Nuprl Lemma : norm-pair_wf_sq
∀[A,B:Type].
(∀[Na:sq-id-fun(A)]. ∀[Nb:sq-id-fun(B)]. (norm-pair(Na;Nb) ∈ sq-id-fun(A × B))) supposing
((B ⊆r Base) and
(A ⊆r Base) and
value-type(B) and
value-type(A))
Proof
Definitions occuring in Statement :
norm-pair: norm-pair(Na;Nb),
sq-id-fun: sq-id-fun(T),
value-type: value-type(T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
product: x:A × B[x],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq-id-fun: sq-id-fun(T),
norm-pair: norm-pair(Na;Nb),
has-value: (a)↓,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
prop: ℙ
Lemmas referenced :
subtype_base_sq,
value-type-has-value,
sq-id-fun_wf,
subtype_rel_wf,
base_wf,
value-type_wf,
product_subtype_base,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
independent_isectElimination,
hypothesis,
functionExtensionality,
productElimination,
sqequalRule,
callbyvalueReduce,
setEquality,
sqequalIntensionalEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :isect_memberEquality_alt,
Error :inhabitedIsType,
universeEquality,
Error :dependent_set_memberEquality_alt,
independent_pairEquality,
Error :lambdaEquality_alt,
setElimination,
rename,
Error :lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation
Latex:
\mforall{}[A,B:Type].
(\mforall{}[Na:sq-id-fun(A)]. \mforall{}[Nb:sq-id-fun(B)]. (norm-pair(Na;Nb) \mmember{} sq-id-fun(A \mtimes{} B))) supposing
((B \msubseteq{}r Base) and
(A \msubseteq{}r Base) and
value-type(B) and
value-type(A))
Date html generated:
2019_06_20-AM-11_27_23
Last ObjectModification:
2018_10_06-AM-09_00_29
Theory : call!by!value_2
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