Nuprl Lemma : up-set-lattice_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[whole:fset(T)]. ∀[le:T ⟶ T ⟶ ℙ].
up-set-lattice(T;eq;whole;x,y.le[x;y]) ∈ BoundedDistributiveLattice supposing (∀x:T. x ∈ whole) ∧ Trans(T;x,y.le[x;y])
Proof
Definitions occuring in Statement :
up-set-lattice: up-set-lattice(T;eq;whole;x,y.le[x; y]),
bdd-distributive-lattice: BoundedDistributiveLattice,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
trans: Trans(T;x,y.E[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
up-set-lattice: up-set-lattice(T;eq;whole;x,y.le[x; y]),
and: P ∧ Q,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_apply: x[s],
cand: A c∧ B,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
false: False,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
sub-powerset-lattice_wf,
all_wf,
fset-member_wf,
fset_wf,
or_wf,
member-fset-union,
fset-union_wf,
and_wf,
member-fset-intersection,
fset-intersection_wf,
mem_empty_lemma,
false_wf,
trans_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
because_Cache,
functionEquality,
hypothesis,
applyEquality,
universeEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
unionElimination,
inlFormation,
inrFormation,
addLevel,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[whole:fset(T)]. \mforall{}[le:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
up-set-lattice(T;eq;whole;x,y.le[x;y]) \mmember{} BoundedDistributiveLattice
supposing (\mforall{}x:T. x \mmember{} whole) \mwedge{} Trans(T;x,y.le[x;y])
Date html generated:
2020_05_20-AM-08_47_43
Last ObjectModification:
2015_12_28-PM-02_00_32
Theory : lattices
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