Nuprl Lemma : quadratic-1-2-equal
∀[a,b,c:ℝ].
({(quadratic1(a;b;c) = quadratic2(a;b;c)) ∧ (quadratic1(a;b;c) = (-(b)/r(2) * a))}) supposing
(((b * b) = (r(4) * a * c)) and
a ≠ r0)
Proof
Definitions occuring in Statement :
quadratic2: quadratic2(a;b;c),
quadratic1: quadratic1(a;b;c),
rdiv: (x/y),
rneq: x ≠ y,
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
top: Top,
not: ¬A,
false: False,
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
quadratic1: quadratic1(a;b;c),
quadratic2: quadratic2(a;b;c),
prop: ℙ,
uimplies: b supposing a,
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,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
rneq: x ≠ y,
guard: {T}
Lemmas referenced :
real_term_value_minus_lemma,
real_term_value_add_lemma,
radd_functionality,
rdiv_functionality,
rsqrt_functionality,
req_functionality,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
real_wf,
rneq_wf,
req_wf,
rdiv_wf,
quadratic2_wf,
rsub_functionality,
rleq_functionality,
quadratic1_wf,
req_witness,
req_weakening,
itermMinus_wf,
itermAdd_wf,
rminus_wf,
radd_wf,
rsqrt0,
rleq_weakening_equal,
rleq_wf,
rleq_weakening,
req_inversion,
rsqrt_wf,
req-iff-rsub-is-0,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
itermSubtract_wf,
rsub_wf,
req-implies-req,
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,
lambdaEquality,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
isect_memberEquality,
independent_pairEquality,
isect_memberFormation,
applyEquality,
dependent_set_memberEquality,
inrFormation,
because_Cache,
independent_isectElimination,
baseClosed,
imageMemberEquality,
independent_pairFormation,
productElimination,
independent_functionElimination,
hypothesis,
natural_numberEquality,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
inlFormation,
thin,
unionElimination,
sqequalHypSubstitution,
cut,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[a,b,c:\mBbbR{}].
(\{(quadratic1(a;b;c) = quadratic2(a;b;c)) \mwedge{} (quadratic1(a;b;c) = (-(b)/r(2) * a))\}) supposing
(((b * b) = (r(4) * a * c)) and
a \mneq{} r0)
Date html generated:
2018_05_22-PM-02_24_07
Last ObjectModification:
2018_05_21-AM-00_46_26
Theory : reals
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