Nuprl Lemma : isect2-record
∀[T1,T2:Atom ⟶ Type].
((record(x.T1[x]) ⋂ record(x.T2[x]) ⊆r record(x.T1[x] ⋂ T2[x]))
∧ (record(x.T1[x] ⋂ T2[x]) ⊆r record(x.T1[x]) ⋂ record(x.T2[x])))
Proof
Definitions occuring in Statement :
record: record(x.T[x]),
isect2: T1 ⋂ T2,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
function: x:A ⟶ B[x],
atom: Atom,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
record: record(x.T[x]),
isect2: T1 ⋂ T2,
subtype_rel: A ⊆r B,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff
Lemmas referenced :
subtype_rel_dep_function,
isect2_wf,
isect2_subtype_rel,
isect2_subtype_rel2,
bool_wf,
isect2_decomp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
lambdaEquality,
isect_memberEquality,
hypothesisEquality,
applyEquality,
sqequalHypSubstitution,
unionElimination,
thin,
lemma_by_obid,
isectElimination,
atomEquality,
because_Cache,
independent_isectElimination,
lambdaFormation,
functionEquality,
productElimination,
independent_pairEquality,
axiomEquality,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
equalityElimination
Latex:
\mforall{}[T1,T2:Atom {}\mrightarrow{} Type].
((record(x.T1[x]) \mcap{} record(x.T2[x]) \msubseteq{}r record(x.T1[x] \mcap{} T2[x]))
\mwedge{} (record(x.T1[x] \mcap{} T2[x]) \msubseteq{}r record(x.T1[x]) \mcap{} record(x.T2[x])))
Date html generated:
2016_05_15-PM-06_43_24
Last ObjectModification:
2015_12_27-PM-00_10_08
Theory : general
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