Nuprl Lemma : quadratic-1-2-equal-implies
∀[a,b,c:ℝ].
((b * b) = (r(4) * a * c)) supposing
((quadratic1(a;b;c) = quadratic2(a;b;c)) and
(r0 ≤ ((b * b) - r(4) * a * c)) and
a ≠ r0)
Proof
Definitions occuring in Statement :
quadratic2: quadratic2(a;b;c),
quadratic1: quadratic1(a;b;c),
rneq: x ≠ y,
rleq: x ≤ y,
rsub: x - y,
req: x = y,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
quadratic1: quadratic1(a;b;c),
quadratic2: quadratic2(a;b;c),
guard: {T},
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
rneq: x ≠ y,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rsqrt-is-zero,
radd-rminus,
real_term_value_minus_lemma,
real_term_value_add_lemma,
itermMinus_wf,
itermAdd_wf,
radd-rminus-assoc,
radd-preserves-req,
req_functionality,
rmul_preserves_req,
fractions-req,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
itermSubtract_wf,
quadratic2_wf,
quadratic1_wf,
req_witness,
req-implies-req,
equal_wf,
rneq_wf,
real_wf,
rleq_wf,
rsqrt_wf,
rminus_wf,
radd_wf,
rdiv_wf,
req_wf,
rsub_wf,
rless_wf,
rmul-zero-both,
rmul_comm,
rmul_wf,
rless_functionality,
rless-int,
int-to-real_wf,
rmul_preserves_rless
Rules used in proof :
voidEquality,
voidElimination,
intEquality,
int_eqEquality,
approximateComputation,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
productEquality,
setEquality,
rename,
setElimination,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
lambdaFormation,
inrFormation,
because_Cache,
independent_isectElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
sqequalRule,
productElimination,
independent_functionElimination,
hypothesis,
natural_numberEquality,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
inlFormation,
thin,
unionElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,b,c:\mBbbR{}].
((b * b) = (r(4) * a * c)) supposing
((quadratic1(a;b;c) = quadratic2(a;b;c)) and
(r0 \mleq{} ((b * b) - r(4) * a * c)) and
a \mneq{} r0)
Date html generated:
2018_05_22-PM-02_24_23
Last ObjectModification:
2018_05_21-AM-00_47_04
Theory : reals
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