Nuprl Lemma : l_member-alt-def
∀[T:Type]. ∀L:T List. ∀x:T. ((x ∈ L) ⇐⇒ ∃i:ℕ||L||. (x = L[i] ∈ T))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
prop: ℙ,
cand: A c∧ B,
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
lelt_wf,
equal_wf,
select_wf,
int_seg_properties,
length_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
exists_wf,
nat_wf,
less_than_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
setElimination,
rename,
dependent_set_memberEquality,
hypothesisEquality,
hypothesis,
cut,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
addLevel,
levelHypothesis,
independent_isectElimination,
natural_numberEquality,
cumulativity,
dependent_functionElimination,
unionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
productEquality,
applyEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}x:T. ((x \mmember{} L) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||L||. (x = L[i]))
Date html generated:
2017_04_17-AM-07_25_28
Last ObjectModification:
2017_02_27-PM-04_04_19
Theory : list_1
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