Nuprl Lemma : per-product-elim
∀[A:Type]. ∀[B:per-function(A;a.Type)]. ∀[p:per-product(A;a.B[a])].
uand(p ~ <fst(p), snd(p)>;uand(fst(p) ∈ A;snd(p) ∈ B[fst(p)]))
Proof
Definitions occuring in Statement :
per-product: per-product(A;a.B[a]),
per-function: per-function(A;a.B[a]),
uand: uand(A;B),
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
member: t ∈ T,
pair: <a, b>,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
type-function: type-function{i:l}(A),
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
per-product: per-product(A;a.B[a]),
uand: uand(A;B),
has-value: (a)↓,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
label: ...$L... t,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
top: Top
Lemmas referenced :
per-function_wf_type,
per-function-type-apply,
per-function_wf,
per-product_wf,
istype-universe,
uand_wf,
equal-wf-base,
has-value_wf_base,
is-exception_wf,
ispair-implies-sq,
istype-sqequal,
apply_wf_type-function,
equal_wf,
member_wf,
subtype_rel_self,
iff_weakening_equal,
istype-top,
istype-void,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
sqequalRule,
promote_hyp,
Error :universeIsType,
hypothesis,
instantiate,
universeEquality,
pointwiseFunctionality,
sqequalIntensionalEquality,
baseApply,
closedConclusion,
baseClosed,
isectEquality,
equalityTransitivity,
equalitySymmetry,
pertypeElimination,
axiomSqleEquality,
divergentSqle,
sqleReflexivity,
axiomSqEquality,
independent_isectElimination,
sqequalExtensionalEquality,
independent_pairFormation,
Error :lambdaFormation_alt,
imageMemberEquality,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
Error :inhabitedIsType,
natural_numberEquality,
productElimination,
independent_functionElimination,
rename,
isaxiomCases,
Error :isect_memberEquality_alt,
voidElimination,
axiomEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:per-function(A;a.Type)]. \mforall{}[p:per-product(A;a.B[a])].
uand(p \msim{} <fst(p), snd(p)>uand(fst(p) \mmember{} A;snd(p) \mmember{} B[fst(p)]))
Date html generated:
2019_06_20-AM-11_30_16
Last ObjectModification:
2018_11_23-PM-00_51_09
Theory : per!type
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