Nuprl Lemma : q-linear-sum
∀[X,Y:ℕ ⟶ ℚ]. ∀[k:ℕ]. ∀[y:ℚ List].
q-linear(k;j.X[j] + Y[j];y) = (q-linear(k;j.X[j];y) + q-linear(k;j.Y[j];y)) ∈ ℚ supposing k ≤ ||y||
Proof
Definitions occuring in Statement :
q-linear: q-linear(k;i.X[i];y),
qadd: r + s,
rationals: ℚ,
length: ||as||,
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+
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,
le_wf,
length_wf,
rationals_wf,
list_wf,
equal_wf,
squash_wf,
true_wf,
q-linear-base,
qadd_wf,
nat_wf,
q-linear_wf,
false_wf,
iff_weakening_equal,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
qmul_wf,
select_wf,
decidable__lt,
q-linear-unroll,
qmul_over_plus_qrng,
qmul_comm_qrng,
mon_assoc_q,
qadd_ac_1_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
imageElimination,
universeEquality,
functionExtensionality,
dependent_set_memberEquality,
because_Cache,
imageMemberEquality,
baseClosed,
productElimination,
unionElimination,
functionEquality
Latex:
\mforall{}[X,Y:\mBbbN{} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[y:\mBbbQ{} List].
q-linear(k;j.X[j] + Y[j];y) = (q-linear(k;j.X[j];y) + q-linear(k;j.Y[j];y)) supposing k \mleq{} ||y||
Date html generated:
2018_05_22-AM-00_17_35
Last ObjectModification:
2017_07_26-PM-06_53_31
Theory : rationals
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