Nuprl Lemma : extend-type-property
∀[T:Type]. ((T ⊆r (T)+) ∧ respects-equality((T)+;T) ∧ (∀X:Type. (respects-equality(X;T) ⇒ (X ⊆r (T)+))))
Proof
Definitions occuring in Statement :
extend-type: (T)+,
subtype_rel: A ⊆r B,
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
extend-type: (T)+,
so_lambda: λ2x y.t[x; y],
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
cand: A c∧ B,
respects-equality: respects-equality(S;T),
quotient: x,y:A//B[x; y]
Lemmas referenced :
istype-universe,
extend-type_wf,
quotient-member-eq,
base_wf,
iff_wf,
equal-wf-base,
equal-wf-T-base,
extend-type-equiv,
istype-base,
respects-equality_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
independent_pairFormation,
cut,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
universeEquality,
hypothesis,
Error :lambdaEquality_alt,
Error :universeIsType,
hypothesisEquality,
pointwiseFunctionalityForEquality,
sqequalRule,
productEquality,
because_Cache,
functionEquality,
Error :inhabitedIsType,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
Error :lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
sqequalBase,
productElimination,
pertypeElimination,
promote_hyp,
Error :productIsType,
Error :functionIsType,
axiomEquality
Latex:
\mforall{}[T:Type]
((T \msubseteq{}r (T)+) \mwedge{} respects-equality((T)+;T) \mwedge{} (\mforall{}X:Type. (respects-equality(X;T) {}\mRightarrow{} (X \msubseteq{}r (T)+))))
Date html generated:
2019_06_20-PM-00_33_29
Last ObjectModification:
2018_11_25-PM-06_55_25
Theory : quot_1
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