Nuprl Lemma : fset-ac-le-distributive-constrained
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[P:fset(T) ⟶ 𝔹].
∀[a,b,c:{ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])} ].
(glb(P;a;lub(P;b;c))
= lub(P;glb(P;a;b);glb(P;a;c))
∈ {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])} )
supposing ∀x,y:fset(T). (y ⊆ x ⇒ (↑(P x)) ⇒ (↑(P y)))
Proof
Definitions occuring in Statement :
fset-constrained-ac-glb: glb(P;ac1;ac2),
fset-constrained-ac-lub: lub(P;ac1;ac2),
fset-antichain: fset-antichain(eq;ac),
fset-all: fset-all(s;x.P[x]),
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
fset-all: fset-all(s;x.P[x]),
sq_stable: SqStable(P),
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sq_type: SQType(T),
f-proper-subset: xs ⊆≠ ys,
top: Top,
decidable: Dec(P),
fset-constrained-ac-lub: lub(P;ac1;ac2),
fset-ac-lub: fset-ac-lub(eq;ac1;ac2),
or: P ∨ Q,
true: True,
fset-union: x ⋃ y,
l-union: as ⋃ bs,
reduce: reduce(f;k;as),
list_ind: list_ind,
squash: ↓T,
false: False,
exists: ∃x:A. B[x],
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
fset-ac-le: fset-ac-le(eq;ac1;ac2),
fset-constrained-ac-glb: glb(P;ac1;ac2),
guard: {T},
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
least-upper-bound: least-upper-bound(T;x,y.R[x; y];a;b;c),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
cand: A c∧ B,
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
decidable__assert,
sq_stable_from_decidable,
equal_wf,
bool_subtype_base,
subtype_base_sq,
assert_elim,
f-subset_transitivity,
member-fset-filter,
assert-deq-f-subset,
fset-extensionality,
mem_empty_lemma,
fset-member_witness,
implies-member-fset-minimals,
decidable__squash_exists_fset,
decidable__f-proper-subset,
member-fset-union,
squash_wf,
true_wf,
istype-universe,
fset-union-commutes,
subtype_rel_self,
iff_weakening_equal,
assert-f-proper-subset-dec,
iff_transitivity,
f-proper-subset_wf,
istype-void,
member-fset-constrained-image-iff,
member-f-union,
istype-assert,
assert_witness,
equal-wf-T-base,
not_wf,
assert-fset-null,
member-fset-minimals,
assert_of_bnot,
fset-member_wf,
isect_wf,
uall_wf,
fset-union_wf,
fset-constrained-image_wf,
f-union_wf,
f-proper-subset-dec_wf,
fset-minimals_wf,
iff_wf,
deq-f-subset_wf,
fset-filter_wf,
fset-null_wf,
bnot_wf,
iff_weakening_uiff,
deq-fset_wf,
fset-all-iff,
fset-ac-le-implies,
fset-ac-le_transitivity,
fset-constrained-ac-glb-is-glb,
fset-constrained-ac-lub_wf,
fset-ac-order-constrained,
fset-ac-le_wf,
least-upper-bound-unique,
fset-constrained-ac-glb_wf,
fset-constrained-ac-lub-is-lub,
deq_wf,
bool_wf,
f-subset_wf,
all_wf,
set_wf,
fset_wf,
fset-all_wf,
fset-antichain_wf,
assert_wf
Rules used in proof :
levelHypothesis,
hyp_replacement,
applyLambdaEquality,
independent_pairEquality,
unionElimination,
instantiate,
natural_numberEquality,
Error :isect_memberFormation_alt,
voidElimination,
Error :functionIsTypeImplies,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
Error :dependent_pairFormation_alt,
imageMemberEquality,
Error :functionIsType,
Error :lambdaFormation_alt,
Error :equalityIstype,
Error :universeIsType,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
sqequalBase,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
imageElimination,
baseClosed,
addLevel,
independent_functionElimination,
productElimination,
functionExtensionality,
cumulativity,
lambdaFormation,
dependent_functionElimination,
independent_isectElimination,
setEquality,
rename,
setElimination,
universeEquality,
equalitySymmetry,
equalityTransitivity,
functionEquality,
because_Cache,
axiomEquality,
isect_memberEquality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
productEquality,
hypothesis,
independent_pairFormation,
dependent_set_memberEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[P:fset(T) {}\mrightarrow{} \mBbbB{}].
\mforall{}[a,b,c:\{ac:fset(fset(T))| (\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\} ].
(glb(P;a;lub(P;b;c)) = lub(P;glb(P;a;b);glb(P;a;c)))
supposing \mforall{}x,y:fset(T). (y \msubseteq{} x {}\mRightarrow{} (\muparrow{}(P x)) {}\mRightarrow{} (\muparrow{}(P y)))
Date html generated:
2019_06_20-PM-02_14_12
Last ObjectModification:
2019_06_20-PM-02_07_11
Theory : finite!sets
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