Nuprl Lemma : assert-bl-exists
∀[T:Type]. ∀L:T List. ∀P:{x:T| (x ∈ L)} ⟶ 𝔹. (↑(∃x∈L.P[x])_b ⇐⇒ (∃x∈L. ↑P[x]))
Proof
Definitions occuring in Statement :
bl-exists: (∃x∈L.P[x])_b,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
bl-exists: (∃x∈L.P[x])_b,
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
false: False,
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
btrue: tt,
true: True
Lemmas referenced :
list-subtype,
bool_subtype_base,
subtype_base_sq,
assert_elim,
bor_wf,
assert_of_bor,
equal_wf,
or_wf,
cons_wf,
cons_member,
nil_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
length_of_nil_lemma,
false_wf,
bool_wf,
l_exists_wf,
l_exists_iff,
reduce_cons_lemma,
reduce_nil_lemma,
list_wf,
and_wf,
exists_wf,
l_member_wf,
bl-exists_wf,
assert_wf,
iff_wf,
list_induction
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
functionEquality,
universeEquality,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
orFunctionality,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality,
cumulativity,
productEquality,
unionElimination,
inlFormation,
inrFormation,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
levelHypothesis,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}. (\muparrow{}(\mexists{}x\mmember{}L.P[x])\_b \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. \muparrow{}P[x]))
Date html generated:
2016_05_14-PM-02_10_05
Last ObjectModification:
2016_01_15-AM-07_59_52
Theory : list_1
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