Nuprl Lemma : fset-ac-glb-is-glb
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[ac1,ac2:{ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ].
greatest-lower-bound({ac:fset(fset(T))| ↑fset-antichain(eq;ac)} ;ac1,ac2.fset-ac-le(eq;ac1;ac2);ac1;ac2;fset-ac-glb(eq\000C;ac1;ac2))
Proof
Definitions occuring in Statement :
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-antichain: fset-antichain(eq;ac),
fset: fset(T),
deq: EqDecider(T),
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
assert: ↑b,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
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],
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
not: ¬A,
squash: ↓T,
false: False,
exists: ∃x:A. B[x],
guard: {T},
top: Top,
true: True,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
reduce: reduce(f;k;as),
list_ind: list_ind
Lemmas referenced :
fset-ac-le_wf,
assert_witness,
fset-null_wf,
fset_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
fset-ac-glb_wf,
assert_wf,
fset-antichain_wf,
set_wf,
deq_wf,
fset-all-iff,
deq-fset_wf,
iff_weakening_uiff,
fset-all_wf,
fset-minimals_wf,
f-proper-subset-dec_wf,
f-union_wf,
fset-image_wf,
fset-union_wf,
uall_wf,
isect_wf,
fset-member_wf,
assert_of_bnot,
bool_wf,
all_wf,
iff_wf,
f-subset_wf,
member-fset-minimals,
assert-fset-null,
not_wf,
equal-wf-T-base,
member-f-union,
member-fset-image-iff,
member-fset-filter,
assert-deq-f-subset,
f-subset-union,
mem_empty_lemma,
squash_wf,
true_wf,
fset-union-commutes,
iff_weakening_equal,
fset-ac-le_transitivity,
fset-minimals-ac-le,
fset-ac-le-implies,
fset-extensionality,
fset-member_witness,
false_wf,
f-union-subset,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
independent_functionElimination,
setEquality,
dependent_functionElimination,
isect_memberEquality,
universeEquality,
functionEquality,
functionExtensionality,
independent_isectElimination,
addLevel,
impliesFunctionality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
imageElimination,
hyp_replacement,
applyLambdaEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageMemberEquality,
dependent_pairFormation,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[ac1,ac2:\{ac:fset(fset(T))| \muparrow{}fset-antichain(eq;ac)\} ].
greatest-lower-bound(\{ac:fset(fset(T))|
\muparrow{}fset-antichain(eq;ac)\} ;ac1,ac2.fset-ac-le(eq;ac1;ac2);ac1;ac2;fset-ac-glb(\000Ceq;ac1;ac2))
Date html generated:
2017_04_17-AM-09_24_17
Last ObjectModification:
2017_02_27-PM-05_27_17
Theory : finite!sets
Home
Index