Nuprl Lemma : first-success-is-inl
∀[T:Type]. ∀[A:T ⟶ Type]. ∀[f:x:T ⟶ (A[x]?)]. ∀[L:T List]. ∀[j:ℕ||L||]. ∀[a:A[L[j]]].
(first-success(f;L) = (inl <j, a>) ∈ (i:ℕ||L|| × A[L[i]]?)
⇐⇒ j < ||L|| ∧ ((f L[j]) = (inl a) ∈ (A[L[j]]?)) ∧ (∀x∈firstn(j;L).↑isr(f x)))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
l_all: (∀x∈L.P[x]),
first-success: first-success(f;L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
isr: isr(x),
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
inl: inl x,
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
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 ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
guard: {T},
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x]),
or: P ∨ Q,
first-success: first-success(f;L),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3],
cons: [a / b],
decidable: Dec(P),
isr: isr(x),
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
firstn: firstn(n;as),
list_ind: list_ind,
uiff: uiff(P;Q),
pi2: snd(t),
pi1: fst(t),
cand: A c∧ B,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
outl: outl(x),
isl: isl(x),
true: True,
unit: Unit,
bool: 𝔹,
bnot: ¬bb,
istype: istype(T),
subtract: n - m
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
member-less_than,
int_seg_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_witness,
list-cases,
length_of_nil_lemma,
stuck-spread,
istype-base,
list_ind_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
select_wf,
firstn_wf,
decidable__lt,
length_wf,
bfalse_wf,
btrue_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
le_wf,
length_of_cons_lemma,
list_ind_cons_lemma,
istype-nat,
list_wf,
unit_wf2,
istype-universe,
satisfiable-full-omega-tt,
equal-wf-base-T,
less_than_wf,
equal_wf,
l_all_wf,
nil_wf,
assert_wf,
isr_wf,
l_member_wf,
int_seg_wf,
non_neg_length,
add-is-int-iff,
false_wf,
cons_wf,
istype-false,
lelt_wf,
first0,
subtype_rel_list,
top_wf,
l_all_nil,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
bool_cases,
lt_int_wf,
bnot_wf,
not_wf,
istype-assert,
l_all_cons,
outl_wf,
true_wf,
first-success_wf,
assert-bnot,
bool_cases_sqequal,
iff_weakening_equal,
select_cons_tl,
squash_wf,
subtype_rel_wf,
add-subtract-cancel,
select-cons-tl,
subtype_rel-equal,
add-member-int_seg2,
equal_functionality_wrt_subtype_rel2,
subtype_rel_self,
subtype_rel_union,
select-cons,
assert_of_le_int,
le_int_wf,
l_all_wf_nil,
btrue_neq_bfalse,
select-cons-hd,
isr-first-success
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
independent_pairEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
unionElimination,
baseClosed,
promote_hyp,
hypothesis_subsumption,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
applyEquality,
instantiate,
baseApply,
closedConclusion,
intEquality,
sqequalBase,
functionIsType,
unionIsType,
universeEquality,
isect_memberFormation,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
isect_memberEquality,
voidEquality,
computeAll,
unionEquality,
productEquality,
inlEquality,
dependent_pairEquality,
functionExtensionality,
cumulativity,
setEquality,
productIsType,
addEquality,
inlEquality_alt,
dependent_pairEquality_alt,
pointwiseFunctionality,
setIsType,
Error :memTop,
hyp_replacement,
equalityIsType1,
equalityIsType2,
equalityElimination,
isectIsType,
imageMemberEquality,
equalityIsType3
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} Type]. \mforall{}[f:x:T {}\mrightarrow{} (A[x]?)]. \mforall{}[L:T List]. \mforall{}[j:\mBbbN{}||L||]. \mforall{}[a:A[L[j]]].
(first-success(f;L) = (inl <j, a>) \mLeftarrow{}{}\mRightarrow{} j < ||L|| \mwedge{} ((f L[j]) = (inl a)) \mwedge{} (\mforall{}x\mmember{}firstn(j;L).\muparrow{}isr(f x\000C)))
Date html generated:
2020_05_19-PM-09_41_49
Last ObjectModification:
2019_12_31-PM-07_21_56
Theory : list_1
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