Nuprl Lemma : Spread-family-ext
∀[P,Pos:Type]. ∀[f:Pos ⟶ ℤ]. ∀[Mv:ℤ ⟶ P ⟶ P ⟶ Type].
Spread-family(P;Pos;f;n,p,q.Mv[n;p;q])
≡ λp.(a:Pos × (q:P ⟶ Mv[f a;p;q] ⟶ (Spread-family(P;Pos;f;n,p,q.Mv[n;p;q]) q)))
Proof
Definitions occuring in Statement :
Spread-family: Spread-family(P;Pos;f;n,p,q.Mv[n; p; q]),
ext-family: F ≡ G,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Spread-family: Spread-family(P;Pos;f;n,p,q.Mv[n; p; q]),
so_apply: x[s1;s2;s3],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
ext-family: F ≡ G,
all: ∀x:A. B[x],
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
type-family-continuous: type-family-continuous{i:l}(P;H),
sub-family: F ⊆ G,
isect-family: ⋂a:A. F[a],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
top: Top,
pi1: fst(t),
pi2: snd(t),
family-monotone: family-monotone{i:l}(P;H),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
corec-family-ext,
nat_wf,
false_wf,
le_wf,
pair-eta,
equal_wf,
subtype_rel_self,
subtype_rel_wf,
subtype_rel_product,
subtype_rel_dep_function,
sub-family_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
productEquality,
cumulativity,
functionEquality,
applyEquality,
functionExtensionality,
intEquality,
universeEquality,
independent_isectElimination,
independent_pairFormation,
hypothesis,
sqequalRule,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
lambdaFormation,
isectEquality,
dependent_set_memberEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
independent_functionElimination,
dependent_pairEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[P,Pos:Type]. \mforall{}[f:Pos {}\mrightarrow{} \mBbbZ{}]. \mforall{}[Mv:\mBbbZ{} {}\mrightarrow{} P {}\mrightarrow{} P {}\mrightarrow{} Type].
Spread-family(P;Pos;f;n,p,q.Mv[n;p;q]) \mequiv{} \mlambda{}p.(a:Pos \mtimes{} (q:P
{}\mrightarrow{} Mv[f a;p;q]
{}\mrightarrow{} (Spread-family(P;Pos;f;n,p,q.Mv[n;p;q])
q)))
Date html generated:
2017_04_17-AM-09_29_00
Last ObjectModification:
2017_02_27-PM-05_28_54
Theory : spread
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