Nuprl Lemma : fset-ac-le-face-lattice0
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[i:T]. ∀[s:fset(fset(T + T))].
(fset-all(s;x.inl i ∈b x) ⇐⇒ fset-ac-le(union-deq(T;T;eq;eq);s;(i=0)))
Proof
Definitions occuring in Statement :
face-lattice0: (x=0),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
deq-fset-member: a ∈b s,
fset: fset(T),
union-deq: union-deq(A;B;a;b),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
inl: inl x,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
subtype_rel: A ⊆r B,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
false: False,
fset-member: a ∈ s,
assert: ↑b,
ifthenelse: if b then t else f fi ,
deq-member: x ∈b L,
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
bfalse: ff,
face-lattice0: (x=0),
all: ∀x:A. B[x],
guard: {T},
cand: A c∧ B,
deq-fset-member: a ∈b s,
squash: ↓T,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
f-subset: xs ⊆ ys
Lemmas referenced :
fl-point-sq,
fset_wf,
assert_wf,
fset-antichain_wf,
union-deq_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
face-lattice0_wf,
deq-fset-member_wf,
fset-ac-le_wf,
deq_wf,
fset-all-iff,
deq-fset_wf,
iff_weakening_uiff,
uall_wf,
isect_wf,
fset-member_wf,
assert_of_bnot,
assert-fset-null,
not_wf,
equal-wf-T-base,
fset-singleton_wf,
equal_wf,
member-fset-filter,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
member-fset-singleton,
assert-deq-f-subset,
f-singleton-subset,
fset-ac-le-implies2,
sq_stable__assert,
assert-deq-fset-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
lambdaEquality,
setElimination,
rename,
hypothesisEquality,
setEquality,
unionEquality,
productEquality,
independent_pairFormation,
lambdaFormation,
cumulativity,
because_Cache,
applyEquality,
independent_functionElimination,
inlEquality,
universeEquality,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
addLevel,
impliesFunctionality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
dependent_functionElimination,
functionEquality,
imageElimination,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[i:T]. \mforall{}[s:fset(fset(T + T))].
(fset-all(s;x.inl i \mmember{}\msubb{} x) \mLeftarrow{}{}\mRightarrow{} fset-ac-le(union-deq(T;T;eq;eq);s;(i=0)))
Date html generated:
2018_05_22-PM-09_55_06
Last ObjectModification:
2018_05_20-PM-10_12_47
Theory : lattices
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