Nuprl Lemma : filter-mFOL-freevars-of-rename
∀z,z':ℤ. ∀fmla':mFOL().
((¬(z' ∈ mFOL-boundvars(fmla')))
⇒ (¬(z' ∈ mFOL-freevars(fmla')))
⇒ (filter(λx.(¬b(x =z z'));mFOL-freevars(mFOL-rename(fmla';z;z')))
= filter(λx.(¬b(x =z z));mFOL-freevars(fmla'))
∈ (ℤ List)))
Proof
Definitions occuring in Statement :
mFOL-rename: mFOL-rename(fmla;old;new),
mFOL-boundvars: mFOL-boundvars(fmla),
mFOL-freevars: mFOL-freevars(fmla),
mFOL: mFOL(),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
bnot: ¬bb,
eq_int: (i =z j),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
lambda: λx.A[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s],
mFOL-freevars: mFOL-freevars(fmla),
mFOL-rename: mFOL-rename(fmla;old;new),
mFOatomic: name(vars),
mFOL_ind: mFOL_ind,
not: ¬A,
false: False,
mFOL-boundvars: mFOL-boundvars(fmla),
nil: [],
it: ⋅,
mFOconnect: mFOconnect(knd;left;right),
mFOquant: mFOquant(isall;var;body),
guard: {T},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
assert: ↑b,
nequal: a ≠ b ∈ T ,
true: True,
cand: A c∧ B
Lemmas referenced :
mFOL-induction,
not_wf,
l_member_wf,
mFOL-boundvars_wf,
mFOL-freevars_wf,
equal-wf-base,
list_wf,
filter_wf5,
mFOL-rename_wf,
bnot_wf,
eq_int_wf,
list_subtype_base,
int_subtype_base,
mFOL_wf,
mFOatomic_wf,
istype-void,
nil_wf,
istype-atom,
mFOconnect_wf,
mFOquant_wf,
bool_wf,
istype-int,
member-remove-repeats,
int-deq_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
list-cases,
map_nil_lemma,
remove_repeats_nil_lemma,
filter_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
map_cons_lemma,
remove_repeats_cons_lemma,
intdeq_reduce_lemma,
filter_cons_lemma,
istype-nat,
cons_member,
cons_wf,
eqtt_to_assert,
assert_of_eq_int,
length_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
ifthenelse_wf,
bfalse_wf,
swap-filter-filter,
remove-repeats_wf,
map_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
filter_trivial,
iff_transitivity,
assert_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-assert,
l_all_iff,
band-idempotent,
filter-filter,
val-union-l-union,
int-valueall-type,
member-union,
l-union_wf,
istype-universe,
filter-union,
deq_wf,
member-insert,
member_filter,
istype-true
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
functionEquality,
intEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
setIsType,
inhabitedIsType,
universeIsType,
applyEquality,
independent_isectElimination,
independent_functionElimination,
functionIsType,
equalityIstype,
because_Cache,
sqequalBase,
equalitySymmetry,
dependent_functionElimination,
productElimination,
voidElimination,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
axiomEquality,
functionIsTypeImplies,
unionElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
instantiate,
equalityTransitivity,
applyLambdaEquality,
imageElimination,
baseApply,
closedConclusion,
baseClosed,
inrFormation_alt,
inlFormation_alt,
cumulativity,
equalityElimination,
imageMemberEquality,
functionExtensionality_alt,
universeEquality,
unionIsType,
productIsType
Latex:
\mforall{}z,z':\mBbbZ{}. \mforall{}fmla':mFOL().
((\mneg{}(z' \mmember{} mFOL-boundvars(fmla')))
{}\mRightarrow{} (\mneg{}(z' \mmember{} mFOL-freevars(fmla')))
{}\mRightarrow{} (filter(\mlambda{}x.(\mneg{}\msubb{}(x =\msubz{} z'));mFOL-freevars(mFOL-rename(fmla';z;z')))
= filter(\mlambda{}x.(\mneg{}\msubb{}(x =\msubz{} z));mFOL-freevars(fmla'))))
Date html generated:
2020_05_20-AM-09_08_32
Last ObjectModification:
2019_12_31-PM-04_58_27
Theory : minimal-first-order-logic
Home
Index