Nuprl Lemma : hd-filter
∀[T:Type]
∀P:T ⟶ 𝔹. ∀as:T List.
((∃a:T. ((a ∈ as) ∧ (↑P[a])))
⇒ ((hd(filter(λa.P[a];as)) ∈ T) c∧ ((hd(filter(λa.P[a];as)) ∈ as) ∧ (↑P[hd(filter(λa.P[a];as))]))))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
hd: hd(l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
cand: A c∧ B,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_apply: x[s],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
cons: [a / b],
sq_stable: SqStable(P)
Lemmas referenced :
filter_type,
length-filter-pos,
l_exists_iff,
l_member_wf,
assert_wf,
exists_wf,
list_wf,
bool_wf,
hd_wf,
subtype_rel_list,
decidable__le,
length_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
less_than_wf,
equal_wf,
hd_member,
set_wf,
list-cases,
product_subtype_list,
assert_elim,
null_wf3,
top_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
member_filter,
sq_stable__assert
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
hypothesis,
setElimination,
rename,
setEquality,
productElimination,
independent_functionElimination,
productEquality,
functionEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
unionElimination,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
hypothesis_subsumption,
addLevel,
levelHypothesis,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}as:T List.
((\mexists{}a:T. ((a \mmember{} as) \mwedge{} (\muparrow{}P[a])))
{}\mRightarrow{} ((hd(filter(\mlambda{}a.P[a];as)) \mmember{} T)
c\mwedge{} ((hd(filter(\mlambda{}a.P[a];as)) \mmember{} as) \mwedge{} (\muparrow{}P[hd(filter(\mlambda{}a.P[a];as))]))))
Date html generated:
2018_05_21-PM-06_50_26
Last ObjectModification:
2017_07_26-PM-04_57_29
Theory : general
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