Nuprl Lemma : squash_thru_uequiv_rel
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ]. ((↓UniformEquivRel(T;x,y.E[x;y])) ⇒ UniformEquivRel(T;x,y.↓E[x;y]))
Proof
Definitions occuring in Statement :
uequiv_rel: UniformEquivRel(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 :
uequiv_rel: UniformEquivRel(T;x,y.E[x; y]),
utrans: UniformlyTrans(T;x,y.E[x; y]),
usym: UniformlySym(T;x,y.E[x; y]),
urefl: UniformlyRefl(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
squash: ↓T,
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
squash_wf,
uall_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
imageMemberEquality,
hypothesisEquality,
baseClosed,
hypothesis,
independent_pairFormation,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
lambdaEquality,
dependent_functionElimination,
because_Cache,
isect_memberEquality,
productEquality,
functionEquality,
universeEquality,
independent_pairEquality,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((\mdownarrow{}UniformEquivRel(T;x,y.E[x;y])) {}\mRightarrow{} UniformEquivRel(T;x,y.\mdownarrow{}E[x;y]))
Date html generated:
2016_10_21-AM-09_42_04
Last ObjectModification:
2016_08_01-PM-09_49_08
Theory : rel_1
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