Nuprl Lemma : equiv_rel_squash2
∀[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,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
squash: ↓T,
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
guard: {T}
Lemmas referenced :
squash_wf,
equiv_rel_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
because_Cache,
independent_pairFormation,
Error :universeIsType,
extract_by_obid,
isectElimination,
applyEquality,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
dependent_functionElimination,
independent_pairEquality,
Error :functionIsTypeImplies,
Error :functionIsType,
universeEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((\mdownarrow{}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_50
Last ObjectModification:
2019_03_18-PM-07_09_34
Theory : rel_1
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