Nuprl Lemma : not-choice-baire-to-nat-ssq
¬(∀P:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ ℙ. (∀t:(ℕ ⟶ ℕ) ⟶ ℕ. (↓P[t]) ⇐⇒ ↓∀t:(ℕ ⟶ ℕ) ⟶ ℕ. P[t]))
Proof
Definitions occuring in Statement :
nat: ℕ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
iff: P ⇐⇒ Q,
guard: {T},
squash: ↓T,
unsquashed-WCP: unsquashed-WCP,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
exists_wf,
nat_wf,
all_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
strong-continuity2-implies-weak-skolem,
iff_wf,
squash_wf,
quotient-implies-squash,
unsquashed-weak-continuity-false,
decidable__le,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
introduction,
extract_by_obid,
isectElimination,
functionEquality,
because_Cache,
natural_numberEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
instantiate,
cumulativity,
universeEquality,
productElimination,
imageElimination,
dependent_pairFormation,
unionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality
Latex:
\mneg{}(\mforall{}P:((\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbP{}. (\mforall{}t:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. (\mdownarrow{}P[t]) \mLeftarrow{}{}\mRightarrow{} \mdownarrow{}\mforall{}t:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. P[t]))
Date html generated:
2017_04_17-AM-10_02_11
Last ObjectModification:
2017_02_27-PM-05_53_38
Theory : continuity
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