Nuprl Lemma : not-bl-exists-eq-bl-all
∀[L,P:Top].  (¬b(∃x∈L.P[x])_b ~ (∀x∈L.¬bP[x])_b)
Proof
Definitions occuring in Statement : 
bl-exists: (∃x∈L.P[x])_b
, 
bl-all: (∀x∈L.P[x])_b
, 
bnot: ¬bb
, 
uall: ∀[x:A]. B[x]
, 
top: Top
, 
so_apply: x[s]
, 
sqequal: s ~ t
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
bl-all: (∀x∈L.P[x])_b
, 
bl-exists: (∃x∈L.P[x])_b
, 
reduce: reduce(f;k;as)
, 
list_ind: list_ind, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
false: False
, 
ge: i ≥ j 
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
all: ∀x:A. B[x]
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
nat_plus: ℕ+
, 
subtype_rel: A ⊆r B
, 
has-value: (a)↓
, 
sq_type: SQType(T)
, 
guard: {T}
, 
uiff: uiff(P;Q)
, 
btrue: tt
, 
bor: p ∨bq
, 
bfalse: ff
, 
band: p ∧b q
Lemmas referenced : 
has-value-band-type, 
bottom_diverge, 
has-value-implies-dec-isaxiom-2, 
assert_of_bnot, 
eqff_to_assert, 
eqtt_to_assert, 
bool_subtype_base, 
bool_wf, 
subtype_base_sq, 
bool_cases, 
isl_wf, 
injection-eta, 
has-value-bor-type, 
has-value-implies-dec-ispair-2, 
union-value-type, 
value-type-has-value, 
top_wf, 
is-exception_wf, 
has-value_wf_base, 
int_subtype_base, 
has-value-bnot-type, 
fun_exp_unroll_1, 
int_term_value_subtract_lemma, 
int_formula_prop_not_lemma, 
itermSubtract_wf, 
intformnot_wf, 
subtract_wf, 
decidable__le, 
bottom-sqle, 
strictness-decide, 
strictness-apply, 
fun_exp0_lemma, 
base_wf, 
less_than_wf, 
ge_wf, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_and_lemma, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
nat_properties
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalRule, 
thin, 
sqequalSqle, 
fixpointLeast, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
intWeakElimination, 
lambdaFormation, 
natural_numberEquality, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
independent_functionElimination, 
axiomSqleEquality, 
unionElimination, 
dependent_set_memberEquality, 
divergentSqle, 
baseApply, 
closedConclusion, 
baseClosed, 
applyEquality, 
because_Cache, 
equalityTransitivity, 
equalitySymmetry, 
decideExceptionCases, 
callbyvalueCallbyvalue, 
callbyvalueReduce, 
callbyvalueExceptionCases, 
exceptionSqequal, 
sqleReflexivity, 
sqequalAxiom, 
unionEquality, 
instantiate, 
cumulativity, 
productElimination
Latex:
\mforall{}[L,P:Top].    (\mneg{}\msubb{}(\mexists{}x\mmember{}L.P[x])\_b  \msim{}  (\mforall{}x\mmember{}L.\mneg{}\msubb{}P[x])\_b)
Date html generated:
2016_05_14-PM-02_11_22
Last ObjectModification:
2016_01_15-AM-08_03_46
Theory : list_1
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