Nuprl Lemma : W-to-not-not-sig
∀[A:Type]. ∀[B:A ⟶ Type]. (W(A;a.B[a]) ⇒ (¬¬(a:A × (¬B[a]))))
Proof
Definitions occuring in Statement :
W: W(A;a.B[a]),
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],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x]
Lemmas referenced :
not_wf,
W_wf,
W-induction,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
lemma_by_obid,
isectElimination,
productEquality,
hypothesisEquality,
applyEquality,
lambdaEquality,
sqequalRule,
universeEquality,
dependent_functionElimination,
because_Cache,
functionEquality,
cumulativity,
isect_memberEquality,
rename,
dependent_pairEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. (W(A;a.B[a]) {}\mRightarrow{} (\mneg{}\mneg{}(a:A \mtimes{} (\mneg{}B[a]))))
Date html generated:
2016_05_14-AM-06_17_36
Last ObjectModification:
2015_12_26-PM-00_03_45
Theory : co-recursion
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