Nuprl Lemma : subtract-rpolynomials-same-degree
∀[n:ℕ]. ∀[a,b:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ]. (((Σi≤n. a_i * x^i) - (Σi≤n. b_i * x^i)) = (Σi≤n. λi.((a i) - b i)_i * x^i))
Proof
Definitions occuring in Statement :
rpolynomial: (Σi≤n. a_i * x^i),
rsub: x - y,
req: x = y,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
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,
nat: ℕ,
implies: P ⇒ Q,
int_seg: {i..j-},
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
lelt: i ≤ j < k,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rsub_wf,
rpolynomial_wf,
int_seg_wf,
real_wf,
istype-nat,
ifthenelse_wf,
le_int_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eqtt_to_assert,
assert_of_le_int,
req_weakening,
intformand_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
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,
subtract-rpolynomials,
rpolynomial_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
applyEquality,
universeIsType,
natural_numberEquality,
addEquality,
setElimination,
rename,
independent_functionElimination,
sqequalRule,
isect_memberEquality_alt,
because_Cache,
isectIsTypeImplies,
inhabitedIsType,
functionIsType,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
lambdaFormation_alt,
equalityElimination,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}].
(((\mSigma{}i\mleq{}n. a\_i * x\^{}i) - (\mSigma{}i\mleq{}n. b\_i * x\^{}i)) = (\mSigma{}i\mleq{}n. \mlambda{}i.((a i) - b i)\_i * x\^{}i))
Date html generated:
2019_10_29-AM-10_14_44
Last ObjectModification:
2019_01_06-PM-05_25_06
Theory : reals
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