Nuprl Lemma : equiv-on-quotient
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
(EquivRel(T;x,y.x R y)
⇒ (∀Q:(x,y:T//(x R y)) ⟶ (x,y:T//(x R y)) ⟶ ℙ. (EquivRel(x,y:T//(x R y);u,v.u Q v) ⇒ EquivRel(T;x,y.x Q y))))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
uimplies: b supposing a,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
prop: ℙ
Lemmas referenced :
subtype_quotient,
equiv_rel_wf,
quotient_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
independent_pairFormation,
productElimination,
thin,
promote_hyp,
cut,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
functionExtensionality,
cumulativity,
independent_isectElimination,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(EquivRel(T;x,y.x R y)
{}\mRightarrow{} (\mforall{}Q:(x,y:T//(x R y)) {}\mrightarrow{} (x,y:T//(x R y)) {}\mrightarrow{} \mBbbP{}
(EquivRel(x,y:T//(x R y);u,v.u Q v) {}\mRightarrow{} EquivRel(T;x,y.x Q y))))
Date html generated:
2016_10_21-AM-09_44_00
Last ObjectModification:
2016_08_08-PM-08_46_13
Theory : quot_1
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