Nuprl Lemma : equiv_rel_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])))
  
⇒ EquivRel(x,y:T/per/(↑E2[x;y]);x,y.↑E1[x;y]))
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
, 
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_quotient
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
cut, 
introduction, 
extract_by_obid, 
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{}  EquivRel(x,y:T/per/(\muparrow{}E2[x;y]);x,y.\muparrow{}E1[x;y]))
Date html generated:
2019_06_20-PM-00_33_41
Last ObjectModification:
2018_08_21-PM-10_49_29
Theory : per-quotient
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