Nuprl Lemma : compact-product
∀[T:Type]. ∀[S:T ⟶ Type]. (compact-type(T) ⇒ (∀t:T. compact-type(S[t])) ⇒ compact-type(t:T × S[t]))
Proof
Definitions occuring in Statement :
compact-type: compact-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],
implies: P ⇒ Q,
compact-type: compact-type(T),
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
isr: isr(x),
not: ¬A,
false: False,
guard: {T},
squash: ↓T,
true: True,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bool_wf,
all_wf,
compact-type_wf,
isr_wf,
exists_wf,
equal-wf-T-base,
or_wf,
equal-wf-base,
btrue_neq_bfalse,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
functionEquality,
productEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
thin,
lambdaEquality,
universeEquality,
rename,
dependent_functionElimination,
because_Cache,
unionElimination,
productElimination,
inlFormation,
dependent_pairEquality,
baseClosed,
independent_functionElimination,
voidElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
inrFormation,
imageElimination,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[S:T {}\mrightarrow{} Type].
(compact-type(T) {}\mRightarrow{} (\mforall{}t:T. compact-type(S[t])) {}\mRightarrow{} compact-type(t:T \mtimes{} S[t]))
Date html generated:
2017_10_01-AM-08_28_57
Last ObjectModification:
2017_07_26-PM-04_23_44
Theory : basic
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