Nuprl Lemma : bl-exists-first
∀[A:Type]. ∀P:A ⟶ 𝔹. ∀L:A List. (↑(∃x∈L.P[x])_b ⇐⇒ ∃i:ℕ||L||. ((↑P[L[i]]) ∧ (∀j:ℕi. (¬↑P[L[j]]))))
Proof
Definitions occuring in Statement :
bl-exists: (∃x∈L.P[x])_b,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
and: 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],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
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,
less_than: a < b,
squash: ↓T,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
cons: [a / b],
cand: A c∧ B,
subtract: n - m
Lemmas referenced :
list_induction,
iff_wf,
l_exists_wf,
l_member_wf,
assert_wf,
exists_wf,
int_seg_wf,
length_wf,
select_wf,
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,
all_wf,
not_wf,
list_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
assert-bl-exists,
bl-exists_wf,
bool_wf,
l_exists_nil,
l_exists_wf_nil,
cons_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
l_exists_cons,
decidable__assert,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
lelt_wf,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
select-cons-tl,
add-subtract-cancel,
decidable__equal_int,
assert_functionality_wrt_uiff,
squash_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
natural_numberEquality,
productEquality,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
baseClosed,
addLevel,
impliesFunctionality,
functionEquality,
universeEquality,
addEquality,
dependent_set_memberEquality,
imageMemberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
inlFormation,
inrFormation
Latex:
\mforall{}[A:Type]
\mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List. (\muparrow{}(\mexists{}x\mmember{}L.P[x])\_b \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||L||. ((\muparrow{}P[L[i]]) \mwedge{} (\mforall{}j:\mBbbN{}i. (\mneg{}\muparrow{}P[L[j]]))))
Date html generated:
2017_04_17-AM-08_04_08
Last ObjectModification:
2017_02_27-PM-04_34_38
Theory : list_1
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