Nuprl Lemma : compact-type-compact-type2
∀[T:Type]. (compact-type(T) ∧ T ⇐⇒ compact-type2(T))
Proof
Definitions occuring in Statement :
compact-type2: compact-type2(T),
compact-type: compact-type(T),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
compact-type: compact-type(T),
compact-type2: compact-type2(T),
p-selector: p-selector(T;x;p),
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
not: ¬A,
false: False,
sq_stable: SqStable(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
top: Top
Lemmas referenced :
compact-type_wf,
compact-type2_wf,
bool_wf,
exists_wf,
equal-wf-T-base,
squash_wf,
true_wf,
equal_wf,
bfalse_wf,
iff_weakening_equal,
btrue_neq_bfalse,
sq_stable__all,
sq_stable__equal,
all_wf,
btrue_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
hypothesis,
universeEquality,
dependent_functionElimination,
unionElimination,
functionEquality,
dependent_set_memberFormation,
lambdaEquality,
applyEquality,
functionExtensionality,
baseClosed,
because_Cache,
rename,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
setElimination,
axiomEquality,
equalityElimination,
inlFormation,
dependent_pairFormation,
inrFormation,
equalityUniverse,
levelHypothesis,
isect_memberEquality,
voidEquality
Latex:
\mforall{}[T:Type]. (compact-type(T) \mwedge{} T \mLeftarrow{}{}\mRightarrow{} compact-type2(T))
Date html generated:
2017_10_01-AM-08_28_49
Last ObjectModification:
2017_07_26-PM-04_23_41
Theory : basic
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