Nuprl Lemma : per-quot_elim
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. (EquivRel(T;x,y.E[x;y]) ⇒ (∀a,b:T. (a = b ∈ (x,y:T/per/E[x;y]) ⇐⇒ ↓E[a;b])))
Proof
Definitions occuring in Statement :
per-quotient: x,y:T/per/E[x; y],
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
per-quotient: x,y:T/per/E[x; y],
quotient: x,y:A//B[x; y]
Lemmas referenced :
quot_elim
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalRule,
sqequalReflexivity,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
hypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (EquivRel(T;x,y.E[x;y]) {}\mRightarrow{} (\mforall{}a,b:T. (a = b \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}E[a;b])))
Date html generated:
2019_06_20-PM-00_33_33
Last ObjectModification:
2018_08_21-PM-10_54_14
Theory : per-quotient
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