Nuprl Lemma : select_listify_id
∀[T:Type]. ∀[n:ℕ]. ∀[f:ℕn ⟶ T]. ∀[i:ℕn]. ((f)[ℕn][i] = (f i) ∈ T)
Proof
Definitions occuring in Statement :
select: L[n],
listify: listify(f;m;n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
sq_type: SQType(T),
not: ¬A,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
guard: {T},
lelt: i ≤ j < k,
false: False,
or: P ∨ Q,
le: A ≤ B,
int_seg: {i..j-},
and: P ∧ Q,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
int_lower: {...i},
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
decidable: Dec(P),
listify: listify(f;m;n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
select: L[n],
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P)
Lemmas referenced :
int_seg_wf,
nat_wf,
not-lt-2,
subtype_base_sq,
minus-add,
le-add-cancel-alt,
mul-commutes,
less-iff-le,
le-add-cancel,
mul-associates,
mul-distributes,
less_than_wf,
omega-shadow,
mul-distributes-right,
two-mul,
zero-mul,
add-mul-special,
add-commutes,
add-swap,
zero-add,
one-mul,
add-zero,
minus-zero,
add-associates,
minus-one-mul-top,
add_functionality_wrt_le,
not-le-2,
minus-one-mul,
int_subtype_base,
set_subtype_base,
add-is-int-iff,
int_lower_wf,
lelt_wf,
le_transitivity,
base_wf,
subtype_rel-equal,
less_than_irreflexivity,
less_than_transitivity1,
le_weakening2,
less_than_transitivity2,
length_wf_nat,
listify_length,
non_neg_length,
subtract_wf,
le_reflexive,
int_seg_subtype,
subtype_rel_dep_function,
listify_wf,
select_wf,
equal_wf,
all_wf,
le_wf,
int_lower_ind,
int_seg_properties,
int_lower_properties,
nat_properties,
decidable__le,
decidable__lt,
le_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
lt_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
stuck-spread,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
decidable__int_equal,
iff_weakening_equal,
condition-implies-le,
false_wf,
select_cons_hd,
true_wf,
squash_wf,
subtype_rel_self,
not-equal-implies-less,
select_cons_tl,
mul-swap,
minus-minus,
le-add-cancel2,
not-equal-2,
sq_stable__le
Rules used in proof :
Error :universeIsType,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
Error :functionIsType,
functionEquality,
universeEquality,
Error :isect_memberFormation_alt,
sqequalRule,
isect_memberEquality,
axiomEquality,
imageMemberEquality,
voidEquality,
minusEquality,
multiplyEquality,
intEquality,
baseClosed,
closedConclusion,
baseApply,
addEquality,
dependent_set_memberEquality,
functionExtensionality,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
sqequalIntensionalEquality,
dependent_pairFormation,
voidElimination,
independent_functionElimination,
unionElimination,
productElimination,
lambdaFormation,
independent_pairFormation,
independent_isectElimination,
applyEquality,
cumulativity,
lambdaEquality,
dependent_functionElimination,
instantiate,
equalityElimination,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} T]. \mforall{}[i:\mBbbN{}n]. ((f)[\mBbbN{}n][i] = (f i))
Date html generated:
2019_06_20-PM-00_40_58
Last ObjectModification:
2018_09_26-PM-02_18_36
Theory : list_0
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