Nuprl Lemma : notAC20
∀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 :
qsquash: ⇃T,
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
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],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
choice-principle: ChoicePrinciple(T),
iff: P ⇐⇒ Q,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
qsquash: ⇃T,
true: True,
cand: A c∧ B,
quotient: x,y:A//B[x; y],
squash: ↓T,
guard: {T}
Lemmas referenced :
squash_wf,
member_wf,
equal-wf-base,
quotient-member-eq,
prop-truncation-quot,
equiv_rel_true,
true_wf,
quotient_wf,
not-choice-baire-to-nat,
exists_wf,
qsquash_wf,
nat_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
because_Cache,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
instantiate,
lemma_by_obid,
isectElimination,
functionEquality,
applyEquality,
lambdaEquality,
cumulativity,
hypothesisEquality,
universeEquality,
sqequalRule,
functionExtensionality,
independent_pairFormation,
independent_isectElimination,
dependent_functionElimination,
rename,
introduction,
promote_hyp,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
pointwiseFunctionality,
pertypeElimination,
productElimination,
productEquality,
imageElimination,
imageMemberEquality,
baseClosed,
dependent_pairFormation
Latex:
\mforall{}T:Type
(\00D9T
{}\mRightarrow{} (\mneg{}(\mforall{}P:((\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}
((\mforall{}n:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \00D9\mexists{}m:T. (P n m))
{}\mRightarrow{} \00D9\mexists{}f:((\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} T. \mforall{}n:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. (P n (f n))))))
Date html generated:
2016_05_14-PM-09_42_52
Last ObjectModification:
2016_04_05-PM-05_12_24
Theory : continuity
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