Nuprl Lemma : sub-powerset-lattice_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[whole:fset(T)]. ∀[P:fset(T) ⟶ ℙ].
sub-powerset-lattice(T;eq;whole;P) ∈ BoundedDistributiveLattice
supposing (∀x:T. x ∈ whole) ∧ (∀a,b:fset(T). ((P a) ⇒ (P b) ⇒ ((P a ⋃ b) ∧ (P a ⋂ b)))) ∧ (P {}) ∧ (P whole)
Proof
Definitions occuring in Statement :
sub-powerset-lattice: sub-powerset-lattice(T;eq;whole;P),
bdd-distributive-lattice: BoundedDistributiveLattice,
empty-fset: {},
fset-intersection: a ⋂ b,
fset-union: x ⋃ y,
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
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,
sub-powerset-lattice: sub-powerset-lattice(T;eq;whole;P),
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
so_apply: x[s1;s2],
squash: ↓T,
label: ...$L... t,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
fset-union: x ⋃ y,
l-union: as ⋃ bs,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
uiff: uiff(P;Q)
Lemmas referenced :
mk-bounded-distributive-lattice_wf,
fset_wf,
fset-intersection_wf,
fset-union_wf,
empty-fset_wf,
equal_wf,
fset-intersection-commutes,
iff_weakening_equal,
trivial-equal,
set_wf,
fset-union-commutes,
fset-intersection-associative,
fset-union-associative,
fset-absorption1,
fset-absorption2,
fset-distributive,
all_wf,
fset-member_wf,
deq_wf,
fset-extensionality,
fset-member_witness,
iff_weakening_uiff,
member-fset-intersection,
uiff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
setEquality,
because_Cache,
applyEquality,
hypothesisEquality,
hypothesis,
lambdaEquality,
lambdaFormation,
functionExtensionality,
cumulativity,
setElimination,
rename,
dependent_set_memberEquality,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
equalitySymmetry,
imageElimination,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
natural_numberEquality,
equalityTransitivity,
isect_memberEquality,
axiomEquality,
independent_pairFormation,
productEquality,
functionEquality,
universeEquality,
independent_pairEquality,
addLevel
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[whole:fset(T)]. \mforall{}[P:fset(T) {}\mrightarrow{} \mBbbP{}].
sub-powerset-lattice(T;eq;whole;P) \mmember{} BoundedDistributiveLattice
supposing (\mforall{}x:T. x \mmember{} whole)
\mwedge{} (\mforall{}a,b:fset(T). ((P a) {}\mRightarrow{} (P b) {}\mRightarrow{} ((P a \mcup{} b) \mwedge{} (P a \mcap{} b))))
\mwedge{} (P \{\})
\mwedge{} (P whole)
Date html generated:
2020_05_20-AM-08_47_37
Last ObjectModification:
2017_07_28-AM-09_15_07
Theory : lattices
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