Nuprl Lemma : approx-fixpoint-unit-ball-2
∀n:ℕ. ∀f:{f:B(n) ⟶ B(n)| (∀x,y:B(n). (req-vec(n;x;y) ⇒ req-vec(n;f x;f y))) ∧ (¬(∀x:B(n). f x ≠ x))} . ∀e:{e:ℝ|
r0 < e} .
∃p:B(n). (↓d(f p;p) < e)
Proof
Definitions occuring in Statement :
real-unit-ball: B(n),
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
req-vec: req-vec(n;x;y),
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
and: P ∧ Q,
implies: P ⇒ Q,
real-unit-ball: B(n),
subtype_rel: A ⊆r B,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
le: A ≤ B,
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
real-ball: B(n;r),
rneq: x ≠ y,
rdiv: (x/y),
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:A [B[x]],
rge: x ≥ y
Lemmas referenced :
find-approx-fp_wf,
real_wf,
rless_wf,
int-to-real_wf,
real-unit-ball_wf,
req-vec_wf,
real-vec-sep_wf,
istype-void,
istype-nat,
real-vec-dist_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
rless-int,
istype-le,
real-ball-uniform-continuity,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
subtype_rel_self,
subtype_rel_set,
real-vec_wf,
rleq_wf,
real-vec-norm_wf,
nat_wf,
set_subtype_base,
le_wf,
decidable__le,
sq_stable__rless,
rless_functionality,
real-vec-dist-dim0,
req_weakening,
subtype_rel_dep_function,
real-ball_wf,
rdiv_wf,
rmul_preserves_rless,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
rinv_wf2,
req_transitivity,
rmul-rinv3,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rless-int-fractions2,
nat_plus_properties,
int_term_value_mul_lemma,
rleq_weakening_rless,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd-preserves-rless,
rminus_wf,
radd_wf,
itermAdd_wf,
itermMinus_wf,
real_term_value_add_lemma,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
setIsType,
universeIsType,
isectElimination,
natural_numberEquality,
functionIsType,
because_Cache,
sqequalRule,
productIsType,
setElimination,
rename,
applyEquality,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
independent_pairFormation,
productElimination,
promote_hyp,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
dependent_pairFormation_alt,
imageMemberEquality,
baseClosed,
voidElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
imageElimination,
closedConclusion,
inrFormation_alt,
multiplyEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f:\{f:B(n) {}\mrightarrow{} B(n)|
(\mforall{}x,y:B(n). (req-vec(n;x;y) {}\mRightarrow{} req-vec(n;f x;f y))) \mwedge{} (\mneg{}(\mforall{}x:B(n). f x \mneq{} x))\} .
\mforall{}e:\{e:\mBbbR{}| r0 < e\} .
\mexists{}p:B(n). (\mdownarrow{}d(f p;p) < e)
Date html generated:
2019_10_30-AM-11_29_30
Last ObjectModification:
2019_07_30-PM-00_58_58
Theory : real!vectors
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