Nuprl Lemma : strong-continuity-rel
∀P:(ℕ ⟶ ℕ) ⟶ ℕ ⟶ ℙ. ∀F:∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. (P f n)).
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)
∀f:ℕ ⟶ ℕ
∃n:ℕ. ∃k:ℕn. ((P f k) ∧ ((M n f) = (inl k) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ ((M m f) = (inl k) ∈ (ℕ?))))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
prop: ℙ,
all: ∀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,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
true: True,
quotient: x,y:A//B[x; y],
squash: ↓T
Lemmas referenced :
axiom-choice-1X-quot,
nat_wf,
prop-truncation-quot,
exists_wf,
int_seg_wf,
unit_wf2,
all_wf,
int_seg_subtype_nat,
false_wf,
equal_wf,
subtype_rel_dep_function,
subtype_rel_self,
subtype_rel_union,
assert_wf,
isl_wf,
quotient_wf,
true_wf,
equiv_rel_true,
strong-continuity2-no-inner-squash-bound,
less_than_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
lelt_wf,
quotient-member-eq,
equal-wf-base,
member_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
hypothesisEquality,
independent_functionElimination,
isectElimination,
functionEquality,
because_Cache,
natural_numberEquality,
setElimination,
rename,
unionEquality,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
inlEquality,
cumulativity,
universeEquality,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}. \mforall{}F:\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. (P f n)).
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}n?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}
\mexists{}n:\mBbbN{}
\mexists{}k:\mBbbN{}n. ((P f k) \mwedge{} ((M n f) = (inl k)) \mwedge{} (\mforall{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} ((M m f) = (inl k))))))
Date html generated:
2017_04_17-AM-10_02_22
Last ObjectModification:
2017_02_27-PM-05_54_40
Theory : continuity
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