Nuprl Lemma : select-qv-add
∀[as,bs:Top List]. ∀[i:ℕdimension(as)]. (qv-add(as;bs)[i] ~ as[i] + bs[i]) supposing dimension(as) = dimension(bs) ∈ ℤ
Proof
Definitions occuring in Statement :
qv-add: qv-add(as;bs),
qv-dim: dimension(as),
qadd: r + s,
select: L[n],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
natural_number: $n,
int: ℤ,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
qv-dim: dimension(as),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
qv-add: qv-add(as;bs),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
ifthenelse: if b then t else f fi ,
btrue: tt,
int_seg: {i..j-},
lelt: i ≤ j < k,
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bfalse: ff,
uiff: uiff(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,
int_seg_wf,
length_wf,
top_wf,
equal_wf,
list_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
length_of_nil_lemma,
null_nil_lemma,
reduce_tl_nil_lemma,
stuck-spread,
base_wf,
int_seg_properties,
equal-wf-base-T,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
length_of_cons_lemma,
null_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
add-is-int-iff,
false_wf,
decidable__lt,
lelt_wf,
select-cons-tl
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
unionElimination,
baseClosed,
productElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
instantiate,
cumulativity,
imageElimination,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[as,bs:Top List].
\mforall{}[i:\mBbbN{}dimension(as)]. (qv-add(as;bs)[i] \msim{} as[i] + bs[i]) supposing dimension(as) = dimension(bs)
Date html generated:
2018_05_22-AM-00_20_02
Last ObjectModification:
2017_07_26-PM-06_54_36
Theory : rationals
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