Nuprl Lemma : all-accessible-iff-induction
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(∀t:T. accessible(T;x,y.R[x;y];t) ⇐⇒ ∀[P:T ⟶ ℙ]. ((∀t:T. ((∀x:T. (R[x;t] ⇒ P[x])) ⇒ P[t])) ⇒ (∀t:T. P[t])))
Proof
Definitions occuring in Statement :
accessible: accessible(T;x,y.R[x; y];t),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
rev_implies: P ⇐ Q
Lemmas referenced :
accessible-induction,
all_wf,
accessible_wf,
uall_wf,
accessible-iff
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
lambdaFormation,
sqequalRule,
independent_functionElimination,
dependent_functionElimination,
lambdaEquality,
functionEquality,
applyEquality,
cumulativity,
universeEquality,
instantiate,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(\mforall{}t:T. accessible(T;x,y.R[x;y];t)
\mLeftarrow{}{}\mRightarrow{} \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}t:T. ((\mforall{}x:T. (R[x;t] {}\mRightarrow{} P[x])) {}\mRightarrow{} P[t])) {}\mRightarrow{} (\mforall{}t:T. P[t])))
Date html generated:
2016_05_14-AM-06_18_50
Last ObjectModification:
2015_12_26-PM-00_02_43
Theory : co-recursion
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