Nuprl Lemma : general-uniform-continuity-from-fan
∀[B:ℕ ⟶ Type]
⇃(∀i:ℕ. ∀K:B[i] ⟶ ℕ. (∃Bnd:ℕ [(∀t:B[i]. ((K t) ≤ Bnd))]))
⇒ (∀[T:Type]
∀F:(i:ℕ ⟶ B[i]) ⟶ T
(⇃(∃M:n:ℕ ⟶ (i:ℕn ⟶ B[i]) ⟶ (T?) [(∀f:i:ℕ ⟶ B[i]
((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (T?)))
∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (T?) supposing ↑isl(M n f))))])
⇒ ⇃(∃n:ℕ. ∀f,g:i:ℕ ⟶ B[i]. ((f = g ∈ (i:ℕn ⟶ B[i])) ⇒ ((F f) = (F g) ∈ T)))))
supposing ∀i:ℕ. B[i]
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
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 :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
isl: isl(x),
less_than': less_than'(a;b),
istype: istype(T),
sq_exists: ∃x:A [B[x]],
so_apply: x[s1;s2],
guard: {T},
so_lambda: λ2x y.t[x; y],
bounded-type: Bounded(T),
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
true: True,
sq_stable: SqStable(P),
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
sq_type: SQType(T),
outl: outl(x)
Lemmas referenced :
implies-quotient-true2,
sq_exists_wf,
nat_wf,
int_seg_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
unit_wf2,
equal_wf,
assert_wf,
istype-nat,
exists_wf,
all_wf,
int_seg_subtype_nat,
istype-false,
subtype_rel_dep_function,
le_wf,
trivial-quotient-true,
simple_more_general_fan_theorem-ext,
btrue_wf,
bfalse_wf,
istype-assert,
quotient_wf,
true_wf,
equiv_rel_true,
istype-universe,
squash_wf,
isl_wf,
assert_functionality_wrt_uiff,
decidable__assert,
decidable__exists_int_seg,
sq_stable_from_decidable,
sq_stable__all,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
le-add-cancel,
add_functionality_wrt_le,
less-iff-le,
sq_stable__le,
zero-add,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
add-commutes,
minus-one-mul,
condition-implies-le,
not-le-2,
int_seg_subtype,
iff_weakening_equal,
subtype_rel_self,
bool_subtype_base,
bool_wf,
subtype_base_sq,
outl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
hypothesis,
natural_numberEquality,
setElimination,
rename,
because_Cache,
applyEquality,
hypothesisEquality,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
unionEquality,
productEquality,
inlEquality_alt,
isectEquality,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
functionIsType,
unionIsType,
closedConclusion,
promote_hyp,
imageMemberEquality,
baseClosed,
setIsType,
productIsType,
isectIsType,
instantiate,
universeEquality,
voidEquality,
minusEquality,
addEquality,
applyLambdaEquality,
cumulativity
Latex:
\mforall{}[B:\mBbbN{} {}\mrightarrow{} Type]
\00D9(\mforall{}i:\mBbbN{}. \mforall{}K:B[i] {}\mrightarrow{} \mBbbN{}. (\mexists{}Bnd:\mBbbN{} [(\mforall{}t:B[i]. ((K t) \mleq{} Bnd))]))
{}\mRightarrow{} (\mforall{}[T:Type]
\mforall{}F:(i:\mBbbN{} {}\mrightarrow{} B[i]) {}\mrightarrow{} T
(\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{}n {}\mrightarrow{} B[i]) {}\mrightarrow{} (T?) [(\mforall{}f:i:\mBbbN{} {}\mrightarrow{} B[i]
((\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))))])
{}\mRightarrow{} \00D9(\mexists{}n:\mBbbN{}. \mforall{}f,g:i:\mBbbN{} {}\mrightarrow{} B[i]. ((f = g) {}\mRightarrow{} ((F f) = (F g))))))
supposing \mforall{}i:\mBbbN{}. B[i]
Date html generated:
2020_05_19-PM-10_04_49
Last ObjectModification:
2019_11_22-AM-10_53_20
Theory : continuity
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