Nuprl Lemma : continuous-monotone-depproduct
∀[F:Type ⟶ Type]. ∀[G:T:Type ⟶ F[T] ⟶ Type].
(ContinuousMonotone(T.x:F[T] × G[T;x])) supposing
((∀X:ℕ ⟶ Type. ∀x:⋂n:ℕ. F[X n]. ((⋂n:ℕ. G[X n;x]) ⊆r G[⋂n:ℕ. (X n);x])) and
(∀A,B:Type. ((A ⊆r B) ⇒ (∀x:F[A]. (G[A;x] ⊆r G[B;x])))) and
ContinuousMonotone(T.F[T]))
Proof
Definitions occuring in Statement :
continuous-monotone: ContinuousMonotone(T.F[T]),
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
type-continuous: Continuous(T.F[T]),
subtype_rel: A ⊆r B,
type-monotone: Monotone(T.F[T]),
and: P ∧ Q,
continuous-monotone: ContinuousMonotone(T.F[T]),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
pi2: snd(t),
pi1: fst(t),
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ
Lemmas referenced :
continuous-monotone_wf,
subtype_rel_wf,
nat_wf,
all_wf,
subtype_rel_product,
pi2_wf,
pi1_wf,
equal_wf,
le_wf,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
independent_isectElimination,
cumulativity,
universeEquality,
functionExtensionality,
isectEquality,
lambdaEquality,
applyEquality,
functionEquality,
extract_by_obid,
instantiate,
equalitySymmetry,
equalityTransitivity,
because_Cache,
hypothesis,
axiomEquality,
hypothesisEquality,
isectElimination,
isect_memberEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
productEquality,
natural_numberEquality,
dependent_set_memberEquality,
dependent_pairEquality
Latex:
\mforall{}[F:Type {}\mrightarrow{} Type]. \mforall{}[G:T:Type {}\mrightarrow{} F[T] {}\mrightarrow{} Type].
(ContinuousMonotone(T.x:F[T] \mtimes{} G[T;x])) supposing
((\mforall{}X:\mBbbN{} {}\mrightarrow{} Type. \mforall{}x:\mcap{}n:\mBbbN{}. F[X n]. ((\mcap{}n:\mBbbN{}. G[X n;x]) \msubseteq{}r G[\mcap{}n:\mBbbN{}. (X n);x])) and
(\mforall{}A,B:Type. ((A \msubseteq{}r B) {}\mRightarrow{} (\mforall{}x:F[A]. (G[A;x] \msubseteq{}r G[B;x])))) and
ContinuousMonotone(T.F[T]))
Date html generated:
2019_06_20-PM-00_27_42
Last ObjectModification:
2018_09_16-PM-01_30_20
Theory : subtype_1
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