Nuprl Lemma : quotient-of-quotient
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
(EquivRel(T;x,y.x R y)
⇒ (∀Q:(x,y:T//(x R y)) ⟶ (x,y:T//(x R y)) ⟶ ℙ
(EquivRel(x,y:T//(x R y);u,v.u Q v) ⇒ u,v:x,y:T//(x R y)//(u Q v) ≡ x,y:T//(x Q y))))
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
ext-eq: A ≡ B,
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
istype: istype(T),
ext-eq: A ≡ B,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
infix_ap: x f y,
guard: {T}
Lemmas referenced :
equiv-on-quotient,
quotient_wf,
infix_ap_wf,
subtype_rel_dep_function,
subtype_quotient,
equal-wf-base,
quotient_subtype_quotient,
subtype_rel_self,
equiv_rel_wf,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
isectElimination,
because_Cache,
sqequalRule,
Error :lambdaEquality_alt,
instantiate,
cumulativity,
universeEquality,
applyEquality,
Error :inhabitedIsType,
independent_isectElimination,
functionEquality,
functionExtensionality,
Error :universeIsType,
independent_pairFormation,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType1,
productEquality,
Error :functionIsType,
hyp_replacement
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(EquivRel(T;x,y.x R y)
{}\mRightarrow{} (\mforall{}Q:(x,y:T//(x R y)) {}\mrightarrow{} (x,y:T//(x R y)) {}\mrightarrow{} \mBbbP{}
(EquivRel(x,y:T//(x R y);u,v.u Q v) {}\mRightarrow{} u,v:x,y:T//(x R y)//(u Q v) \mequiv{} x,y:T//(x Q y))))
Date html generated:
2019_06_20-PM-00_33_09
Last ObjectModification:
2018_09_30-PM-00_36_21
Theory : quot_1
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