Nuprl Lemma : spread-ext
∀[Pos:Type]. ∀[Mv:Pos ⟶ Type]. Spread(Pos;a.Mv[a]) ≡ a:Pos × (Mv[a] ⟶ Spread(Pos;a.Mv[a]))
Proof
Definitions occuring in Statement :
Spread: Spread(Pos;a.Mv[a]),
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Spread: Spread(Pos;a.Mv[a]),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
continuous-monotone: ContinuousMonotone(T.F[T]),
and: P ∧ Q,
type-monotone: Monotone(T.F[T]),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
strong-type-continuous: Continuous+(T.F[T]),
type-continuous: Continuous(T.F[T]),
guard: {T},
ext-eq: A ≡ B
Lemmas referenced :
corec-ext,
subtype_rel_product,
subtype_rel_function,
subtype_rel_wf,
strong-continuous-depproduct,
strong-continuous-function,
continuous-id,
subtype_rel_weakening,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
productEquality,
hypothesisEquality,
functionEquality,
applyEquality,
universeEquality,
independent_isectElimination,
independent_pairFormation,
because_Cache,
hypothesis,
lambdaFormation,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
isectEquality,
cumulativity,
productElimination,
independent_pairEquality
Latex:
\mforall{}[Pos:Type]. \mforall{}[Mv:Pos {}\mrightarrow{} Type]. Spread(Pos;a.Mv[a]) \mequiv{} a:Pos \mtimes{} (Mv[a] {}\mrightarrow{} Spread(Pos;a.Mv[a]))
Date html generated:
2016_05_14-PM-03_56_19
Last ObjectModification:
2015_12_26-PM-05_48_18
Theory : spread
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