Nuprl Lemma : partial-type-continuous
∀[X:ℕ ⟶ {T:Type| value-type(T)} ]. ((⋂n:ℕ. partial(X[n])) ⊆r partial(⋂n:ℕ. X[n]))
Proof
Definitions occuring in Statement :
partial: partial(T),
nat: ℕ,
value-type: value-type(T),
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
base-partial: base-partial(T),
partial: partial(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
not: ¬A,
false: False,
guard: {T},
quotient: x,y:A//B[x; y],
per-partial: per-partial(T;x;y),
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
uiff: uiff(P;Q),
has-value: (a)↓,
squash: ↓T,
true: True,
sq_stable: SqStable(P)
Lemmas referenced :
istype-nat,
partial_wf,
istype-universe,
value-type_wf,
istype-base,
base-partial_wf,
nat_wf,
per-partial_wf,
per-partial-equiv_rel,
quotient-member-eq,
has-value_wf_base,
is-exception_wf,
istype-void,
le_wf,
istype-false,
partial-not-exception,
member_wf,
squash_wf,
true_wf,
equal-wf-base,
false_wf,
equal_functionality_wrt_subtype_rel2,
sq_stable__value-type,
termination-equality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
isectIsType,
extract_by_obid,
hypothesis,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
setElimination,
rename,
setIsType,
instantiate,
universeEquality,
sqequalRule,
axiomEquality,
functionIsType,
lambdaFormation_alt,
equalityIstype,
sqequalBase,
equalitySymmetry,
isectEquality,
because_Cache,
inhabitedIsType,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
isect_memberEquality_alt,
equalityTransitivity,
independent_pairFormation,
productIsType,
pertypeElimination,
promote_hyp,
productElimination,
equalityElimination,
equalityIsType1,
natural_numberEquality,
axiomSqleEquality,
pointwiseFunctionality,
imageElimination,
imageMemberEquality,
baseClosed,
productEquality,
functionExtensionality,
lambdaEquality,
setEquality,
cumulativity,
dependent_set_memberEquality,
lambdaFormation
Latex:
\mforall{}[X:\mBbbN{} {}\mrightarrow{} \{T:Type| value-type(T)\} ]. ((\mcap{}n:\mBbbN{}. partial(X[n])) \msubseteq{}r partial(\mcap{}n:\mBbbN{}. X[n]))
Date html generated:
2020_05_19-PM-09_36_42
Last ObjectModification:
2020_01_04-PM-07_57_15
Theory : partial_1
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