Nuprl Lemma : add-rpolynomials
∀[n,m:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[b:ℕm + 1 ⟶ ℝ]. ∀[x:ℝ].
((Σi≤n. a_i * x^i) + (Σi≤m. b_i * x^i)) = (Σi≤n. λi.if i ≤z m then (a i) + (b i) else a i fi _i * x^i) supposing m ≤ n
Proof
Definitions occuring in Statement :
rpolynomial: (Σi≤n. a_i * x^i),
req: x = y,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rpolynomial: (Σi≤n. a_i * x^i),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
guard: {T},
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
true: True,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
req_int_terms: t1 ≡ t2
Lemmas referenced :
req_witness,
radd_wf,
rpolynomial_wf,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_wf,
real_wf,
istype-nat,
rsum-split-shift,
rmul_wf,
rnexp_wf,
int_seg_subtype_nat,
istype-false,
decidable__le,
rsum_wf,
subtract_wf,
int_seg_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
le_wf,
req_functionality,
radd_functionality,
req_weakening,
ifthenelse_wf,
btrue_wf,
int_seg_subtype,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
bfalse_wf,
rsum_functionality,
rmul-distrib2,
req_inversion,
rsum_linearity1,
req-iff-rsub-is-0,
real_polynomial_null,
int-to-real_wf,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
setElimination,
rename,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
applyEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
dependent_functionElimination,
addEquality,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
functionIsType,
closedConclusion,
promote_hyp,
instantiate,
cumulativity,
minusEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[b:\mBbbN{}m + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}].
((\mSigma{}i\mleq{}n. a\_i * x\^{}i) + (\mSigma{}i\mleq{}m. b\_i * x\^{}i))
= (\mSigma{}i\mleq{}n. \mlambda{}i.if i \mleq{}z m then (a i) + (b i) else a i fi \_i * x\^{}i)
supposing m \mleq{} n
Date html generated:
2019_10_29-AM-10_14_02
Last ObjectModification:
2019_01_06-PM-05_13_03
Theory : reals
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