Nuprl Lemma : weak-continuity-rel-fun
∀P:(ℕ ⟶ ℕ) ⟶ ℙ. ((∀f:ℕ ⟶ ℕ. ⇃(P f)) ⇒ (∀f:ℕ ⟶ ℕ. ⇃(∃k:ℕ. ∀g:ℕ ⟶ ℕ. ((f = g ∈ (ℕk ⟶ ℕ)) ⇒ (P g)))))
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 :
guard: {T},
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
exists: ∃x:A. B[x],
so_apply: x[s],
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
subtype_rel_self,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
equal_wf,
le_wf,
false_wf,
exists_wf,
implies-quotient-true,
weak-continuity-rel,
equiv_rel_true,
true_wf,
quotient_wf,
all_wf,
nat_wf
Rules used in proof :
productElimination,
rename,
setElimination,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
dependent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
universeEquality,
cumulativity,
independent_isectElimination,
hypothesisEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
because_Cache,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut,
functionEquality,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. ((\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(P f)) {}\mRightarrow{} (\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}k:\mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} (P g)))))
Date html generated:
2017_04_17-AM-10_02_29
Last ObjectModification:
2017_04_15-PM-05_16_11
Theory : continuity
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