Nuprl Lemma : int-dot-add-left
∀[as,bs,cs:ℤ List]. as + bs ⋅ cs ~ as ⋅ cs + bs ⋅ cs supposing ||as|| = ||bs|| ∈ ℤ
Proof
Definitions occuring in Statement :
int-vec-add: as + bs,
integer-dot-product: as ⋅ bs,
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
add: n + m,
int: ℤ,
sqequal: s ~ t,
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,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
int-vec-add: as + bs,
nil: [],
it: ⋅,
top: Top,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
subtract: n - m,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
sq_stable: SqStable(P),
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-base,
list_wf,
nat_wf,
list-cases,
int_dot_nil_left_lemma,
length_of_nil_lemma,
product_subtype_list,
spread_cons_lemma,
equal_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
length_of_cons_lemma,
non_neg_length,
length_wf_nat,
subtype_rel-equal,
base_wf,
le_antisymmetry_iff,
length_wf,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
integer-dot-product_wf,
cons_wf,
colength_wf_list,
sq_stable__le,
equal-wf-T-base,
decidable__le,
false_wf,
not-le-2,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
list_subtype_base,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
one-mul,
minus-zero,
int_dot_cons_nil_lemma,
int_dot_cons_lemma,
decidable__equal_int,
not-equal-2,
le-add-cancel2
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,
intEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
unionElimination,
voidEquality,
promote_hyp,
hypothesis_subsumption,
productElimination,
instantiate,
cumulativity,
dependent_pairFormation,
sqequalIntensionalEquality,
addEquality,
minusEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
dependent_set_memberEquality,
independent_pairFormation,
multiplyEquality
Latex:
\mforall{}[as,bs,cs:\mBbbZ{} List]. as + bs \mcdot{} cs \msim{} as \mcdot{} cs + bs \mcdot{} cs supposing ||as|| = ||bs||
Date html generated:
2017_04_14-AM-08_55_52
Last ObjectModification:
2017_02_27-PM-03_39_59
Theory : omega
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