Nuprl Lemma : notAC20-ssq
∀T:Type
((↓T)
⇒ (¬(∀P:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ T ⟶ ℙ
((∀n:(ℕ ⟶ ℕ) ⟶ ℕ. (↓∃m:T. (P n m))) ⇒ (↓∃f:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ T. ∀n:(ℕ ⟶ ℕ) ⟶ ℕ. (P n (f n)))))))
Proof
Definitions occuring in Statement :
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
exists: ∃x:A. B[x],
guard: {T}
Lemmas referenced :
all_wf,
nat_wf,
squash_wf,
exists_wf,
not-choice-baire-to-nat-ssq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
instantiate,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
hypothesisEquality,
universeEquality,
sqequalRule,
lambdaEquality,
applyEquality,
independent_pairFormation,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
dependent_pairFormation,
productElimination
Latex:
\mforall{}T:Type
((\mdownarrow{}T)
{}\mRightarrow{} (\mneg{}(\mforall{}P:((\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}
((\mforall{}n:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. (\mdownarrow{}\mexists{}m:T. (P n m)))
{}\mRightarrow{} (\mdownarrow{}\mexists{}f:((\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} T. \mforall{}n:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. (P n (f n)))))))
Date html generated:
2018_05_21-PM-01_20_24
Last ObjectModification:
2018_05_19-AM-06_32_41
Theory : continuity
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