Nuprl Lemma : select_l_index
∀[T:Type]. ∀[dT:EqDecider(T)]. ∀[L:T List]. ∀[x:T]. L[index(L;x)] = x ∈ T supposing (x ∈ L)
Proof
Definitions occuring in Statement :
l_index: index(L;x),
l_member: (x ∈ l),
select: L[n],
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
l_index: index(L;x),
deq: EqDecider(T),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
eqof: eqof(d),
so_lambda: λ2x.t[x],
so_apply: x[s],
l_member: (x ∈ l),
nat: ℕ,
le: A ≤ B,
cand: A c∧ B,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
l_member_wf,
list_wf,
deq_wf,
mu-bound-property,
length_wf_nat,
select_wf,
int_seg_properties,
length_wf,
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,
int_seg_wf,
safe-assert-deq,
assert_wf,
eqof_wf,
exists_wf,
equal_wf,
lelt_wf,
nat_properties,
deq_property,
l_index_wf,
non_neg_length,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
lambdaEquality,
applyEquality,
setElimination,
rename,
independent_isectElimination,
natural_numberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
addLevel,
existsFunctionality,
levelHypothesis,
existsLevelFunctionality,
dependent_set_memberEquality,
applyLambdaEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[dT:EqDecider(T)]. \mforall{}[L:T List]. \mforall{}[x:T]. L[index(L;x)] = x supposing (x \mmember{} L)
Date html generated:
2017_04_17-AM-09_15_59
Last ObjectModification:
2017_02_27-PM-05_21_24
Theory : decidable!equality
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