Nuprl Lemma : W_ind_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[Q:W(A;a.B[a]) ⟶ ℙ].
∀[F:∀a:A. ∀f:B[a] ⟶ W(A;a.B[a]). ((∀b:B[a]. Q[f b]) ⇒ Q[Wsup(a;f)])]. ∀[w:W(A;a.B[a])].
(W_ind(F;w) ∈ Q[w])
Proof
Definitions occuring in Statement :
W_ind: W_ind(F;w),
Wsup: Wsup(a;b),
W: W(A;a.B[a]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
W-induction1-extract,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
W_ind: W_ind(F;w),
genrec-ap: genrec-ap
Lemmas referenced :
W-induction1-extract,
isect_wf,
W_wf,
subtype_rel_self,
Wsup_wf,
equal_wf,
all_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
instantiate,
extract_by_obid,
hypothesis,
isect_memberFormation,
introduction,
applyEquality,
sqequalRule,
lambdaEquality,
isectElimination,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
thin,
sqequalHypSubstitution,
functionEquality,
cumulativity,
universeEquality,
because_Cache,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
isectEquality,
functionExtensionality,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[Q:W(A;a.B[a]) {}\mrightarrow{} \mBbbP{}]. \mforall{}[F:\mforall{}a:A. \mforall{}f:B[a] {}\mrightarrow{} W(A;a.B[a]).
((\mforall{}b:B[a]. Q[f b]) {}\mRightarrow{} Q[Wsup(a;f)])].
\mforall{}[w:W(A;a.B[a])].
(W\_ind(F;w) \mmember{} Q[w])
Date html generated:
2018_05_21-PM-00_05_39
Last ObjectModification:
2018_05_19-AM-07_00_38
Theory : co-recursion
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