Nuprl Lemma : approx-fixpoint-unit-ball-1
∀n:ℕ. ∀f:{f:B(n) ⟶ B(n)|
(∀e:{e:ℝ| r0 < e} . ∃del:{del:ℝ| r0 < del} . ∀x,y:B(n). ((d(x;y) < del) ⇒ (d(f x;f y) < e)))
∧ (¬(∀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),
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,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat_plus: ℕ+,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
real-unit-ball: B(n),
le: A ≤ B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
ext-eq: A ≡ B
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
approx-fixpoint-unit-ball-0-ext,
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,
istype-void,
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,
sq_stable__ex_nat_plus,
unit-ball-approx_wf,
nat_plus_properties,
decidable__le,
istype-le,
assert_wf,
approx-ball-to-ball_wf,
nat_plus_wf,
decidable__exists-unit-ball-approx,
nat_plus_subtype_nat,
decidable__assert,
squash_wf,
rless_wf,
real-vec-dist_wf,
real_wf,
int-to-real_wf,
real-unit-ball_wf,
real-vec-sep_wf,
istype-nat,
real-unit-ball-0,
sq_stable__rless,
subtype_rel_self,
subtype_rel_set,
real-vec_wf,
rleq_wf,
real-vec-norm_wf,
nat_wf,
set_subtype_base,
le_wf,
rless_functionality,
real-vec-dist-dim0,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
imageElimination,
productEquality,
applyEquality,
imageMemberEquality,
baseClosed,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
setIsType,
functionIsType,
productIsType
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f:\{f:B(n) {}\mrightarrow{} B(n)|
(\mforall{}e:\{e:\mBbbR{}| r0 < e\}
\mexists{}del:\{del:\mBbbR{}| r0 < del\} . \mforall{}x,y:B(n). ((d(x;y) < del) {}\mRightarrow{} (d(f x;f y) < e)))
\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_13
Last ObjectModification:
2019_07_30-PM-00_32_07
Theory : real!vectors
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