Nuprl Lemma : product-valueall-type
∀[A:Type]. ∀[B:A ⟶ Type]. (valueall-type(A) ⇒ (∀a:A. valueall-type(B[a])) ⇒ valueall-type(a:A × B[a]))
Proof
Definitions occuring in Statement :
valueall-type: valueall-type(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
valueall-type: valueall-type(T),
uimplies: b supposing a,
sq_stable: SqStable(P),
all: ∀x:A. B[x],
top: Top,
has-value: (a)↓,
has-valueall: has-valueall(a),
so_apply: x[s],
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
guard: {T}
Lemmas referenced :
sq_stable__has-value,
evalall-pair,
valueall-type-has-valueall,
has-value_wf_base,
is-exception_wf,
equal_wf,
equal-wf-base,
base_wf,
all_wf,
valueall-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
hypothesis,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
callbyvalueReduce,
independent_isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
divergentSqle,
sqleReflexivity,
dependent_functionElimination,
imageMemberEquality,
imageElimination,
productEquality,
lambdaEquality,
axiomSqleEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type].
(valueall-type(A) {}\mRightarrow{} (\mforall{}a:A. valueall-type(B[a])) {}\mRightarrow{} valueall-type(a:A \mtimes{} B[a]))
Date html generated:
2017_04_14-AM-07_14_54
Last ObjectModification:
2017_02_27-PM-02_50_35
Theory : call!by!value_1
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