Nuprl Lemma : first-success_wf
∀[T:Type]. ∀[A:T ⟶ Type]. ∀[f:x:T ⟶ (A[x]?)]. ∀[L:T List]. (first-success(f;L) ∈ i:ℕ||L|| × A[L[i]]?)
Proof
Definitions occuring in Statement :
first-success: first-success(f;L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
unit: Unit,
member: t ∈ T,
function: x:A ⟶ B[x],
product: x:A × B[x],
union: left + right,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
first-success: first-success(f;L),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
cons: [a / b],
colength: colength(L),
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
less_than: a < b,
nat_plus: ℕ+,
exists: ∃x:A. B[x]
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
length_of_nil_lemma,
stuck-spread,
base_wf,
list_ind_nil_lemma,
it_wf,
int_seg_wf,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le_wf,
equal_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
length_of_cons_lemma,
list_ind_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
lelt_wf,
length_wf,
select_wf,
non_neg_length,
add-member-int_seg2,
le-add-cancel2,
subtype_rel-equal,
cons_wf,
select-cons-tl,
decidable__lt,
not-lt-2,
add-subtract-cancel,
list_wf,
unit_wf2,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
omega-shadow,
mul-distributes,
mul-associates,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
baseClosed,
isect_memberEquality,
voidEquality,
inrEquality,
productEquality,
functionExtensionality,
productElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
intEquality,
instantiate,
inlEquality,
dependent_pairEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
unionEquality,
functionEquality,
universeEquality,
multiplyEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} Type]. \mforall{}[f:x:T {}\mrightarrow{} (A[x]?)]. \mforall{}[L:T List].
(first-success(f;L) \mmember{} i:\mBbbN{}||L|| \mtimes{} A[L[i]]?)
Date html generated:
2017_04_14-AM-08_37_30
Last ObjectModification:
2017_02_27-PM-03_29_45
Theory : list_0
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