Nuprl Lemma : respects-equality-quotient1
∀[X,T:Type]. ∀[E:T ⟶ T ⟶ ℙ].
(respects-equality(X;x,y:T//E[x;y])) supposing (respects-equality(X;T) and EquivRel(T;x,y.E[x;y]))
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,
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B
Lemmas referenced :
quotient_wf,
istype-base,
respects-equality_wf,
equiv_rel_wf,
istype-universe,
subtype_rel_self,
change-equality-type,
subtype_quotient
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalRule,
Error :equalityIstype,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
independent_isectElimination,
hypothesis,
because_Cache,
sqequalBase,
equalitySymmetry,
Error :functionIsType,
universeEquality,
instantiate,
pertypeElimination,
promote_hyp,
productElimination,
Error :productIsType,
equalityTransitivity,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[X,T:Type]. \mforall{}[E:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(respects-equality(X;x,y:T//E[x;y])) supposing (respects-equality(X;T) and EquivRel(T;x,y.E[x;y]))
Date html generated:
2019_06_20-PM-00_32_26
Last ObjectModification:
2018_12_13-PM-04_09_24
Theory : quot_1
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