Nuprl Lemma : length_nth_tl
∀[A:Type]. ∀[as:A List]. ∀[n:{0...||as||}]. (||nth_tl(n;as)|| = (||as|| - n) ∈ ℤ)
Proof
Definitions occuring in Statement :
length: ||as||,
nth_tl: nth_tl(n;as),
list: T List,
int_iseg: {i...j},
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
or: P ∨ Q,
cons: [a / b],
less_than': less_than'(a;b),
not: ¬A,
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_stable: SqStable(P),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
subtype_rel: A ⊆r B,
int_iseg: {i...j},
exists: ∃x:A. B[x],
nat_plus: ℕ+,
nth_tl: nth_tl(n;as),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
spread_cons_lemma,
sq_stable__le,
decidable__int_equal,
subtract_wf,
istype-false,
not-equal-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
minus-minus,
le_weakening2,
istype-nat,
list_wf,
istype-universe,
le_int_wf,
equal-wf-base,
bool_wf,
and_wf,
istype-int,
assert_wf,
equal_wf,
zero_ann_a,
add-zero,
minus-zero,
iff_weakening_equal,
trivial-equal,
lt_int_wf,
less_than_wf,
bnot_wf,
less-iff-le,
int_iseg_wf,
length_wf,
non_neg_length,
length_wf_nat,
istype-sqequal,
not-less-implies-equal,
add-is-int-iff,
le_reflexive,
one-mul,
add-mul-special,
zero-mul,
omega-shadow,
two-mul,
mul-distributes-right,
mul-associates,
decidable__le,
not-le-2,
length_of_nil_lemma,
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_nil_lemma,
length_of_cons_lemma,
reduce_tl_cons_lemma,
int_iseg_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
universeIsType,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
promote_hyp,
hypothesis_subsumption,
Error :memTop,
equalityIstype,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
addEquality,
minusEquality,
baseApply,
closedConclusion,
applyEquality,
sqequalBase,
universeEquality,
multiplyEquality,
inlFormation_alt,
inrFormation_alt,
dependent_pairFormation_alt,
productIsType,
equalityElimination,
hyp_replacement
Latex:
\mforall{}[A:Type]. \mforall{}[as:A List]. \mforall{}[n:\{0...||as||\}]. (||nth\_tl(n;as)|| = (||as|| - n))
Date html generated:
2020_05_19-PM-09_37_03
Last ObjectModification:
2020_02_06-PM-09_36_52
Theory : list_0
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