Nuprl Lemma : dec_alt_char
∀[T:Type]. ∀[A:T ⟶ ℙ]. (∀x:T. Dec(A x) ⇐⇒ (∀x:T. SqStable(A x)) ∧ detach_fun(T;A))
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,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
all_wf,
decidable_wf,
and_wf,
sq_stable_wf,
detach_fun_wf,
sq_stable_from_decidable,
exists_det_fun,
exists_det_fun_a
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
hypothesisEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} \mBbbP{}]. (\mforall{}x:T. Dec(A x) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x:T. SqStable(A x)) \mwedge{} detach\_fun(T;A))
Date html generated:
2016_05_15-PM-00_00_30
Last ObjectModification:
2015_12_26-PM-11_26_46
Theory : gen_algebra_1
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