Nuprl Lemma : sq_stable__connex
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ((∀x,y:T. Dec(R[x;y])) ⇒ SqStable(Connex(T;x,y.R[x;y])))
Proof
Definitions occuring in Statement :
connex: Connex(T;x,y.R[x; y]),
sq_stable: SqStable(P),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
connex: Connex(T;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
prop: ℙ
Lemmas referenced :
sq_stable__all,
all_wf,
or_wf,
sq_stable_from_decidable,
decidable__or,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
hypothesis,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
Error :functionIsType,
Error :universeIsType,
Error :inhabitedIsType,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x,y:T. Dec(R[x;y])) {}\mRightarrow{} SqStable(Connex(T;x,y.R[x;y])))
Date html generated:
2019_06_20-PM-00_29_44
Last ObjectModification:
2018_09_26-AM-11_51_41
Theory : rel_1
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