Nuprl Lemma : decidable-predicate-not
∀[T:Type]. ∀[A:T ⟶ ℙ].
((Decidable(A) ⇒ Decidable(¬(A))) ∧ ((Decidable(¬(A)) ∧ (∀t:T. ((¬¬(A t)) ⇒ (A t)))) ⇒ Decidable(A)))
Proof
Definitions occuring in Statement :
predicate-not: ¬(A),
dec-predicate: Decidable(X),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
dec-predicate: Decidable(X),
predicate-not: ¬(A),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
guard: {T}
Lemmas referenced :
decidable__not,
all_wf,
decidable_wf,
not_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
lambdaEquality,
independent_pairFormation,
productElimination,
unionElimination,
inrFormation,
inlFormation,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A:T {}\mrightarrow{} \mBbbP{}].
((Decidable(A) {}\mRightarrow{} Decidable(\mneg{}(A)))
\mwedge{} ((Decidable(\mneg{}(A)) \mwedge{} (\mforall{}t:T. ((\mneg{}\mneg{}(A t)) {}\mRightarrow{} (A t)))) {}\mRightarrow{} Decidable(A)))
Date html generated:
2016_05_14-PM-04_08_56
Last ObjectModification:
2015_12_26-PM-07_54_55
Theory : fan-theorem
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