Nuprl Lemma : free-dist-lattice-with-constraints_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[Cs:T ⟶ fset(fset(T))].
(free-dist-lattice-with-constraints(T;eq;x.Cs[x]) ∈ BoundedDistributiveLattice)
Proof
Definitions occuring in Statement :
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
bdd-distributive-lattice: BoundedDistributiveLattice,
fset: fset(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
assert: ↑b,
ifthenelse: if b then t else f fi ,
fset-contains-none: fset-contains-none(eq;s;x.Cs[x]),
fset-contains-none-of: fset-contains-none-of(eq;s;cs),
fset-null: fset-null(s),
null: null(as),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
empty-fset: {},
nil: [],
it: ⋅,
btrue: tt,
true: True
Lemmas referenced :
constrained-antichain-lattice_wf,
fset-contains-none_wf,
fset_wf,
fset-contains-none-closed-downward,
assert_wf,
f-subset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
because_Cache,
applyEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[Cs:T {}\mrightarrow{} fset(fset(T))].
(free-dist-lattice-with-constraints(T;eq;x.Cs[x]) \mmember{} BoundedDistributiveLattice)
Date html generated:
2020_05_20-AM-08_48_15
Last ObjectModification:
2015_12_28-PM-01_59_06
Theory : lattices
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