Nuprl Lemma : nth_tl_factor
∀T:Type. ∀n:ℕ. ∀as:T List. ((n ≤ ||as||) ⇒ (nth_tl(n;as) = (Π n ≤ i < ||as||. [as[i]]) ∈ (T List)))
Proof
Definitions occuring in Statement :
lapp_imon: <T List,@>,
select: L[n],
length: ||as||,
nth_tl: nth_tl(n;as),
cons: [a / b],
nil: [],
list: T List,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
equal: s = t ∈ T,
mon_itop: Π lb ≤ i < ub. E[i]
Definitions unfolded in proof :
all: ∀x:A. B[x],
nth_tl: nth_tl(n;as),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
prop: ℙ,
bfalse: ff,
guard: {T},
subtract: n - m,
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
subtype_rel: A ⊆r B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
imon: IMonoid,
list: T List,
grp_car: |g|,
pi1: fst(t),
lapp_imon: <T List,@>
Lemmas referenced :
le_int_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
assert_wf,
le_wf,
eqtt_to_assert,
assert_of_le_int,
length_wf,
list_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
int_subtype_base,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract_wf,
nth_tl_wf,
mon_itop_wf,
lapp_imon_wf,
cons_wf,
select_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__lt,
nil_wf,
int_seg_wf,
primrec-wf2,
all_wf,
equal_wf,
nat_properties,
nat_wf,
lapp_fact_b,
squash_wf,
true_wf,
istype-universe,
tl_wf,
length_tl,
iff_weakening_equal,
subtype_rel_self,
grp_car_wf,
imon_wf,
select_tl,
imon_subtype_grp_sig,
itermAdd_wf,
int_term_value_add_lemma,
add-associates,
add-swap,
add-commutes,
zero-add,
mon_itop_shift
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
hypothesis,
inhabitedIsType,
unionElimination,
equalityElimination,
baseClosed,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
universeIsType,
hypothesisEquality,
imageElimination,
voidElimination,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
rename,
setElimination,
baseApply,
closedConclusion,
applyEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
independent_pairFormation,
functionIsType,
setIsType,
functionEquality,
universeEquality,
imageMemberEquality,
instantiate,
dependent_set_memberEquality_alt,
productIsType,
addEquality
Latex:
\mforall{}T:Type. \mforall{}n:\mBbbN{}. \mforall{}as:T List. ((n \mleq{} ||as||) {}\mRightarrow{} (nth\_tl(n;as) = (\mPi{} n \mleq{} i < ||as||. [as[i]])))
Date html generated:
2019_10_16-PM-01_05_32
Last ObjectModification:
2018_10_08-AM-10_53_40
Theory : list_2
Home
Index