Nuprl Lemma : int-dot-select
∀[as,bs:ℤ List]. ∀[i:ℕ]. as ⋅ bs ~ (as[i] * bs[i]) + as\i ⋅ bs\i supposing i < ||as|| ∧ i < ||bs||
Proof
Definitions occuring in Statement :
list-delete: as\i,
integer-dot-product: as ⋅ bs,
select: L[n],
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
multiply: n * m,
add: n + m,
int: ℤ,
sqequal: s ~ 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,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
top: Top,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
squash: ↓T,
sq_stable: SqStable(P),
uiff: uiff(P;Q),
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
less_than: a < b,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
list-delete: as\i,
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
length_wf,
nat_wf,
list_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
list-cases,
nil_wf,
length_of_nil_lemma,
int_dot_nil_left_lemma,
stuck-spread,
base_wf,
product_subtype_list,
spread_cons_lemma,
equal_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
length_of_cons_lemma,
colength_wf_list,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
equal-wf-T-base,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
cons_wf,
int_dot_cons_nil_lemma,
int_dot_cons_lemma,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
lt_int_wf,
assert_of_lt_int,
top_wf,
integer-dot-product_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
not-lt-2,
minus-zero,
le-add-cancel2,
select_wf,
decidable__lt,
select-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
sqequalAxiom,
productEquality,
intEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
unionElimination,
voidEquality,
productElimination,
promote_hyp,
hypothesis_subsumption,
instantiate,
cumulativity,
addEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
dependent_set_memberEquality,
independent_pairFormation,
minusEquality,
equalityElimination,
lessCases,
multiplyEquality,
dependent_pairFormation
Latex:
\mforall{}[as,bs:\mBbbZ{} List]. \mforall{}[i:\mBbbN{}]. as \mcdot{} bs \msim{} (as[i] * bs[i]) + as\mbackslash{}i \mcdot{} bs\mbackslash{}i supposing i < ||as|| \mwedge{} i < ||bs||
Date html generated:
2017_04_14-AM-08_55_57
Last ObjectModification:
2017_02_27-PM-03_40_02
Theory : omega
Home
Index