Nuprl Lemma : exists_det_fun
∀[T:Type]. ∀[A:T ⟶ ℙ]. ((∀x:T. SqStable(A x)) ⇒ (detach_fun(T;A) ⇐⇒ ∀x:T. Dec(A x)))
Proof
Definitions occuring in Statement :
detach_fun: detach_fun(T;A),
sq_stable: SqStable(P),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
all: ∀x:A. B[x],
member: t ∈ T,
rev_implies: P ⇐ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
detach_fun: detach_fun(T;A),
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
isl: isl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
bfalse: ff,
not: ¬A,
false: False
Lemmas referenced :
detach_fun_wf,
all_wf,
decidable_wf,
sq_stable_wf,
detach_fun_properties,
decidable_functionality,
assert_wf,
decidable__assert,
isl_wf,
not_wf,
iff_wf,
equal_wf,
true_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
hypothesisEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
productElimination,
dependent_set_memberFormation,
because_Cache,
unionEquality,
unionElimination,
natural_numberEquality,
voidElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x:T. SqStable(A x)) {}\mRightarrow{} (detach\_fun(T;A) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:T. Dec(A x)))
Date html generated:
2016_05_15-PM-00_00_26
Last ObjectModification:
2015_12_26-PM-11_26_54
Theory : gen_algebra_1
Home
Index