Nuprl Lemma : fset-ac-order-constrained
∀[T:Type]
∀eq:EqDecider(T). ∀P:fset(T) ⟶ 𝔹.
Order({ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])} ;ac1,ac2.fset-ac-le(eq;ac1;ac2))
Proof
Definitions occuring in Statement :
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-antichain: fset-antichain(eq;ac),
fset-all: fset-all(s;x.P[x]),
fset: fset(T),
deq: EqDecider(T),
order: Order(T;x,y.R[x; y]),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
order: Order(T;x,y.R[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
trans: Trans(T;x,y.E[x; y]),
implies: P ⇒ Q,
anti_sym: AntiSym(T;x,y.R[x; y]),
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-all: fset-all(s;x.P[x]),
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
top: Top,
false: False,
f-proper-subset: xs ⊆≠ ys
Lemmas referenced :
fset-ac-le_weakening,
set_wf,
fset_wf,
assert_wf,
fset-antichain_wf,
fset-all_wf,
fset-ac-le_transitivity,
fset-ac-le_wf,
bool_wf,
deq_wf,
assert_witness,
fset-null_wf,
fset-filter_wf,
bnot_wf,
deq-f-subset_wf,
all_wf,
iff_wf,
f-subset_wf,
fset-extensionality,
deq-fset_wf,
fset-ac-le-implies,
fset-member_witness,
fset-member_wf,
decidable__fset-member,
empty-fset_wf,
mem_empty_lemma,
member-fset-filter,
assert-deq-f-subset,
f-subset_transitivity,
and_wf,
equal_wf,
assert-fset-antichain,
decidable__equal_fset,
decidable-equal-deq,
f-subset_antisymmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
hypothesis,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
functionEquality,
dependent_functionElimination,
independent_pairEquality,
setEquality,
independent_functionElimination,
axiomEquality,
universeEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
voidElimination,
voidEquality,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}P:fset(T) {}\mrightarrow{} \mBbbB{}.
Order(\{ac:fset(fset(T))|
(\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\} ;ac1,ac2.fset-ac-le(eq;ac1;ac2))
Date html generated:
2016_10_21-AM-10_48_13
Last ObjectModification:
2016_07_12-AM-05_53_25
Theory : finite!sets
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