Nuprl Lemma : rel_plus_iff
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀x,z:T. (x R+ z ⇐⇒ ∃y:T. ((x (R^*) y) ∧ (y R z)))
Proof
Definitions occuring in Statement :
rel_plus: R+,
rel_star: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
infix_ap: x f y,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
cand: A c∧ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B
Lemmas referenced :
rel_plus_wf,
exists_wf,
and_wf,
rel_star_wf,
rel_plus_implies,
rel_star_weakening,
rel-plus-rel-star,
rel-star-rel-plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
applyEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
hypothesis,
productElimination,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
because_Cache,
independent_isectElimination,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}x,z:T. (x R\msupplus{} z \mLeftarrow{}{}\mRightarrow{} \mexists{}y:T. ((x rel\_star(T; R) y) \mwedge{} (y R z)))
Date html generated:
2016_05_14-PM-03_53_47
Last ObjectModification:
2015_12_26-PM-06_56_33
Theory : relations2
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