Nuprl Lemma : quadratic2-zero
∀[a,b,c:ℝ]. (quadratic2(a;b;c) = r0) supposing ((b ≤ r0) and (c = r0) and a ≠ r0)
Proof
Definitions occuring in Statement :
quadratic2: quadratic2(a;b;c),
rneq: x ≠ y,
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
quadratic2: quadratic2(a;b;c),
assert: ↑b,
ifthenelse: if b then t else f fi ,
nonneg-poly: nonneg-poly(p),
bl-all: (∀x∈L.P[x])_b,
reduce: reduce(f;k;as),
list_ind: list_ind,
int_term_to_ipoly: int_term_to_ipoly(t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
add_ipoly: add_ipoly(p;q),
add-ipoly-prepend: add-ipoly-prepend(p;q;l),
itermMultiply: left (*) right,
mul_ipoly: mul_ipoly(p;q),
itermVar: vvar,
cons: [a / b],
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
nil: [],
it: ⋅,
mul-mono-poly: mul-mono-poly(m;p),
mul-monomials: mul-monomials(m1;m2),
merge-int-accum: merge-int-accum(as;bs),
eager-accum: eager-accum(x,a.f[x; a];y;l),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
insert-int: insert-int(x;l),
minus-poly: minus-poly(p),
map: map(f;as),
itermConstant: "const",
rev-append: rev(as) + bs,
list_accum: list_accum,
band: p ∧b q,
nonneg-monomial: nonneg-monomial(m),
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
even-int-list: even-int-list(L),
bor: p ∨bq,
null: null(as),
tl: tl(l),
pi2: snd(t),
eq_int: (i =z j),
hd: hd(l),
pi1: fst(t),
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
guard: {T},
rdiv: (x/y)
Lemmas referenced :
sq_stable__req,
quadratic2_wf,
rmul_preserves_rless,
int-to-real_wf,
rless-int,
rless_wf,
rmul_wf,
rleq_wf,
req_wf,
rneq_wf,
real_wf,
rsub_wf,
real-term-nonneg,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
req-iff-rsub-is-0,
rdiv_wf,
rminus_wf,
rsqrt_wf,
square-nonneg,
rabs_wf,
rleq_functionality,
req_weakening,
rsub_functionality,
rmul_functionality,
req_functionality,
real_polynomial_null,
rless_functionality,
rdiv_functionality,
rsqrt_functionality,
rsqrt_square,
rinv_wf2,
itermMinus_wf,
rabs-of-nonpos,
req_transitivity,
rinv-of-rmul,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
independent_functionElimination,
unionElimination,
inlFormation_alt,
dependent_functionElimination,
natural_numberEquality,
productElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
universeIsType,
inrFormation_alt,
imageElimination,
inhabitedIsType,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
applyEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
approximateComputation
Latex:
\mforall{}[a,b,c:\mBbbR{}]. (quadratic2(a;b;c) = r0) supposing ((b \mleq{} r0) and (c = r0) and a \mneq{} r0)
Date html generated:
2019_10_30-AM-07_59_47
Last ObjectModification:
2019_10_10-AM-10_51_23
Theory : reals
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