Nuprl Lemma : equiv_rel_quotient
∀[T:Type]. ∀[E1,E2:T ⟶ T ⟶ 𝔹].
(EquivRel(T;x,y.↑E2[x;y])
⇒ EquivRel(T;x,y.↑E1[x;y])
⇒ (∀x,y:T. ((↑E2[x;y]) ⇒ (↑E1[x;y])))
⇒ EquivRel(x,y:T//(↑E2[x;y]);x,y.↑E1[x;y]))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
cand: A c∧ B,
sym: Sym(T;x,y.E[x; y]),
prop: ℙ,
trans: Trans(T;x,y.E[x; y]),
so_lambda: λ2x.t[x],
so_apply: x[s],
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
equiv_rel-wf-quotient,
quotient_wf,
assert_wf,
assert_witness,
all_wf,
equiv_rel_wf,
bool_wf,
subtype_quotient,
equal-wf-base,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
independent_pairFormation,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
because_Cache,
functionEquality,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
productEquality,
dependent_functionElimination,
rename
Latex:
\mforall{}[T:Type]. \mforall{}[E1,E2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}].
(EquivRel(T;x,y.\muparrow{}E2[x;y])
{}\mRightarrow{} EquivRel(T;x,y.\muparrow{}E1[x;y])
{}\mRightarrow{} (\mforall{}x,y:T. ((\muparrow{}E2[x;y]) {}\mRightarrow{} (\muparrow{}E1[x;y])))
{}\mRightarrow{} EquivRel(x,y:T//(\muparrow{}E2[x;y]);x,y.\muparrow{}E1[x;y]))
Date html generated:
2017_04_14-AM-07_39_44
Last ObjectModification:
2017_02_27-PM-03_12_30
Theory : quot_1
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