Nuprl Lemma : approx-fixpoint
∀I:{I:Interval| icompact(I)} . ∀n:ℕ. ∀f:{f:I^n ⟶ I^n| ∀a,b:I^n. (req-vec(n;a;b) ⇒ req-vec(n;f a;f b))} .
((¬(∀x:I^n. f x ≠ x)) ⇒ (∀e:{e:ℝ| r0 < e} . ∃x:I^n. (d(f x;x) < e)))
Proof
Definitions occuring in Statement :
real-vec-sep: a ≠ b,
real-vec-dist: d(x;y),
interval-vec: I^n,
req-vec: req-vec(n;x;y),
icompact: icompact(I),
interval: Interval,
rless: x < y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: 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,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
not: ¬A,
subtype_rel: A ⊆r B,
interval-vec: I^n,
false: False,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
rev_uimplies: rev_uimplies(P;Q),
real-vec-sep: a ≠ b,
rneq: x ≠ y,
or: P ∨ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
iff: P ⇐⇒ Q
Lemmas referenced :
approx-zero,
real_wf,
rless_wf,
int-to-real_wf,
interval-vec_wf,
real-vec-sep_wf,
istype-void,
req-vec_wf,
istype-nat,
interval_wf,
icompact_wf,
real-vec-dist_wf,
sq_stable__req,
req_wf,
req_weakening,
req_functionality,
real-vec-dist_functionality,
rneq_wf,
real-vec-dist-nonneg,
rless_transitivity2,
rless_transitivity1,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
rabs_wf,
rless_functionality,
rabs-of-nonneg
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
setIsType,
universeIsType,
isectElimination,
natural_numberEquality,
functionIsType,
setElimination,
rename,
applyEquality,
lambdaEquality_alt,
because_Cache,
dependent_set_memberEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
productElimination,
voidElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt
Latex:
\mforall{}I:\{I:Interval| icompact(I)\} . \mforall{}n:\mBbbN{}. \mforall{}f:\{f:I\^{}n {}\mrightarrow{} I\^{}n|
\mforall{}a,b:I\^{}n. (req-vec(n;a;b) {}\mRightarrow{} req-vec(n;f a;f b))\} .
((\mneg{}(\mforall{}x:I\^{}n. f x \mneq{} x)) {}\mRightarrow{} (\mforall{}e:\{e:\mBbbR{}| r0 < e\} . \mexists{}x:I\^{}n. (d(f x;x) < e)))
Date html generated:
2019_10_30-AM-10_14_39
Last ObjectModification:
2019_06_28-PM-01_52_05
Theory : real!vectors
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