Nuprl Lemma : altWind-induction
∀[A:𝕌']. ∀[B:A ⟶ Type]. ∀[P:altW(A;a.B[a]) ⟶ ℙ].
((∀w:altW(A;a.B[a]). ((∀b:coW-dom(a.B[a];w). P[altW-item(w;b)]) ⇒ P[w])) ⇒ (∀w:altW(A;a.B[a]). P[w]))
Proof
Definitions occuring in Statement :
altW-item: altW-item(w;b),
altW: altW(A;a.B[a]),
coW-dom: coW-dom(a.B[a];w),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
altW: altW(A;a.B[a]),
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
altW-item_wf,
coW-dom_wf,
all_wf,
altW_wf,
altWind_wf
Rules used in proof :
universeEquality,
because_Cache,
setElimination,
functionEquality,
cumulativity,
instantiate,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
rename,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:\mBbbU{}']. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:altW(A;a.B[a]) {}\mrightarrow{} \mBbbP{}].
((\mforall{}w:altW(A;a.B[a]). ((\mforall{}b:coW-dom(a.B[a];w). P[altW-item(w;b)]) {}\mRightarrow{} P[w]))
{}\mRightarrow{} (\mforall{}w:altW(A;a.B[a]). P[w]))
Date html generated:
2018_07_29-AM-09_22_30
Last ObjectModification:
2018_07_26-PM-09_37_15
Theory : co-recursion
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