Nuprl Lemma : implies-k-continuous
∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ ℕk ⟶ Type].
(k-Monotone(T.F[T])
⇒ (∀i,j:ℕk. ∀Z:ℕk ⟶ Type. Continuous(X.F[λi.if (i =z j) then X else Z i fi ] i))
⇒ k-Continuous(T.F[T]))
Proof
Definitions occuring in Statement :
k-continuous: k-Continuous(T.F[T]),
k-monotone: k-Monotone(T.F[T]),
type-continuous: Continuous(T.F[T]),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
uimplies: b supposing a,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
k-continuous: k-Continuous(T.F[T]),
k-subtype: A ⊆ B,
decidable: Dec(P),
not: ¬A,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
k-monotone: k-Monotone(T.F[T])
Lemmas referenced :
k-continuous-iff-all-k-1,
int_seg_wf,
all_wf,
type-continuous_wf,
eq_int_wf,
bool_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
k-monotone_wf,
nat_wf,
k-subtype_wf,
decidable__le,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
implies-k-1-continuous
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
cumulativity,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
universeEquality,
productElimination,
independent_functionElimination,
instantiate,
unionElimination,
equalityElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
voidElimination,
isect_memberEquality,
axiomEquality,
dependent_set_memberEquality,
addEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
voidEquality,
intEquality,
minusEquality
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[F:(\mBbbN{}k {}\mrightarrow{} Type) {}\mrightarrow{} \mBbbN{}k {}\mrightarrow{} Type].
(k-Monotone(T.F[T])
{}\mRightarrow{} (\mforall{}i,j:\mBbbN{}k. \mforall{}Z:\mBbbN{}k {}\mrightarrow{} Type. Continuous(X.F[\mlambda{}i.if (i =\msubz{} j) then X else Z i fi ] i))
{}\mRightarrow{} k-Continuous(T.F[T]))
Date html generated:
2018_05_21-PM-00_10_20
Last ObjectModification:
2017_10_18-PM-02_41_08
Theory : co-recursion
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