Nuprl Lemma : member_firstn
∀[T:Type]. ∀L:T List. ∀n:ℕ. ∀x:T. ((x ∈ firstn(n;L)) ⇐⇒ ∃i:ℕ. ((i < n ∧ i < ||L||) ∧ (x = L[i] ∈ T)))
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
firstn: firstn(n;as),
so_apply: x[s],
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
uimplies: b supposing a,
and: P ∧ Q,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
nat_plus: ℕ+,
cand: A c∧ B,
cons: [a / b],
less_than': less_than'(a;b),
le: A ≤ B,
less_than: a < b,
squash: ↓T,
subtract: n - m,
sq_type: SQType(T),
true: True
Lemmas referenced :
istype-universe,
list_wf,
istype-less_than,
length_of_cons_lemma,
list_ind_cons_lemma,
istype-base,
stuck-spread,
length_of_nil_lemma,
list_ind_nil_lemma,
istype-nat,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
select_wf,
equal_wf,
length_wf,
less_than_wf,
exists_wf,
firstn_wf,
l_member_wf,
iff_wf,
nat_wf,
all_wf,
list_induction,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
satisfiable-full-omega-tt,
intformless_wf,
int_formula_prop_less_lemma,
equal-wf-T-base,
bnot_wf,
le_int_wf,
assert_wf,
int_subtype_base,
le_wf,
set_subtype_base,
bool_wf,
equal-wf-base,
lt_int_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
cons_wf,
istype-le,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
cons_member,
false_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
intformeq_wf,
itermAdd_wf,
add-is-int-iff,
nat_plus_properties,
decidable__lt,
length_wf_nat,
add_nat_plus,
select-cons-tl,
add-associates,
add-swap,
add-commutes,
zero-add,
subtype_base_sq,
decidable__equal_int,
iff_weakening_equal,
subtype_rel_self,
select_cons_tl,
true_wf,
squash_wf
Rules used in proof :
universeEquality,
instantiate,
equalityIstype,
productIsType,
functionIsType,
baseClosed,
inhabitedIsType,
universeIsType,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
natural_numberEquality,
dependent_functionElimination,
productElimination,
independent_isectElimination,
productEquality,
rename,
setElimination,
because_Cache,
hypothesis,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaFormation,
cumulativity,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
lambdaEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
applyEquality,
closedConclusion,
baseApply,
equalityElimination,
addEquality,
dependent_set_memberEquality_alt,
promote_hyp,
pointwiseFunctionality,
applyLambdaEquality,
imageElimination,
hyp_replacement,
inlFormation_alt,
imageMemberEquality,
inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}n:\mBbbN{}. \mforall{}x:T. ((x \mmember{} firstn(n;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}. ((i < n \mwedge{} i < ||L||) \mwedge{} (x = L[i])))
Date html generated:
2019_10_15-AM-10_22_27
Last ObjectModification:
2019_08_05-PM-01_56_49
Theory : list_1
Home
Index