Nuprl Lemma : respects-equality-quotient
∀[X,T:Type]. ∀[E1:X ⟶ X ⟶ ℙ]. ∀[E2:T ⟶ T ⟶ ℙ].
(respects-equality(x,y:X//E1[x;y];x,y:T//E2[x;y])) supposing
((∀x,y:X. (E1[x;y] ⇒ (x ∈ T) ⇒ ((y ∈ T) ∧ E2[x;y]))) and
respects-equality(X;T) and
EquivRel(X;x,y.E1[x;y]) and
EquivRel(T;x,y.E2[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],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
implies: P ⇒ Q,
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
quotient: x,y:A//B[x; y],
cand: A c∧ B
Lemmas referenced :
sq_stable__respects-equality,
quotient_wf,
istype-base,
subtype_rel_self,
respects-equality_wf,
equiv_rel_wf,
istype-universe,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
because_Cache,
independent_isectElimination,
hypothesis,
independent_functionElimination,
Error :lambdaFormation_alt,
Error :equalityIstype,
Error :universeIsType,
sqequalBase,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
Error :functionIsType,
instantiate,
dependent_functionElimination,
Error :productIsType,
universeEquality,
Error :inhabitedIsType,
pertypeElimination,
promote_hyp,
productElimination,
equalityTransitivity
Latex:
\mforall{}[X,T:Type]. \mforall{}[E1:X {}\mrightarrow{} X {}\mrightarrow{} \mBbbP{}]. \mforall{}[E2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(respects-equality(x,y:X//E1[x;y];x,y:T//E2[x;y])) supposing
((\mforall{}x,y:X. (E1[x;y] {}\mRightarrow{} (x \mmember{} T) {}\mRightarrow{} ((y \mmember{} T) \mwedge{} E2[x;y]))) and
respects-equality(X;T) and
EquivRel(X;x,y.E1[x;y]) and
EquivRel(T;x,y.E2[x;y]))
Date html generated:
2019_06_20-PM-00_32_24
Last ObjectModification:
2018_11_29-PM-07_01_37
Theory : quot_1
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