Nuprl Lemma : equiv_rel_squash
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. (EquivRel(T;x,y.E[x;y]) ⇒ EquivRel(T;x,y.↓E[x;y]))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
squash: ↓T,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
squash: ↓T,
prop: ℙ,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
equiv_rel_squash2,
equiv_rel_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
Error :universeIsType,
independent_functionElimination,
hypothesis,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
productElimination,
independent_pairEquality,
imageElimination,
Error :functionIsTypeImplies,
Error :functionIsType,
because_Cache,
universeEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (EquivRel(T;x,y.E[x;y]) {}\mRightarrow{} EquivRel(T;x,y.\mdownarrow{}E[x;y]))
Date html generated:
2019_06_20-PM-00_28_51
Last ObjectModification:
2019_03_18-PM-07_10_11
Theory : rel_1
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