Nuprl Lemma : squash_thru_equiv_rel
∀[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 :
squash: ↓T,
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T}
Lemmas referenced :
all_wf,
squash_wf
Rules used in proof :
isect_memberEquality,
independent_pairEquality,
productElimination,
dependent_functionElimination,
baseClosed,
imageMemberEquality,
imageElimination,
independent_pairFormation,
lambdaFormation,
isect_memberFormation,
universeEquality,
functionEquality,
lambdaEquality,
sqequalRule,
productEquality,
because_Cache,
hypothesis,
cumulativity,
functionExtensionality,
applyEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
hypothesisEquality,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
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_29_10
Last ObjectModification:
2018_08_07-PM-00_51_06
Theory : rel_1
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