Nuprl Lemma : strong-continuity2_biject_retract
∀[T,S,U:Type].
∀r:ℕ ⟶ U
((U ⊆r ℕ)
⇒ (∀x:U. ((r x) = x ∈ U))
⇒ (∀g:S ⟶ U
(Bij(S;U;g)
⇒ (∀F:(ℕ ⟶ T) ⟶ S
(strong-continuity2(T;g o F)
⇒ (∃M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (S?)
∀f:ℕ ⟶ T
((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (S?)))
∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (S?) supposing ↑isl(M n f)))))))))
Proof
Definitions occuring in Statement :
strong-continuity2: strong-continuity2(T;F),
biject: Bij(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
member: t ∈ T,
compose: f o g,
exists: ∃x:A. B[x],
strong-continuity2: strong-continuity2(T;F),
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
nat: ℕ,
uimplies: b supposing a,
isl: isl(x),
pi1: fst(t),
squash: ↓T,
uiff: uiff(P;Q),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
le: A ≤ B,
less_than': less_than'(a;b),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
sq_type: SQType(T)
Lemmas referenced :
subtype_rel_wf,
equal_wf,
all_wf,
biject_wf,
nat_wf,
compose_wf,
strong-continuity2_wf,
biject-inverse,
unit_wf2,
int_seg_wf,
istype-nat,
istype-assert,
btrue_wf,
bfalse_wf,
squash_wf,
true_wf,
istype-universe,
inl-one-one,
iff_weakening_equal,
not-0-eq-1,
inr-one-one,
btrue_neq_bfalse,
subtype_rel_self,
subtype_rel_function,
int_seg_subtype_nat,
istype-false,
istype-void,
subtype_base_sq,
bool_wf,
bool_subtype_base
Rules used in proof :
universeEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
rename,
hypothesis,
independent_functionElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
sqequalRule,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
Error :inlEquality_alt,
Error :universeIsType,
Error :inrEquality_alt,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
Error :functionIsType,
natural_numberEquality,
setElimination,
Error :productIsType,
because_Cache,
Error :unionIsType,
Error :isectIsType,
independent_pairEquality,
Error :dependent_pairEquality_alt,
axiomEquality,
Error :isect_memberEquality_alt,
imageElimination,
instantiate,
unionEquality,
independent_isectElimination,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
voidElimination,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
cumulativity
Latex:
\mforall{}[T,S,U:Type].
\mforall{}r:\mBbbN{} {}\mrightarrow{} U
((U \msubseteq{}r \mBbbN{})
{}\mRightarrow{} (\mforall{}x:U. ((r x) = x))
{}\mRightarrow{} (\mforall{}g:S {}\mrightarrow{} U
(Bij(S;U;g)
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} T) {}\mrightarrow{} S
(strong-continuity2(T;g o F)
{}\mRightarrow{} (\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} (S?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} T
((\mexists{}n:\mBbbN{}. ((M n f) = (inl (F f))))
\mwedge{} (\mforall{}n:\mBbbN{}. (M n f) = (inl (F f)) supposing \muparrow{}isl(M n f)))))))))
Date html generated:
2019_06_20-PM-02_50_31
Last ObjectModification:
2019_02_07-PM-01_06_04
Theory : continuity
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