Nuprl Lemma : streamless-implies-not-not-enum
∀[T:Type]. (streamless(T) ⇒ (¬¬(∃L:T List. ∀x:T. (x ∈ L))))
Proof
Definitions occuring in Statement :
streamless: streamless(T),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
prop: ℙ,
squash: ↓T,
top: Top,
exists: ∃x:A. B[x],
no_repeats: no_repeats(T;l),
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
and: P ∧ Q,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than: a < b,
true: True,
iff: P ⇐⇒ Q,
select: L[n],
cons: [a / b],
l_member: (x ∈ l),
cand: A c∧ B,
streamless: streamless(T),
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
streamless-dec-equal,
basic-bar-induction,
not_wf,
no_repeats_wf,
list_wf,
or_wf,
exists_wf,
all_wf,
l_member_wf,
decidable__not,
decidable__no_repeats,
append_wf,
cons_wf,
nil_wf,
nat_wf,
no_repeats_nil,
streamless_wf,
decidable__l_member,
equal_wf,
select_wf,
length-append,
length_of_cons_lemma,
length_of_nil_lemma,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
less_than_wf,
length_wf,
subtype_rel_list,
top_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
length_nil,
non_neg_length,
length_cons,
length_append,
decidable__lt,
intformless_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__equal_int,
le_wf,
equal-wf-base,
int_subtype_base,
zero-add,
subtract_wf,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf,
length-singleton,
squash_wf,
true_wf,
iff_weakening_equal,
select-append,
imax_wf,
add_nat_wf,
imax_nat,
map_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
upto_wf,
map-length,
length_upto,
ifthenelse_wf,
le_int_wf,
assert_of_le_int,
add_functionality_wrt_eq,
imax_unfold,
lelt_wf,
select-map,
select-upto
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
dependent_functionElimination,
unionElimination,
inrFormation,
voidElimination,
inlFormation,
imageElimination,
imageMemberEquality,
baseClosed,
functionEquality,
isect_memberEquality,
voidEquality,
universeEquality,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
applyEquality,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
dependent_set_memberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
productEquality,
addEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. (streamless(T) {}\mRightarrow{} (\mneg{}\mneg{}(\mexists{}L:T List. \mforall{}x:T. (x \mmember{} L))))
Date html generated:
2018_05_21-PM-09_02_43
Last ObjectModification:
2017_07_26-PM-06_25_36
Theory : general
Home
Index