Nuprl Lemma : quadratic-formula2
∀a,b,c:ℝ.
(a ≠ r0
⇒ (r0 ≤ ((b * b) - r(4) * a * c))
⇒ (∀x:ℝ
((((a * x^2) + (b * x) + c) = r0)
⇒ ((¬¬((x = quadratic1(a;b;c)) ∨ (x = quadratic2(a;b;c))))
∧ ((((r(2) * a) * x) + b ≠ r0 ∨ (r0 < ((b * b) - r(4) * a * c)))
⇒ ((x = quadratic1(a;b;c)) ∨ (x = quadratic2(a;b;c))))))))
Proof
Definitions occuring in Statement :
quadratic2: quadratic2(a;b;c),
quadratic1: quadratic1(a;b;c),
rneq: x ≠ y,
rleq: x ≤ y,
rless: x < y,
rnexp: x^k1,
rsub: x - y,
req: x = y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rneq: x ≠ y,
or: P ∨ Q,
member: t ∈ T,
uall: ∀[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,
uimplies: b supposing a,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q),
prop: ℙ,
guard: {T},
quadratic2: quadratic2(a;b;c),
quadratic1: quadratic1(a;b;c),
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
cand: A c∧ B,
rsub: x - y,
stable: Stable{P},
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
nat_plus: ℕ+,
sq_exists: ∃x:{A| B[x]},
rless: x < y
Lemmas referenced :
rmul_preserves_rless,
int-to-real_wf,
rless-int,
rless_functionality,
rmul_wf,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rless_wf,
rsqrt_wf,
rsub_wf,
rleq_wf,
set_wf,
real_wf,
req_wf,
radd_wf,
rnexp_wf,
false_wf,
le_wf,
equal_wf,
rneq_wf,
squash_wf,
sq_stable__and,
sq_stable__rleq,
sq_stable__req,
req_witness,
req_functionality,
radd_functionality,
req_weakening,
rmul_functionality,
rnexp2,
req_transitivity,
itermAdd_wf,
real_term_value_add_lemma,
radd-preserves-req,
rminus_wf,
itermMinus_wf,
real_term_value_minus_lemma,
rminus_functionality,
rsqrt-unique2,
iff_weakening_equal,
or_wf,
true_wf,
not_wf,
req_inversion,
rdiv_wf,
radd_comm,
radd-zero-both,
rmul-zero-both,
radd-int,
rmul-distrib2,
rmul-identity1,
radd-assoc,
rminus-as-rmul,
rmul_comm,
rmul-rdiv-cancel2,
uiff_transitivity,
rmul_preserves_req,
stable_req,
rless_irreflexivity,
rleq_weakening,
rless_transitivity1,
radd-rminus-assoc,
radd-ac,
rmul-rdiv-cancel,
radd-preserves-rless,
rmul-is-positive,
rless_transitivity2,
rless-cases,
rleq_weakening_rless,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
intformless_wf,
satisfiable-full-omega-tt,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
unionElimination,
thin,
inlFormation,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
isectElimination,
natural_numberEquality,
hypothesis,
independent_functionElimination,
productElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
independent_isectElimination,
computeAll,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
inrFormation,
dependent_set_memberEquality,
productEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
imageElimination,
minusEquality,
setEquality,
universeEquality,
applyEquality,
orFunctionality,
addLevel,
functionEquality,
addEquality,
promote_hyp,
impliesFunctionality,
dependent_pairFormation
Latex:
\mforall{}a,b,c:\mBbbR{}.
(a \mneq{} r0
{}\mRightarrow{} (r0 \mleq{} ((b * b) - r(4) * a * c))
{}\mRightarrow{} (\mforall{}x:\mBbbR{}
((((a * x\^{}2) + (b * x) + c) = r0)
{}\mRightarrow{} ((\mneg{}\mneg{}((x = quadratic1(a;b;c)) \mvee{} (x = quadratic2(a;b;c))))
\mwedge{} ((((r(2) * a) * x) + b \mneq{} r0 \mvee{} (r0 < ((b * b) - r(4) * a * c)))
{}\mRightarrow{} ((x = quadratic1(a;b;c)) \mvee{} (x = quadratic2(a;b;c))))))))
Date html generated:
2017_10_03-AM-10_45_55
Last ObjectModification:
2017_07_28-AM-08_19_26
Theory : reals
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