-union,
l_member_wf,
l-union_wf,
decidable__l_member,
decidable__equal_int,
append_wf,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
istype-void,
imax-list_wf,
cons_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
non_neg_length,
length_of_cons_lemma,
null_wf3,
subtype_rel_list,
top_wf,
eqtt_to_assert,
assert_of_null,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
length_of_nil_lemma,
nil_wf,
length_wf_nat,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
imax-list-lb,
decidable__le,
nat_properties,
istype-le,
istype-less_than,
intformeq_wf,
int_formula_prop_eq_lemma,
member_append,
mFOquant_wf,
mFOL-rename_wf,
filter-mFOL-freevars-of-rename,
b-union_wf,
FOAssignment_wf,
FOStruct+_wf,
bfalse_wf,
FOSatWith+_wf,
update-assignment_wf,
subtype_rel_FOAssignment,
filter_wf5,
bnot_wf,
eq_int_wf,
eq_int_eq_true,
assert_elim,
iff_transitivity,
not_wf,
assert_of_bnot,
assert_of_eq_int,
istype-assert,
member_filter,
mFOL-freevars-of-rename,
l_all_iff,
neg_assert_of_eq_int,
FOL-abstract-rename,
filter-contains,
deq-member_wf,
assert-deq-member,
insert_wf,
mFOL-boundvars-of-rename,
member-insert,
FOQuantifier+_wf
Rules used in proof :
cut,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
closedConclusion,
intEquality,
productEquality,
cumulativity,
hypothesisEquality,
applyEquality,
independent_isectElimination,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
inhabitedIsType,
universeIsType,
independent_functionElimination,
lambdaFormation_alt,
functionIsType,
setIsType,
productIsType,
equalityIstype,
sqequalBase,
dependent_set_memberFormation_alt,
independent_pairFormation,
baseApply,
baseClosed,
imageElimination,
natural_numberEquality,
imageMemberEquality,
dependent_functionElimination,
setElimination,
rename,
productElimination,
universeEquality,
functionEquality,
unionElimination,
voidElimination,
dependent_pairFormation_alt,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality_alt,
addEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
approximateComputation,
voidEquality,
isect_memberEquality,
equalityElimination,
dependent_set_memberEquality_alt,
minusEquality,
unionIsType,
inlFormation_alt,
inrFormation_alt,
tokenEquality,
hyp_replacement,
functionExtensionality,
setEquality
Latex:
\mforall{}fmla:mFOL(). \mforall{}L:\mBbbZ{} List.
(\mexists{}fmla':mFOL() [((mFOL-freevars(fmla') = mFOL-freevars(fmla))
\mwedge{} (FOL-abstract(fmla') = FOL-abstract(fmla))
\mwedge{} l\_disjoint(\mBbbZ{};L;mFOL-boundvars(fmla')))])
Date html generated:
2019_10_16-AM-11_40_42
Last ObjectModification:
2018_12_08-PM-03_43_07
Theory : minimal-first-order-logic
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