Nuprl Lemma : filter_is_nil_implies2
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. (∀x∈L.¬↑P[x]) supposing filter(P;L) = [] ∈ ({x:T| (x ∈ L)} List)
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
filter: filter(P;l),
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
l_all: (∀x∈L.P[x]),
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B
Lemmas referenced :
equal-wf-T-base,
list_wf,
l_member_wf,
filter_wf2,
set_wf,
bool_wf,
assert_wf,
select_wf,
list-subtype,
int_seg_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
length_wf,
member_filter,
select_member,
sqequal-nil,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
and_wf,
equal_wf,
null_wf,
btrue_neq_bfalse,
nil_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
hypothesis,
functionExtensionality,
applyEquality,
baseClosed,
functionEquality,
because_Cache,
sqequalRule,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
dependent_set_memberEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}]. (\mforall{}x\mmember{}L.\mneg{}\muparrow{}P[x]) supposing filter(P;L) = []
Date html generated:
2017_04_14-AM-09_26_20
Last ObjectModification:
2017_02_27-PM-04_00_25
Theory : list_1
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