Nuprl Lemma : member-firstn
∀[T:Type]. ∀L:T List. ∀n:ℕ||L|| + 1. ∀x:T. ((x ∈ firstn(n;L)) ⇐⇒ ∃i:ℕn. (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,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
so_apply: x[s]
Lemmas referenced :
member_firstn,
int_seg_subtype_nat,
length_wf,
false_wf,
lelt_wf,
equal_wf,
select_wf,
int_seg_properties,
nat_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,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
l_member_wf,
firstn_wf,
less_than_wf,
exists_wf,
int_seg_wf,
add-is-int-iff,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
natural_numberEquality,
addEquality,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
productElimination,
independent_functionElimination,
dependent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality,
unionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
productEquality,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}n:\mBbbN{}||L|| + 1. \mforall{}x:T. ((x \mmember{} firstn(n;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. (x = L[i]))
Date html generated:
2017_04_17-AM-07_51_53
Last ObjectModification:
2017_02_27-PM-04_25_01
Theory : list_1
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