Nuprl Lemma : deq-f-subset_wf
∀[T:Type]. ∀[eq:EqDecider(T)]. (deq-f-subset(eq) ∈ {d:fset(T) ⟶ fset(T) ⟶ 𝔹| ∀x,y:fset(T). (x ⊆ y ⇐⇒ ↑(d x y))} )
Proof
Definitions occuring in Statement :
deq-f-subset: deq-f-subset(eq),
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
deq-f-subset: deq-f-subset(eq),
member: t ∈ T,
uall: ∀[x:A]. B[x],
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff_weakening_uiff,
decidable__fset-member,
assert-deq-fset-member,
decidable__assert,
fset-all-iff,
fset-null: fset-null(s),
null: null(as),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
so_lambda: λ2x.t[x],
so_apply: x[s],
isl: isl(x),
rev_implies: P ⇐ Q,
false: False,
not: ¬A,
bfalse: ff,
true: True,
btrue: tt,
assert: ↑b,
and: P ∧ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
implies: P ⇒ Q,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
deq_wf,
lifting-strict-spread,
istype-void,
strict4-apply,
lifting-strict-decide,
strict4-decide,
iff_wf,
all_wf,
assert_wf,
equal_wf,
not_wf,
isl_wf,
f-subset_wf,
decidable_wf,
fset_wf,
decidable__f-subset,
decidable__all_fset,
decidable_functionality,
iff_preserves_decidability,
iff_weakening_uiff,
decidable__fset-member,
assert-deq-fset-member,
decidable__assert,
fset-all-iff
Rules used in proof :
universeEquality,
because_Cache,
isect_memberEquality,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
extract_by_obid,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
functionExtensionality,
natural_numberEquality,
unionElimination,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
functionEquality,
isectEquality,
instantiate,
applyEquality,
lambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)].
(deq-f-subset(eq) \mmember{} \{d:fset(T) {}\mrightarrow{} fset(T) {}\mrightarrow{} \mBbbB{}| \mforall{}x,y:fset(T). (x \msubseteq{} y \mLeftarrow{}{}\mRightarrow{} \muparrow{}(d x y))\} )
Date html generated:
2019_06_20-PM-01_59_21
Last ObjectModification:
2019_01_09-PM-03_28_01
Theory : finite!sets
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