Nuprl Lemma : norm-pair_wf
∀[A:Type]. ∀[B:A ⟶ Type].
(∀[Na:id-fun(A)]. ∀[Nb:⋂a:A. id-fun(B[a])]. (norm-pair(Na;Nb) ∈ id-fun(a:A × B[a]))) supposing
((∀a:A. value-type(B[a])) and
value-type(A))
Proof
Definitions occuring in Statement :
norm-pair: norm-pair(Na;Nb),
id-fun: id-fun(T),
value-type: value-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
member: t ∈ T,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
has-value: (a)↓,
norm-pair: norm-pair(Na;Nb),
top: Top,
id-fun: id-fun(T),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
value-type_wf,
all_wf,
id-fun_wf,
and_wf,
subtype_rel-equal,
set_wf,
set-value-type,
equal_wf,
value-type-has-value,
top_wf
Rules used in proof :
universeEquality,
cumulativity,
because_Cache,
axiomEquality,
independent_functionElimination,
dependent_functionElimination,
productEquality,
applyLambdaEquality,
independent_pairFormation,
dependent_set_memberEquality,
dependent_pairEquality,
rename,
setElimination,
lambdaFormation,
functionEquality,
isectEquality,
equalitySymmetry,
equalityTransitivity,
applyEquality,
lambdaEquality,
independent_isectElimination,
hypothesisEquality,
setEquality,
isectElimination,
callbyvalueReduce,
productElimination,
thin,
hypothesis,
extract_by_obid,
voidEquality,
voidElimination,
isect_memberEquality,
functionExtensionality,
sqequalRule,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
(\mforall{}[Na:id-fun(A)]. \mforall{}[Nb:\mcap{}a:A. id-fun(B[a])]. (norm-pair(Na;Nb) \mmember{} id-fun(a:A \mtimes{} B[a]))) supposing
((\mforall{}a:A. value-type(B[a])) and
value-type(A))
Date html generated:
2018_07_25-PM-01_29_53
Last ObjectModification:
2018_07_14-PM-01_22_37
Theory : call!by!value_2
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