Nuprl Lemma : W-induction
∀[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 :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B
Lemmas referenced :
W_wf,
all_wf,
Wsup_wf,
W_ind_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesis,
cumulativity,
functionEquality,
because_Cache,
universeEquality,
introduction,
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:
2016_05_14-AM-06_15_39
Last ObjectModification:
2015_12_26-PM-00_04_45
Theory : co-recursion
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