Nuprl Lemma : free-dist-lattice_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. (free-dist-lattice(T; eq) ∈ BoundedDistributiveLattice)
Proof
Definitions occuring in Statement :
free-dist-lattice: free-dist-lattice(T; eq),
bdd-distributive-lattice: BoundedDistributiveLattice,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
free-dist-lattice: free-dist-lattice(T; eq),
prop: ℙ,
all: ∀x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
fset-antichain: fset-antichain(eq;ac),
fset-pairwise: fset-pairwise(x,y.R[x; y];s),
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,
empty-fset: {},
nil: [],
it: ⋅,
btrue: tt,
true: True,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
implies: P ⇒ Q,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
guard: {T},
top: Top
Lemmas referenced :
mk-bounded-distributive-lattice-from-order,
fset_wf,
assert_wf,
fset-antichain_wf,
fset-ac-glb_wf,
fset-ac-lub_wf,
empty-fset_wf,
fset-antichain-singleton,
fset-singleton_wf,
fset-ac-le_wf,
fset-ac-order,
fset-ac-lub-is-lub,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
set_wf,
fset-ac-glb-is-glb,
fset-ac-le-singleton-empty,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
equal_wf,
squash_wf,
true_wf,
fset-ac-le-distributive,
iff_weakening_equal,
deq_wf,
empty-fset-ac-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
sqequalRule,
independent_isectElimination,
independent_pairFormation,
productElimination,
independent_pairEquality,
applyEquality,
independent_functionElimination,
isect_memberEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
functionEquality,
functionExtensionality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
axiomEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. (free-dist-lattice(T; eq) \mmember{} BoundedDistributiveLattice)
Date html generated:
2020_05_20-AM-08_44_57
Last ObjectModification:
2017_07_28-AM-09_14_21
Theory : lattices
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