Nuprl Lemma : non_null_iff_length
∀[L:Top List]. uiff(¬↑null(L);0 < ||L||)
Proof
Definitions occuring in Statement :
length: ||as||,
null: null(as),
list: T List,
assert: ↑b,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
top: Top,
not: ¬A,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
true: True,
false: False,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
cons: [a / b],
top: Top,
bfalse: ff,
nat_plus: ℕ+,
guard: {T},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x]
Lemmas referenced :
top_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
not_wf,
true_wf,
less_than_wf,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
equal_wf,
length_wf,
member-less_than,
assert_wf,
null_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
sqequalRule,
independent_pairFormation,
independent_functionElimination,
natural_numberEquality,
voidElimination,
lambdaFormation,
imageElimination,
productElimination,
lambdaEquality,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
pointwiseFunctionality,
baseApply,
closedConclusion,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
addEquality,
independent_pairEquality
Latex:
\mforall{}[L:Top List]. uiff(\mneg{}\muparrow{}null(L);0 < ||L||)
Date html generated:
2017_04_17-AM-07_36_09
Last ObjectModification:
2017_02_27-PM-04_10_22
Theory : list_1
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