Nuprl Lemma : disjoint-quotient_subtype
∀[A,B:Type].
∀[E:(A + B) ⟶ (A + B) ⟶ ℙ]
(x,y:A + B//E[x;y]) ⊆r (a1,a2:A//E[inl a1;inl a2] + (b1,b2:B//E[inr b1 ;inr b2 ]))
supposing EquivRel(A + B;x,y.E[x;y])
supposing ¬(A ∧ B)
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
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]),
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s1;s2],
trans: Trans(T;x,y.E[x; y]),
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
quotient: x,y:A//B[x; y],
guard: {T},
not: ¬A,
cand: A c∧ B,
false: False
Lemmas referenced :
quotient_wf,
quotient-member-eq,
equal_wf,
equal-wf-base,
equiv_rel_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
independent_pairFormation,
productElimination,
thin,
promote_hyp,
lambdaFormation,
hypothesisEquality,
applyEquality,
functionExtensionality,
unionEquality,
cumulativity,
inlEquality,
inrEquality,
lambdaEquality,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
sqequalRule,
independent_isectElimination,
hypothesis,
pertypeElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
rename,
unionElimination,
dependent_functionElimination,
independent_functionElimination,
productEquality,
universeEquality,
axiomEquality,
isect_memberEquality,
functionEquality,
voidElimination
Latex:
\mforall{}[A,B:Type].
\mforall{}[E:(A + B) {}\mrightarrow{} (A + B) {}\mrightarrow{} \mBbbP{}]
(x,y:A + B//E[x;y]) \msubseteq{}r (a1,a2:A//E[inl a1;inl a2] + (b1,b2:B//E[inr b1 ;inr b2 ]))
supposing EquivRel(A + B;x,y.E[x;y])
supposing \mneg{}(A \mwedge{} B)
Date html generated:
2017_04_14-AM-07_39_55
Last ObjectModification:
2017_02_27-PM-03_11_18
Theory : quot_1
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