Nuprl Lemma : select_nth_tl
∀[T:Type]. ∀[as:T List]. ∀[n:{0...||as||}]. ∀[i:ℕ||as|| - n]. (nth_tl(n;as)[i] = as[i + n] ∈ T)
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
nth_tl: nth_tl(n;as),
list: T List,
int_iseg: {i...j},
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
int_iseg: {i...j},
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
and: P ∧ Q,
lelt: i ≤ j < k,
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,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
so_apply: x[s],
nth_tl: nth_tl(n;as),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
cand: A c∧ B,
ge: i ≥ j ,
less_than: a < b,
subtract: n - m
Lemmas referenced :
list_induction,
uall_wf,
int_iseg_wf,
length_wf,
int_seg_wf,
subtract_wf,
equal_wf,
select_wf,
nth_tl_wf,
int_seg_properties,
int_iseg_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,
less_than_wf,
squash_wf,
true_wf,
length_nth_tl,
iff_weakening_equal,
decidable__lt,
intformless_wf,
itermSubtract_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
list_wf,
length_of_nil_lemma,
nil_wf,
length_of_cons_lemma,
cons_wf,
le_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
le_wf,
lt_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
reduce_tl_cons_lemma,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
add-zero,
add-is-int-iff,
false_wf,
non_neg_length,
lelt_wf,
select_cons_tl,
add-commutes,
add-swap,
add-associates
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
cumulativity,
hypothesis,
because_Cache,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
addEquality,
axiomEquality,
lambdaFormation,
equalityElimination,
instantiate,
productEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
dependent_set_memberEquality,
minusEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as:T List]. \mforall{}[n:\{0...||as||\}]. \mforall{}[i:\mBbbN{}||as|| - n]. (nth\_tl(n;as)[i] = as[i + n])
Date html generated:
2017_04_14-AM-09_25_37
Last ObjectModification:
2017_02_27-PM-04_00_06
Theory : list_1
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