Nuprl Lemma : member-exists2
∀[T:Type]. ∀L:T List. (∃x:T. (x ∈ L) ⇐⇒ 0 < ||L||)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
length: ||as||,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
ge: i ≥ j ,
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
l_member_wf,
exists_wf,
member-exists,
iff_wf,
nil_wf,
list_wf,
equal_wf,
not_wf,
length_of_not_nil,
less_than_wf,
less_than'_wf,
decidable__le,
ge_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
length_wf,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
sqequalHypSubstitution,
lemma_by_obid,
dependent_functionElimination,
thin,
natural_numberEquality,
isectElimination,
hypothesisEquality,
hypothesis,
unionElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
introduction,
imageElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addLevel,
impliesFunctionality,
independent_functionElimination,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. (\mexists{}x:T. (x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} 0 < ||L||)
Date html generated:
2016_05_14-PM-01_30_26
Last ObjectModification:
2016_01_15-AM-08_27_37
Theory : list_1
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