Nuprl Lemma : not-nullset
∀[p:FinProbSpace]. ¬nullset(p;λs.True) supposing ¬¬Konig(||p||)
Proof
Definitions occuring in Statement :
Konig: Konig(k),
nullset: nullset(p;S),
finite-prob-space: FinProbSpace,
length: ||as||,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
true: True,
lambda: λx.A[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
finite-prob-space: FinProbSpace,
nullset: nullset(p;S),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
qdiv: (r/s),
qmul: r * s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
qinv: 1/r,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
Konig: Konig(k),
p-outcome: Outcome,
p-open: p-open(p),
int_seg: {i..j-},
nat: ℕ,
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
p-measure-le: measure(C) ≤ q,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
lt_int: i <z j,
qeq: qeq(r;s),
eq_int: (i =z j),
random-variable: RandomVariable(p;n),
p-open-member: s ∈ C
Lemmas referenced :
nullset_wf,
true_wf,
nat_wf,
p-outcome_wf,
not_wf,
Konig_wf,
length_wf_nat,
rationals_wf,
finite-prob-space_wf,
qinv-positive,
qless-int,
qdiv_wf,
int_nzero-rational,
subtype_base_sq,
int_subtype_base,
equal_wf,
nequal_wf,
qless_wf,
all_wf,
p-open-member_wf,
p-measure-le_wf,
equal-wf-base,
double-negation-hyp-elim,
eq_int_wf,
int_seg_wf,
assert_of_eq_int,
subtype_rel_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
assert_wf,
le_wf,
nat_properties,
decidable__equal_int,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
less_than_wf,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
length_wf,
decidable__le,
natural_number_wf_p-outcome,
nequal-le-implies,
decidable__lt,
neg_assert_of_eq_int,
exists_wf,
expectation-constant,
int-subtype-rationals,
subtype_rel_set,
int-equal-in-rationals,
int_seg_subtype_nat,
equal-wf-T-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
functionEquality,
sqequalRule,
dependent_functionElimination,
because_Cache,
setElimination,
rename,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
applyEquality,
independent_isectElimination,
productElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
addLevel,
instantiate,
cumulativity,
intEquality,
dependent_pairFormation,
promote_hyp,
productEquality,
functionExtensionality,
dependent_pairEquality,
unionElimination,
applyLambdaEquality,
approximateComputation,
int_eqEquality,
voidEquality,
equalityElimination,
hyp_replacement,
allFunctionality,
levelHypothesis,
allLevelFunctionality,
existsFunctionality
Latex:
\mforall{}[p:FinProbSpace]. \mneg{}nullset(p;\mlambda{}s.True) supposing \mneg{}\mneg{}Konig(||p||)
Date html generated:
2018_05_22-AM-00_37_04
Last ObjectModification:
2018_05_16-PM-01_03_03
Theory : randomness
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