Nuprl Lemma : length-list-delete
∀[T:Type]. ∀[as:T List]. ∀[i:ℕ]. ||as\i|| = (||as|| - 1) ∈ ℤ supposing i < ||as||
Proof
Definitions occuring in Statement :
list-delete: as\i,
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
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,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
list-delete: as\i,
nil: [],
it: ⋅,
subtract: n - m,
all: ∀x:A. B[x],
cons: [a / b],
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
false: False,
guard: {T},
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
not: ¬A,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
nat_plus: ℕ+
Lemmas referenced :
list_induction,
uall_wf,
nat_wf,
isect_wf,
less_than_wf,
length_wf,
equal_wf,
list-delete_wf,
subtract_wf,
list_wf,
length_of_nil_lemma,
nil_wf,
length_of_cons_lemma,
spread_cons_lemma,
cons_wf,
less_than_transitivity1,
less_than_irreflexivity,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
add-associates,
add-swap,
add-commutes,
zero-add,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__le,
false_wf,
not-le-2,
not-lt-2,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add_functionality_wrt_le,
le-add-cancel,
le_wf,
non_neg_length,
length_wf_nat,
set_subtype_base,
int_subtype_base,
less-iff-le,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
omega-shadow,
mul-distributes,
mul-associates,
mul-commutes,
le-add-cancel-alt,
squash_wf,
true_wf,
add_functionality_wrt_eq,
iff_weakening_equal,
nat_properties,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
setElimination,
rename,
cumulativity,
intEquality,
because_Cache,
natural_numberEquality,
independent_functionElimination,
voidEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_functionElimination,
voidElimination,
universeEquality,
independent_isectElimination,
unionElimination,
equalityElimination,
productElimination,
lessCases,
sqequalAxiom,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
minusEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
dependent_set_memberEquality,
addEquality,
sqequalIntensionalEquality,
multiplyEquality
Latex:
\mforall{}[T:Type]. \mforall{}[as:T List]. \mforall{}[i:\mBbbN{}]. ||as\mbackslash{}i|| = (||as|| - 1) supposing i < ||as||
Date html generated:
2017_04_14-AM-08_55_37
Last ObjectModification:
2017_02_27-PM-03_39_30
Theory : omega
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