Nuprl Lemma : constrained-antichain-lattice_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:fset(T) ⟶ 𝔹].
constrained-antichain-lattice(T;eq;P) ∈ BoundedDistributiveLattice
supposing (∀x,y:fset(T). (y ⊆ x ⇒ (↑(P x)) ⇒ (↑(P y)))) ∧ (↑(P {}))
Proof
Definitions occuring in Statement :
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
bdd-distributive-lattice: BoundedDistributiveLattice,
empty-fset: {},
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
guard: {T},
squash: ↓T,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
fset-all: fset-all(s;x.P[x]),
true: True,
btrue: tt,
it: ⋅,
nil: [],
empty-fset: {},
list_ind: list_ind,
reduce: reduce(f;k;as),
filter: filter(P;l),
fset-filter: {x ∈ s | P[x]},
null: null(as),
fset-null: fset-null(s),
fset-pairwise: fset-pairwise(x,y.R[x; y];s),
fset-antichain: fset-antichain(eq;ac),
ifthenelse: if b then t else f fi ,
assert: ↑b,
cand: A c∧ B,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
top: Top
Lemmas referenced :
deq_wf,
iff_weakening_equal,
fset-ac-le-distributive-constrained,
true_wf,
squash_wf,
equal_wf,
f-subset_wf,
iff_wf,
all_wf,
bool_wf,
fset-ac-le-singleton-empty,
set_wf,
deq-f-subset_wf,
bnot_wf,
fset-filter_wf,
fset-null_wf,
assert_witness,
fset-ac-order-constrained,
fset-ac-le_wf,
empty-fset_wf,
fset-constrained-ac-lub_wf,
fset-constrained-ac-glb_wf,
fset-all_wf,
fset-antichain_wf,
assert_wf,
fset_wf,
mk-bounded-distributive-lattice-from-order,
fset-member_wf,
member-fset-singleton,
deq-fset_wf,
fset-all-iff,
fset-antichain-singleton,
fset-singleton_wf,
fset-constrained-ac-lub-is-lub,
fset-constrained-ac-glb-is-glb,
empty-fset-ac-le
Rules used in proof :
axiomEquality,
universeEquality,
baseClosed,
imageMemberEquality,
imageElimination,
functionEquality,
equalitySymmetry,
equalityTransitivity,
instantiate,
isect_memberEquality,
independent_functionElimination,
independent_pairEquality,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
because_Cache,
rename,
setElimination,
lambdaFormation,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
productEquality,
hypothesis,
hypothesisEquality,
cumulativity,
setEquality,
isectElimination,
extract_by_obid,
thin,
productElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
applyLambdaEquality,
hyp_replacement,
voidEquality,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:fset(T) {}\mrightarrow{} \mBbbB{}].
constrained-antichain-lattice(T;eq;P) \mmember{} BoundedDistributiveLattice
supposing (\mforall{}x,y:fset(T). (y \msubseteq{} x {}\mRightarrow{} (\muparrow{}(P x)) {}\mRightarrow{} (\muparrow{}(P y)))) \mwedge{} (\muparrow{}(P \{\}))
Date html generated:
2020_05_20-AM-08_47_51
Last ObjectModification:
2020_02_04-PM-02_01_54
Theory : lattices
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