Nuprl Lemma : DAlembert-equation-lemma
∀f,g:ℝ ⟶ ℝ.
((∀x,y:ℝ. ((x = y) ⇒ (f(x) = f(y))))
⇒ (∀x,y:ℝ. ((x = y) ⇒ (g(x) = g(y))))
⇒ ((∀x:ℝ. (f(-(x)) = f(x)))
∧ (∀n:ℤ. ∀y:ℝ. (f(r(n + 1) * y) = ((r(2) * f(y) * f(r(n) * y)) - f(r(n - 1) * y))))
∧ (∀t:ℝ. ((f((t/r(2))) * f((t/r(2)))) = (f(t) + r1/r(2)))))
⇒ ((∀x:ℝ. (g(-(x)) = g(x)))
∧ (∀n:ℤ. ∀y:ℝ. (g(r(n + 1) * y) = ((r(2) * g(y) * g(r(n) * y)) - g(r(n - 1) * y))))
∧ (∀t:ℝ. ((g((t/r(2))) * g((t/r(2)))) = (g(t) + r1/r(2)))))
⇒ (f(r0) = g(r0))
⇒ (∀a:ℝ
((r0 < a)
⇒ (f(a) = g(a))
⇒ (∀x:{x:ℝ| x ∈ [-(a), a]} . (r0 < f(x)))
⇒ (∀x:{x:ℝ| x ∈ [-(a), a]} . (r0 < g(x)))
⇒ (∀x:ℝ. (f(x) = g(x))))))
Proof
Definitions occuring in Statement :
rfun-ap: f(x),
rccint: [l, u],
i-member: r ∈ I,
rdiv: (x/y),
rless: x < y,
rsub: x - y,
req: x = y,
rmul: a * b,
rminus: -(x),
radd: a + b,
int-to-real: r(n),
real: ℝ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
r-ap: f(x),
rfun: I ⟶ℝ,
riiint: (-∞, ∞),
i-member: r ∈ I,
lelt: i ≤ j < k,
int_seg: {i..j-},
int_upper: {i...},
rfun-ap: f(x),
primrec: primrec(n;b;c),
exp: i^n,
le: A ≤ B,
rgt: x > y,
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rneq: x ≠ y,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
guard: {T},
true: True,
less_than': less_than'(a;b),
less_than: a < b,
false: False,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
ge: i ≥ j ,
nat_plus: ℕ+,
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
subtype_rel: A ⊆r B,
sq_exists: ∃x:A [B[x]],
rless: x < y,
nat: ℕ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
dyadic-scaled-rationals-dense,
set_wf,
subtype_rel_dep_function,
functions-equal-on-dense,
riiint_wf,
exists_wf,
rneq_wf,
member_riiint_lemma,
rminus_functionality,
rminus-int,
squash_wf,
rmul_over_rminus,
int_term_value_minus_lemma,
rsub_functionality,
subtract-add-cancel,
rmul-zero-both,
int_term_value_add_lemma,
itermAdd_wf,
subtype_rel_self,
int_formula_prop_eq_lemma,
intformeq_wf,
lelt_wf,
set_subtype_base,
decidable__equal_int,
int_seg_properties,
int_seg_wf,
radd_functionality,
rdiv_functionality,
rnexp2,
rnexp_functionality,
rnexp_wf,
square-req-iff,
false_wf,
exp-ge-1,
le_weakening2,
rinv1,
rfun-ap_functionality,
exp0_lemma,
req_witness,
ge_wf,
rmul-int,
rmul_functionality,
rmul-assoc,
req_functionality,
exp_step,
req-int,
nat_plus_wf,
rmul_preserves_req,
exp-positive,
nat_plus_subtype_nat,
decidable__lt,
fractions-req,
int_term_value_subtract_lemma,
rminus_functionality_wrt_rleq,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
req_weakening,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rmul-rinv3,
req_transitivity,
rmul_comm,
rleq_weakening,
rleq_functionality,
rmul_reverses_rleq_iff,
rmul-identity1,
rleq_weakening_rless,
rmul_preserves_rleq2,
rinv_wf2,
req-iff-rsub-is-0,
itermMinus_wf,
itermMultiply_wf,
itermSubtract_wf,
radd_wf,
subtract_wf,
rsub_wf,
rmul_wf,
req_wf,
rfun-ap_wf,
rless_wf,
all_wf,
rccint_wf,
i-member_wf,
nequal_wf,
true_wf,
equal_wf,
int_subtype_base,
subtype_base_sq,
exp_wf3,
int_nzero-rational,
equal_functionality_wrt_subtype_rel2,
int-subtype-rationals,
subtype_rel_set,
rationals_wf,
equal-wf-T-base,
not_functionality_wrt_implies,
rneq-int,
exp-positive-stronger,
rless-int,
rdiv_wf,
rminus_wf,
rmul_preserves_rleq,
member_rccint_lemma,
nat_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformnot_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
less_than_wf,
exp_wf_nat_plus,
le_wf,
decidable__le,
nat_plus_properties,
real_wf,
int-to-real_wf,
sq_stable__less_than,
nat_properties,
exp_wf2,
rleq-int
Rules used in proof :
inlFormation,
hypothesis_subsumption,
intWeakElimination,
multiplyEquality,
minusEquality,
functionEquality,
functionExtensionality,
productEquality,
setEquality,
cumulativity,
instantiate,
addLevel,
inrFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
independent_isectElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
because_Cache,
unionElimination,
dependent_set_memberEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
addEquality,
rename,
setElimination,
independent_functionElimination,
hypothesis,
hypothesisEquality,
isectElimination,
natural_numberEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}f,g:\mBbbR{} {}\mrightarrow{} \mBbbR{}.
((\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (f(x) = f(y))))
{}\mRightarrow{} (\mforall{}x,y:\mBbbR{}. ((x = y) {}\mRightarrow{} (g(x) = g(y))))
{}\mRightarrow{} ((\mforall{}x:\mBbbR{}. (f(-(x)) = f(x)))
\mwedge{} (\mforall{}n:\mBbbZ{}. \mforall{}y:\mBbbR{}. (f(r(n + 1) * y) = ((r(2) * f(y) * f(r(n) * y)) - f(r(n - 1) * y))))
\mwedge{} (\mforall{}t:\mBbbR{}. ((f((t/r(2))) * f((t/r(2)))) = (f(t) + r1/r(2)))))
{}\mRightarrow{} ((\mforall{}x:\mBbbR{}. (g(-(x)) = g(x)))
\mwedge{} (\mforall{}n:\mBbbZ{}. \mforall{}y:\mBbbR{}. (g(r(n + 1) * y) = ((r(2) * g(y) * g(r(n) * y)) - g(r(n - 1) * y))))
\mwedge{} (\mforall{}t:\mBbbR{}. ((g((t/r(2))) * g((t/r(2)))) = (g(t) + r1/r(2)))))
{}\mRightarrow{} (f(r0) = g(r0))
{}\mRightarrow{} (\mforall{}a:\mBbbR{}
((r0 < a)
{}\mRightarrow{} (f(a) = g(a))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [-(a), a]\} . (r0 < f(x)))
{}\mRightarrow{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [-(a), a]\} . (r0 < g(x)))
{}\mRightarrow{} (\mforall{}x:\mBbbR{}. (f(x) = g(x))))))
Date html generated:
2018_05_22-PM-03_09_01
Last ObjectModification:
2018_05_20-PM-11_41_59
Theory : reals_2
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