Nuprl Lemma : respects-equality-list-type
∀[A,B:Type]. ∀L:A List. ((∀i:ℕ||L||. (L[i] ∈ B)) ⇒ (L ∈ B List)) supposing respects-equality(A;B)
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
top: Top,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
subtract: n - m,
subtype_rel: A ⊆r B,
select: L[n],
int_seg: {i..j-},
lelt: i ≤ j < k,
respects-equality: respects-equality(S;T),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
respects-equality_wf,
istype-universe,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
istype-void,
colength_wf_list,
istype-false,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
add-associates,
istype-int,
add-commutes,
add-swap,
zero-add,
le_weakening2,
istype-nat,
list_wf,
length_of_nil_lemma,
stuck-spread,
istype-base,
nil_wf,
int_seg_wf,
cons_wf,
length_wf,
select_wf,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
add-member-int_seg2,
decidable__le,
subtract_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
non_neg_length,
istype-sqequal,
decidable__lt,
select-cons-tl,
not-lt-2,
le-add-cancel,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
instantiate,
universeEquality,
Error :lambdaFormation_alt,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :functionIsTypeImplies,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
Error :isect_memberEquality_alt,
Error :equalityIstype,
because_Cache,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
cumulativity,
intEquality,
closedConclusion,
imageElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
applyEquality,
minusEquality,
baseApply,
sqequalBase,
Error :functionIsType,
Error :productIsType,
addEquality,
Error :dependent_pairFormation_alt
Latex:
\mforall{}[A,B:Type]. \mforall{}L:A List. ((\mforall{}i:\mBbbN{}||L||. (L[i] \mmember{} B)) {}\mRightarrow{} (L \mmember{} B List)) supposing respects-equality(A;B)
Date html generated:
2019_06_20-PM-00_40_35
Last ObjectModification:
2018_11_28-PM-00_32_55
Theory : list_0
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