Nuprl Lemma : general_length_nth_tl
∀[r:ℕ]. ∀[L:Top List]. (||nth_tl(r;L)|| = if r <z ||L|| then ||L|| - r else 0 fi ∈ ℤ)
Proof
Definitions occuring in Statement :
length: ||as||,
nth_tl: nth_tl(n;as),
list: T List,
nat: ℕ,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
top: Top,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
nth_tl: nth_tl(n;as),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtract: n - m,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
guard: {T},
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
cons: [a / b],
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
list_wf,
top_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
lt_int_wf,
length_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
le_int_wf,
le_wf,
bnot_wf,
non_neg_length,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
equal-wf-base,
int_subtype_base,
bnot_of_le_int,
tl_wf,
nth_tl_wf,
squash_wf,
true_wf,
ifthenelse_wf,
length_tl,
iff_weakening_equal,
list-cases,
reduce_tl_nil_lemma,
nth_tl_nil,
length_of_nil_lemma,
product_subtype_list,
reduce_tl_cons_lemma,
nil_wf,
length_of_null_list,
nth_tl_is_nil,
length_of_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
le_weakening2,
length_nth_tl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
unionElimination,
because_Cache,
baseClosed,
imageElimination,
productElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
applyEquality,
universeEquality,
imageMemberEquality,
promote_hyp,
hypothesis_subsumption,
equalityUniverse,
levelHypothesis,
dependent_set_memberEquality,
addEquality,
minusEquality
Latex:
\mforall{}[r:\mBbbN{}]. \mforall{}[L:Top List]. (||nth\_tl(r;L)|| = if r <z ||L|| then ||L|| - r else 0 fi )
Date html generated:
2018_05_21-PM-06_54_14
Last ObjectModification:
2017_07_26-PM-04_59_10
Theory : general
Home
Index