Nuprl Lemma : wfd-tree-induction
∀[T:Type]
∀P:wfd-tree(T) ⟶ ℙ
(P[Wsup(tt;⋅)] ⇒ (∀f:T ⟶ wfd-tree(T). ((∀b:T. P[f b]) ⇒ P[Wsup(ff;f)])) ⇒ {∀t:wfd-tree(T). P[t]})
Proof
Definitions occuring in Statement :
wfd-tree: wfd-tree(T),
Wsup: Wsup(a;b),
bfalse: ff,
btrue: tt,
it: ⋅,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
wfd-tree: wfd-tree(T),
bool: 𝔹,
ifthenelse: if b then t else f fi ,
unit: Unit,
it: ⋅,
btrue: tt,
prop: ℙ,
bfalse: ff,
guard: {T},
subtype_rel: A ⊆r B,
squash: ↓T,
true: True
Lemmas referenced :
W-induction,
all_wf,
wfd-tree_wf,
Wsup_wf,
bool_wf,
ifthenelse_wf,
bfalse_wf,
btrue_wf,
void_wf,
squash_wf,
true_wf,
W_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
sqequalRule,
hypothesisEquality,
independent_functionElimination,
unionElimination,
equalityElimination,
voidEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
functionEquality,
instantiate,
universeEquality,
voidElimination,
addLevel,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
levelHypothesis
Latex:
\mforall{}[T:Type]
\mforall{}P:wfd-tree(T) {}\mrightarrow{} \mBbbP{}
(P[Wsup(tt;\mcdot{})]
{}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} wfd-tree(T). ((\mforall{}b:T. P[f b]) {}\mRightarrow{} P[Wsup(ff;f)]))
{}\mRightarrow{} \{\mforall{}t:wfd-tree(T). P[t]\})
Date html generated:
2016_10_21-AM-09_47_05
Last ObjectModification:
2016_07_12-AM-05_07_07
Theory : co-recursion
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