Nuprl Lemma : implies-k-1-continuous
∀[k:ℕ]. ∀[F:(ℕk ⟶ Type) ⟶ Type].
((∀[A,B:ℕk ⟶ Type]. F[A] ⊆r F[B] supposing A ⊆ B)
⇒ (∀j:ℕk. ∀Z:ℕk ⟶ Type. Continuous(X.F[λi.if (i =z j) then X else Z i fi ]))
⇒ k-1-continuous{i:l}(k;T.F[T]))
Proof
Definitions occuring in Statement :
k-1-continuous: k-1-continuous{i:l}(k;T.F[T]),
k-subtype: A ⊆ B,
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),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
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: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
k-1-continuous: k-1-continuous{i:l}(k;T.F[T]),
k-intersection: ⋂n. X[n],
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
so_apply: x[s],
int_seg: {i..j-},
top: Top,
ge: i ≥ j ,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
lelt: i ≤ j < k,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
type-continuous: Continuous(T.F[T]),
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
label: ...$L... t,
k-subtype: A ⊆ B,
nat_plus: ℕ+,
less_than: a < b
Lemmas referenced :
all_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,
int_seg_wf,
type-continuous_wf,
ifthenelse_wf,
eq_int_wf,
uall_wf,
subtype_rel_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
subtype_rel_isect-2,
lt_int_wf,
subtype_rel-equal,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__lt,
not-lt-2,
le-add-cancel-alt,
lelt_wf,
squash_wf,
true_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
int_subtype_base,
isect_wf,
less_than_transitivity2,
le_weakening,
not-equal-2,
le_antisymmetry_iff,
le-add-cancel2,
subtype_rel_self,
subtype_rel_isect_general,
member_wf,
imax_unfold,
le_int_wf,
set_subtype_base,
imax_wf,
iff_weakening_equal,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
one-mul,
minus-zero,
add_nat_wf,
subtype_rel_transitivity,
le_reflexive,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates,
mul-swap,
int_seg_properties,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
axiomEquality,
functionEquality,
cumulativity,
universeEquality,
instantiate,
isectEquality,
isect_memberEquality,
voidEquality,
intEquality,
intWeakElimination,
equalityElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hyp_replacement,
sqequalIntensionalEquality,
multiplyEquality,
applyLambdaEquality,
addLevel,
levelHypothesis
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[F:(\mBbbN{}k {}\mrightarrow{} Type) {}\mrightarrow{} Type].
((\mforall{}[A,B:\mBbbN{}k {}\mrightarrow{} Type]. F[A] \msubseteq{}r F[B] supposing A \msubseteq{} B)
{}\mRightarrow{} (\mforall{}j:\mBbbN{}k. \mforall{}Z:\mBbbN{}k {}\mrightarrow{} Type. Continuous(X.F[\mlambda{}i.if (i =\msubz{} j) then X else Z i fi ]))
{}\mRightarrow{} k-1-continuous\{i:l\}(k;T.F[T]))
Date html generated:
2018_05_21-PM-00_10_12
Last ObjectModification:
2017_10_18-PM-02_38_51
Theory : co-recursion
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