Nuprl Lemma : strong-continuity-rel-unique
∀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 = n ∈ 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]
, 
guard: {T}
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
cand: A c∧ B
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, 
implies-quotient-true, 
quotient_wf, 
true_wf, 
equiv_rel_true, 
strong-continuity2-no-inner-squash-unique-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
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, 
productElimination, 
cumulativity, 
universeEquality, 
dependent_pairFormation, 
dependent_set_memberEquality, 
unionElimination, 
equalityTransitivity, 
equalitySymmetry, 
applyLambdaEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll
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  =  n)))))
Date html generated:
2017_04_17-AM-10_02_33
Last ObjectModification:
2017_02_27-PM-05_54_02
Theory : continuity
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