Nuprl Lemma : length_firstn
∀[A:Type]. ∀[as:A List]. ∀[n:{0...||as||}]. (||firstn(n;as)|| ~ n)
Proof
Definitions occuring in Statement :
firstn: firstn(n;as),
length: ||as||,
list: T List,
int_iseg: {i...j},
uall: ∀[x:A]. B[x],
natural_number: $n,
universe: Type,
sqequal: s ~ 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 ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
nil: [],
it: ⋅,
less_than: a < b,
int_iseg: {i...j},
cand: A c∧ B,
firstn: firstn(n;as),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bfalse: ff
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
int_iseg_wf,
length_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
list-cases,
length_of_nil_lemma,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
equal_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
length_of_cons_lemma,
list_wf,
lt_int_wf,
bool_wf,
assert_wf,
le_int_wf,
bnot_wf,
decidable__equal_int,
not-equal-2,
minus-zero,
le_reflexive,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
list_ind_nil_lemma,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
cumulativity,
applyEquality,
because_Cache,
unionElimination,
instantiate,
intEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
productElimination,
voidEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
universeEquality,
productEquality,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[as:A List]. \mforall{}[n:\{0...||as||\}]. (||firstn(n;as)|| \msim{} n)
Date html generated:
2017_04_14-AM-08_47_34
Last ObjectModification:
2017_02_27-PM-03_35_03
Theory : list_0
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