Nuprl Lemma : firstn_factor
∀T:Type. ∀as:T List. ∀n:{0...||as||}. (firstn(n;as) = (Π 0 ≤ i < n. [as[i]]) ∈ (T List))
Proof
Definitions occuring in Statement :
lapp_imon: <T List,@>,
firstn: firstn(n;as),
select: L[n],
length: ||as||,
cons: [a / b],
nil: [],
list: T List,
int_iseg: {i...j},
all: ∀x:A. B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
mon_itop: Π lb ≤ i < ub. E[i]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
int_iseg: {i...j},
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
and: P ∧ Q,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
subtype_rel: A ⊆r B,
so_apply: x[s],
firstn: firstn(n;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
sq_type: SQType(T),
mon_itop: Π lb ≤ i < ub. E[i],
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
grp_id: e,
pi1: fst(t),
pi2: snd(t),
lapp_imon: <T List,@>,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
true: True,
ge: i ≥ j ,
grp_car: |g|,
imon: IMonoid,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y,
grp_op: *,
append: as @ bs,
subtract: n - m
Lemmas referenced :
list_induction,
all_wf,
int_iseg_wf,
length_wf,
equal_wf,
list_wf,
firstn_wf,
mon_itop_wf,
lapp_imon_wf,
cons_wf,
select_wf,
int_seg_properties,
int_iseg_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,
int_formula_prop_less_lemma,
nil_wf,
int_seg_wf,
length_of_nil_lemma,
list_ind_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
lt_int_wf,
bool_wf,
assert_wf,
less_than_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
list_ind_cons_lemma,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
non_neg_length,
grp_car_wf,
imon_wf,
squash_wf,
true_wf,
iff_weakening_equal,
mon_itop_unroll_lo,
grp_op_wf,
select_cons_hd,
select_cons_tl,
mon_itop_shift,
minus-minus,
add-associates,
add-swap,
add-commutes,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
cumulativity,
hypothesis,
because_Cache,
setElimination,
rename,
dependent_functionElimination,
independent_isectElimination,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
independent_functionElimination,
baseClosed,
addEquality,
universeEquality,
instantiate,
productEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
imageElimination,
imageMemberEquality,
functionEquality,
minusEquality
Latex:
\mforall{}T:Type. \mforall{}as:T List. \mforall{}n:\{0...||as||\}. (firstn(n;as) = (\mPi{} 0 \mleq{} i < n. [as[i]]))
Date html generated:
2017_10_01-AM-09_57_32
Last ObjectModification:
2017_03_03-PM-00_59_25
Theory : list_2
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