Nuprl Lemma : strong-continuity-implies3
∀[F:(ℕ ⟶ ℕ) ⟶ ℕ]
(↓∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)
∀f:ℕ ⟶ ℕ. (↓∃n:ℕ. (((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ ((M m f) = (inl (F f)) ∈ (ℕ?)))))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
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,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
exists: ∃x:A. B[x],
so_apply: x[s],
isl: isl(x),
outl: outl(x),
guard: {T},
sq_stable: SqStable(P),
implies: P ⇒ Q,
not: ¬A,
false: False,
less_than': less_than'(a;b),
le: A ≤ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
int_seg: {i..j-},
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
all: ∀x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
pi1: fst(t),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bfalse: ff,
uiff: uiff(P;Q),
less_than: a < b,
sq_type: SQType(T),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
strong-continuity-implies2,
istype-nat,
decidable__lt,
decidable__assert,
decidable__and2,
btrue_neq_bfalse,
equal_wf,
and_wf,
bfalse_wf,
assert_elim,
less_than_wf,
subtype_rel_self,
le_weakening2,
sq_stable__le,
false_wf,
int_seg_subtype,
int_seg_wf,
subtype_rel_function,
le_wf,
unit_wf2,
nat_wf,
isl_wf,
assert_wf,
decidable__exists_int_seg,
nat_properties,
int_seg_subtype_nat,
istype-false,
int_seg_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,
true_wf,
it_wf,
imax_wf,
add_nat_wf,
imax_nat,
istype-le,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
union_subtype_base,
set_subtype_base,
int_subtype_base,
unit_subtype_base,
istype-assert,
decidable_wf,
btrue_wf,
pi1_wf,
subtype_base_sq,
squash_wf,
istype-universe,
iff_weakening_equal,
bool_wf,
bool_subtype_base,
imax_ub,
intformless_wf,
int_formula_prop_less_lemma,
exists_wf,
all_wf,
subtype_rel_union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
hypothesis,
functionIsType,
inhabitedIsType,
functionEquality,
isect_memberEquality,
voidElimination,
applyLambdaEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
unionEquality,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
because_Cache,
dependent_set_memberEquality,
applyEquality,
productEquality,
lambdaEquality,
rename,
setElimination,
natural_numberEquality,
dependent_functionElimination,
instantiate,
lambdaFormation,
dependent_pairFormation_alt,
lambdaEquality_alt,
universeIsType,
functionExtensionality,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
equalityIsType1,
inlEquality_alt,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
productIsType,
inrEquality_alt,
addEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
equalityIstype,
intEquality,
sqequalBase,
dependent_pairEquality_alt,
cumulativity,
universeEquality,
inlFormation_alt,
inrFormation_alt
Latex:
\mforall{}[F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}]
(\mdownarrow{}\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}n?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}
(\mdownarrow{}\mexists{}n:\mBbbN{}. (((M n f) = (inl (F f))) \mwedge{} (\mforall{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} ((M m f) = (inl (F f))))))))
Date html generated:
2020_05_19-PM-10_04_43
Last ObjectModification:
2019_12_17-PM-06_03_12
Theory : continuity
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