Nuprl Lemma : rel_star_closure2
∀[T:Type]. ∀[R,R2:T ⟶ T ⟶ ℙ].
(Refl(T)(R2[_1;_2]) ⇒ Trans(T)(R2[_1;_2]) ⇒ (∀x,y:T. ((x R y) ⇒ R2[x;y])) ⇒ (∀x,y:T. ((x (R^*) y) ⇒ R2[x;y])))
Proof
Definitions occuring in Statement :
rel_star: R^*,
trans: Trans(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
infix_ap: x f y,
so_apply: x[s1;s2],
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
guard: {T},
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
rel_star_closure,
rel_star_wf,
all_wf,
trans_wf,
refl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
dependent_functionElimination,
unionElimination,
applyEquality,
lambdaEquality,
functionEquality,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType,
universeEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Refl(T)(R2[$_{1}$;$_{2}$])
{}\mRightarrow{} Trans(T)(R2[$_{1}$;$_{2}$])
{}\mRightarrow{} (\mforall{}x,y:T. ((x R y) {}\mRightarrow{} R2[x;y]))
{}\mRightarrow{} (\mforall{}x,y:T. ((x rel\_star(T; R) y) {}\mRightarrow{} R2[x;y])))
Date html generated:
2019_06_20-PM-00_30_46
Last ObjectModification:
2018_09_26-PM-00_48_05
Theory : relations
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