Nuprl Lemma : not-not-sig-to-W-implies-stable
∀[A:Type]. ((∀[B:A ⟶ Type]. ((¬¬(a:A × (¬B[a]))) ⇒ W(A;a.B[a]))) ⇒ Stable{A})
Proof
Definitions occuring in Statement :
W: W(A;a.B[a]),
stable: Stable{P},
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
stable: Stable{P},
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
ext-eq: A ≡ B,
and: P ∧ Q
Lemmas referenced :
not_wf,
uall_wf,
W_wf,
false_wf,
W-ext
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
lemma_by_obid,
isectElimination,
hypothesis,
rename,
instantiate,
functionEquality,
cumulativity,
universeEquality,
productEquality,
applyEquality,
because_Cache,
independent_functionElimination,
dependent_pairEquality,
promote_hyp,
productElimination,
hypothesis_subsumption
Latex:
\mforall{}[A:Type]. ((\mforall{}[B:A {}\mrightarrow{} Type]. ((\mneg{}\mneg{}(a:A \mtimes{} (\mneg{}B[a]))) {}\mRightarrow{} W(A;a.B[a]))) {}\mRightarrow{} Stable\{A\})
Date html generated:
2016_05_14-AM-06_17_45
Last ObjectModification:
2015_12_26-PM-00_03_28
Theory : co-recursion
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