Nuprl Lemma : confluent-equiv-is-equiv
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(Refl(T;x,y.R[x;y])
⇒ Trans(T;x,y.R[x;y])
⇒ rel-confluent(T;x,y.R[x;y])
⇒ EquivRel(T;a,b.confluent-equiv(T;x,y.R[x;y]) a b))
Proof
Definitions occuring in Statement :
confluent-equiv: confluent-equiv(T;x,y.R[x; y]),
rel-confluent: rel-confluent(T;x,y.R[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[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]),
all: ∀x:A. B[x],
confluent-equiv: confluent-equiv(T;x,y.R[x; y]),
exists: ∃x:A. B[x],
member: t ∈ T,
cand: A c∧ B,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
prop: ℙ,
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
so_lambda: λ2x y.t[x; y],
guard: {T},
rel-confluent: rel-confluent(T;x,y.R[x; y])
Lemmas referenced :
subtype_rel_self,
rel-confluent_wf,
trans_wf,
refl_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
sqequalRule,
dependent_pairFormation_alt,
hypothesisEquality,
cut,
hypothesis,
productIsType,
universeIsType,
applyEquality,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
productElimination,
lambdaEquality_alt,
inhabitedIsType,
functionIsType,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Refl(T;x,y.R[x;y])
{}\mRightarrow{} Trans(T;x,y.R[x;y])
{}\mRightarrow{} rel-confluent(T;x,y.R[x;y])
{}\mRightarrow{} EquivRel(T;a,b.confluent-equiv(T;x,y.R[x;y]) a b))
Date html generated:
2019_10_15-AM-10_24_48
Last ObjectModification:
2019_08_16-PM-03_12_27
Theory : relations2
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