Nuprl Lemma : unsquashed-weak-continuity-false2
¬(∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ. ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((∀i:ℕn. ((a i) = (b i) ∈ ℕ)) ⇒ ((F a) = (F b) ∈ ℕ)))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
unsquashed-WCP: unsquashed-WCP,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
so_apply: x[s],
nat: ℕ,
pi1: fst(t)
Lemmas referenced :
unsquashed-weak-continuity-false,
all_wf,
nat_wf,
int_seg_wf,
equal_wf,
int_seg_subtype_nat,
false_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
independent_functionElimination,
thin,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
promote_hyp,
productElimination,
dependent_pairFormation,
isectElimination,
functionEquality,
because_Cache,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
voidElimination,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry
Latex:
\mneg{}(\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mexists{}n:\mBbbN{}. \mforall{}b:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mforall{}i:\mBbbN{}n. ((a i) = (b i))) {}\mRightarrow{} ((F a) = (F b))))
Date html generated:
2017_04_17-AM-09_41_08
Last ObjectModification:
2017_02_27-PM-05_36_00
Theory : fan-theorem
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