Nuprl Lemma : cond_rel_star_equiv
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[R1,E:T ⟶ T ⟶ ℙ].
(EquivRel(T)(_1 E _2) ⇒ when P, R1 => E ⇒ R1 preserves P ⇒ when P, R1^* => E)
Proof
Definitions occuring in Statement :
rel_star: R^*,
cond_rel_implies: when P, R1 => R2,
preserved_by: R preserves P,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
cond_rel_implies: when P, R1 => R2,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
preserved_by_wf,
cond_rel_implies_wf,
equiv_rel_wf,
cond_rel_star_monotone,
rel_star_wf,
subtype_rel_self,
rel_star_of_equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType,
universeEquality,
independent_functionElimination,
Error :lambdaFormation_alt,
dependent_functionElimination,
because_Cache,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[R1,E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(EquivRel(T)($_{1}$ E $_{2}$) {}\mRightarrow{} when P, R1 => E {}\mRightarrow{} R1 pre\000Cserves P {}\mRightarrow{} when P, rel\_star(T; R1) => E)
Date html generated:
2019_06_20-PM-00_30_50
Last ObjectModification:
2018_09_26-PM-00_46_13
Theory : relations
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