Nuprl Lemma : weak-continuity-rel
∀P:(ℕ ⟶ ℕ) ⟶ ℕ ⟶ ℙ
((∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. (P f n))) ⇒ (∀f:ℕ ⟶ ℕ. ⇃(∃n,k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) ⇒ (P g n)))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
axiom-choice-1X-quot,
nat_wf,
implies-quotient-true2,
equal_wf,
int_seg_wf,
subtype_rel_function,
int_seg_subtype_nat,
istype-false,
subtype_rel_self,
istype-nat,
quotient_wf,
true_wf,
equiv_rel_true,
strong-continuity2-implies-weak,
implies-quotient-true,
equal-wf-base,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesis,
hypothesisEquality,
independent_functionElimination,
isectElimination,
sqequalRule,
productEquality,
functionEquality,
applyEquality,
natural_numberEquality,
setElimination,
because_Cache,
independent_isectElimination,
independent_pairFormation,
productElimination,
productIsType,
functionIsType,
universeIsType,
lambdaEquality_alt,
inhabitedIsType,
universeEquality,
intEquality,
equalityIstype,
sqequalBase,
equalitySymmetry,
dependent_pairFormation_alt,
instantiate,
equalityTransitivity
Latex:
\mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}
((\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. (P f n))) {}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n,k:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} (P g n)))))
Date html generated:
2020_05_19-PM-10_05_02
Last ObjectModification:
2020_01_04-PM-08_12_33
Theory : continuity
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