Nuprl Lemma : rel-immediate-exists
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
(SWellFounded(R x y) ⇒ (∀x,y:T. Dec(∃z:T. ((R x z) ∧ (R z y)))) ⇒ (∀y:T. ((∃x:T. (R x y)) ⇒ (∃x:T. (R! x y)))))
Proof
Definitions occuring in Statement :
rel-immediate: R!,
strongwellfounded: SWellFounded(R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: 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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
rel_implies: R1 => R2,
infix_ap: x f y,
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
rel-immediate-rel-plus,
exists_wf,
all_wf,
decidable_wf,
and_wf,
strongwellfounded_wf,
rel-rel-plus,
rel_plus_iff,
rel-immediate_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
universeEquality,
functionEquality,
cumulativity,
productElimination,
dependent_functionElimination,
dependent_pairFormation
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(SWellFounded(R x y)
{}\mRightarrow{} (\mforall{}x,y:T. Dec(\mexists{}z:T. ((R x z) \mwedge{} (R z y))))
{}\mRightarrow{} (\mforall{}y:T. ((\mexists{}x:T. (R x y)) {}\mRightarrow{} (\mexists{}x:T. (R! x y)))))
Date html generated:
2016_05_15-PM-04_54_27
Last ObjectModification:
2015_12_27-PM-02_31_49
Theory : general
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