Nuprl Lemma : equiv_rel-wf-per-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])))
⇒ (E1 ∈ (x,y:T/per/(↑E2[x;y])) ⟶ (x,y:T/per/(↑E2[x;y])) ⟶ 𝔹))
Proof
Definitions occuring in Statement :
per-quotient: x,y:T/per/E[x; y],
equiv_rel: EquivRel(T;x,y.E[x; y]),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
per-quotient: x,y:T/per/E[x; y],
quotient: x,y:A//B[x; y]
Lemmas referenced :
equiv_rel-wf-quotient
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalRule,
sqequalReflexivity,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
hypothesis
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{} (E1 \mmember{} (x,y:T/per/(\muparrow{}E2[x;y])) {}\mrightarrow{} (x,y:T/per/(\muparrow{}E2[x;y])) {}\mrightarrow{} \mBbbB{}))
Date html generated:
2019_06_20-PM-00_33_40
Last ObjectModification:
2018_08_21-PM-10_50_34
Theory : per-quotient
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