Nuprl Lemma : respects-equality-function
∀[A,A':Type]. ∀[B:A ⟶ Type]. ∀[B':A' ⟶ Type].
((A' ⊆r A) ⇒ (∀a:A'. respects-equality(B[a];B'[a])) ⇒ respects-equality(a:A ⟶ B[a];a:A' ⟶ B'[a]))
Proof
Definitions occuring in Statement :
subtype_rel: A ⊆r B,
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
istype-base,
respects-equality_wf,
subtype_rel_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
functionExtensionality,
hypothesisEquality,
sqequalRule,
Error :equalityIstype,
Error :functionIsType,
because_Cache,
Error :universeIsType,
applyEquality,
sqequalBase,
equalitySymmetry,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
instantiate,
universeEquality,
pointwiseFunctionalityForEquality,
equalityTransitivity,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
applyLambdaEquality
Latex:
\mforall{}[A,A':Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[B':A' {}\mrightarrow{} Type].
((A' \msubseteq{}r A)
{}\mRightarrow{} (\mforall{}a:A'. respects-equality(B[a];B'[a]))
{}\mRightarrow{} respects-equality(a:A {}\mrightarrow{} B[a];a:A' {}\mrightarrow{} B'[a]))
Date html generated:
2019_06_20-AM-11_13_58
Last ObjectModification:
2018_11_26-AM-00_17_25
Theory : core_2
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