Nuprl Lemma : decidable__quotient_equal
∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ].
(EquivRel(T;x,y.E[x;y]) ⇒ (∀x,y:T. Dec(E[x;y])) ⇒ (∀u,v:x,y:T//E[x;y]. Dec(u = v ∈ (x,y:T//E[x;y]))))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
quotient: x,y:A//B[x; y],
subtype_rel: A ⊆r B,
infix_ap: x f y,
trans: Trans(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
guard: {T},
sym: Sym(T;x,y.E[x; y]),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
istype-universe,
decidable_wf,
equiv_rel_wf,
dec_iff_ex_bvfun,
quotient_wf,
equal_wf,
bool_wf,
subtype_rel_self,
iff_imp_equal_bool,
assert_wf,
assert_witness,
sq_stable__iff,
sq_stable_from_decidable,
decidable__assert,
sq_stable__equal,
squash_wf,
iff_wf,
infix_ap_wf,
quot_elim,
subtype_quotient
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalRule,
Error :functionIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
Error :universeIsType,
applyEquality,
Error :lambdaEquality_alt,
universeEquality,
cumulativity,
functionExtensionality,
because_Cache,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :functionExtensionality_alt,
pointwiseFunctionalityForEquality,
functionEquality,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
rename,
instantiate,
Error :equalityIsType1,
dependent_functionElimination,
Error :productIsType,
Error :equalityIsType4,
independent_pairFormation,
promote_hyp,
Error :dependent_pairFormation_alt,
independent_pairEquality,
axiomEquality,
Error :functionIsTypeImplies,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(EquivRel(T;x,y.E[x;y]) {}\mRightarrow{} (\mforall{}x,y:T. Dec(E[x;y])) {}\mRightarrow{} (\mforall{}u,v:x,y:T//E[x;y]. Dec(u = v)))
Date html generated:
2019_06_20-PM-00_32_18
Last ObjectModification:
2018_10_05-PM-05_44_32
Theory : quot_1
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