Nuprl Lemma : all_quot_elim
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ].
(EquivRel(T;x,y.E[x;y])
⇒ (∀[F:(x,y:T//E[x;y]) ⟶ ℙ]. ((∀w:x,y:T//E[x;y]. SqStable(F w)) ⇒ (∀z:x,y:T//E[x;y]. F[z] ⇐⇒ ∀z:T. F[z]))))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
quotient: x,y:A//B[x; y],
squash: ↓T,
guard: {T},
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_apply: x[s1;s2]
Lemmas referenced :
squash_wf,
equal_wf,
equal-wf-base,
subtype_rel_self,
all_wf,
quotient_wf,
subtype_quotient,
sq_stable_wf,
equiv_rel_wf
Rules used in proof :
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
rename,
imageMemberEquality,
baseClosed,
productEquality,
instantiate,
independent_functionElimination,
dependent_functionElimination,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
applyEquality,
independent_isectElimination,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(EquivRel(T;x,y.E[x;y])
{}\mRightarrow{} (\mforall{}[F:(x,y:T//E[x;y]) {}\mrightarrow{} \mBbbP{}]
((\mforall{}w:x,y:T//E[x;y]. SqStable(F w)) {}\mRightarrow{} (\mforall{}z:x,y:T//E[x;y]. F[z] \mLeftarrow{}{}\mRightarrow{} \mforall{}z:T. F[z]))))
Date html generated:
2019_06_20-PM-01_05_18
Last ObjectModification:
2019_06_20-PM-00_59_36
Theory : quot_1
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