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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
cand: A c∧ B,
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,
fset-all: fset-all(s;x.P[x]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
subtype_rel: A ⊆r B,
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},
uiff: uiff(P;Q),
top: Top
Lemmas referenced :
mk-bounded-distributive-lattice-from-order,
fset_wf,
assert_wf,
fset-antichain_wf,
fset-all_wf,
fset-constrained-ac-glb_wf,
fset-constrained-ac-lub_wf,
empty-fset_wf,
fset-ac-le_wf,
fset-ac-order-constrained,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
set_wf,
fset-ac-le-singleton-empty,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
equal_wf,
squash_wf,
true_wf,
fset-ac-le-distributive-constrained,
iff_weakening_equal,
deq_wf,
fset-singleton_wf,
fset-antichain-singleton,
fset-all-iff,
deq-fset_wf,
member-fset-singleton,
fset-member_wf,
fset-constrained-ac-lub-is-lub,
fset-constrained-ac-glb-is-glb,
empty-fset-ac-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
setEquality,
cumulativity,
hypothesisEquality,
hypothesis,
productEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
lambdaFormation,
setElimination,
rename,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
independent_isectElimination,
dependent_functionElimination,
independent_pairEquality,
independent_functionElimination,
isect_memberEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
functionEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality,
voidElimination,
voidEquality
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:
2017_10_05-AM-00_36_42
Last ObjectModification:
2017_07_28-AM-09_15_09
Theory : lattices
Home
Index