Nuprl Lemma : first_index-positive
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. (0 < index-of-first x in L.P[x] ⇐⇒ (∃x∈L. ↑P[x]))
Proof
Definitions occuring in Statement :
first_index: index-of-first x in L.P[x],
l_exists: (∃x∈L. P[x]),
list: T List,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
first_index: index-of-first x in L.P[x],
member: t ∈ T,
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
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],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
l_exists: (∃x∈L. P[x])
Lemmas referenced :
bool_wf,
list_wf,
l_member_wf,
l_exists_wf,
assert_wf,
exists_wf,
iff_wf,
search_wf,
less_than_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf,
length_wf_nat,
search_property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
cumulativity,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
imageElimination,
addLevel,
impliesFunctionality,
independent_functionElimination,
setEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. (0 < index-of-first x in L.P[x] \mLeftarrow{}{}\mRightarrow{} (\mexists{}x\mmember{}L. \muparrow{}P[x]))
Date html generated:
2016_05_15-PM-03_24_59
Last ObjectModification:
2016_01_16-AM-10_48_21
Theory : general
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