Nuprl Lemma : W-induction1-extract
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[Q:W(A;a.B[a]) ⟶ ℙ].
((∀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]). Q[w]))
Proof
Definitions occuring in Statement :
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,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
it: ⋅,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pW-rec: pW-rec,
genrec-ap: genrec-ap,
W-induction1,
param-W-induction
Lemmas referenced :
W-induction1,
param-W-induction
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[Q:W(A;a.B[a]) {}\mrightarrow{} \mBbbP{}].
((\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)])) {}\mRightarrow{} (\mforall{}w:W(A;a.B[a]). Q[w]))
Date html generated:
2018_05_21-PM-00_05_36
Last ObjectModification:
2018_05_19-AM-07_00_36
Theory : co-recursion
Home
Index