Nuprl Lemma : lub-com
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀[lub:T ⟶ T ⟶ T]
∀[a,b:T]. ((lub a b) = (lub b a) ∈ T) supposing ∀[a,b:T]. least-upper-bound(T;x,y.R[x;y];a;b;lub a b)
supposing Order(T;x,y.R[x;y])
Proof
Definitions occuring in Statement :
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
order: Order(T;x,y.R[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
least-upper-bound-com,
uall_wf,
least-upper-bound_wf,
order_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
lemma_by_obid,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}[lub:T {}\mrightarrow{} T {}\mrightarrow{} T]
\mforall{}[a,b:T]. ((lub a b) = (lub b a))
supposing \mforall{}[a,b:T]. least-upper-bound(T;x,y.R[x;y];a;b;lub a b)
supposing Order(T;x,y.R[x;y])
Date html generated:
2016_05_13-PM-04_18_15
Last ObjectModification:
2015_12_26-AM-11_27_31
Theory : rel_1
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