Nuprl Lemma : imax-list-filter-member
∀L:ℤ List. ∀P:ℤ ⟶ 𝔹. (imax-list(filter(P;L)) ∈ L) supposing ¬(filter(P;L) = [] ∈ (ℤ List))
Proof
Definitions occuring in Statement :
imax-list: imax-list(L),
l_member: (x ∈ l),
filter: filter(P;l),
nil: [],
list: T List,
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
top: Top,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
guard: {T},
decidable: Dec(P),
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q
Lemmas referenced :
equal-wf-T-base,
list_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
list-cases,
length_of_nil_lemma,
not_wf,
equal-wf-base,
product_subtype_list,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
equal_wf,
list_subtype_base,
int_subtype_base,
imax-list-member,
filter_is_sublist,
member_sublist,
member_filter_2,
imax-list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
intEquality,
hypothesis,
applyEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
baseClosed,
unionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
dependent_pairFormation,
int_eqEquality,
computeAll,
functionEquality
Latex:
\mforall{}L:\mBbbZ{} List. \mforall{}P:\mBbbZ{} {}\mrightarrow{} \mBbbB{}. (imax-list(filter(P;L)) \mmember{} L) supposing \mneg{}(filter(P;L) = [])
Date html generated:
2017_04_17-AM-07_40_17
Last ObjectModification:
2017_02_27-PM-04_14_08
Theory : list_1
Home
Index