Nuprl Lemma : NoBallRetraction-implies-BrouwerFPT
∀n:ℕ. BrouwerFPT(n) supposing NoBallRetraction(n)
Proof
Definitions occuring in Statement :
NoBallRetraction: NoBallRetraction(n),
BrouwerFPT: BrouwerFPT(n),
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
NoBallRetraction: NoBallRetraction(n),
not: ¬A,
implies: P ⇒ Q,
false: False,
BrouwerFPT: BrouwerFPT(n),
and: P ∧ Q,
cand: A c∧ B,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
real-unit-ball: B(n),
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
top: Top,
real-vec-sep: a ≠ b,
le: A ≤ B,
less_than': less_than'(a;b),
real-vec: ℝ^n,
int_seg: {i..j-},
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
less_than: a < b,
squash: ↓T,
ext-eq: A ≡ B,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
rev_implies: P ⇐ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
nonneg-poly: nonneg-poly(p),
bl-all: (∀x∈L.P[x])_b,
reduce: reduce(f;k;as),
list_ind: list_ind,
int_term_to_ipoly: int_term_to_ipoly(t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
add_ipoly: add_ipoly(p;q),
add-ipoly-prepend: add-ipoly-prepend(p;q;l),
itermMultiply: left (*) right,
mul_ipoly: mul_ipoly(p;q),
itermVar: vvar,
cons: [a / b],
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
nil: [],
it: ⋅,
mul-mono-poly: mul-mono-poly(m;p),
mul-monomials: mul-monomials(m1;m2),
merge-int-accum: merge-int-accum(as;bs),
eager-accum: eager-accum(x,a.f[x; a];y;l),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
insert-int: insert-int(x;l),
minus-poly: minus-poly(p),
map: map(f;as),
itermConstant: "const",
rev-append: rev(as) + bs,
list_accum: list_accum,
band: p ∧b q,
nonneg-monomial: nonneg-monomial(m),
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
bfalse: ff,
btrue: tt,
even-int-list: even-int-list(L),
bor: p ∨bq,
null: null(as),
tl: tl(l),
pi2: snd(t),
eq_int: (i =z j),
hd: hd(l),
pi1: fst(t),
true: True,
rneq: x ≠ y,
req-vec: req-vec(n;x;y),
real-vec-sub: X - Y,
real-vec-mul: a*X,
real-vec-add: X + Y,
quadratic1: quadratic1(a;b;c),
rdiv: (x/y)
Lemmas referenced :
approx-fixpoint-unit-ball-2,
real-unit-ball_wf,
real-vec-sep_wf,
req-vec_wf,
istype-void,
real_wf,
rless_wf,
int-to-real_wf,
NoBallRetraction_wf,
istype-nat,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
real-unit-ball-0,
real-vec-dist_wf,
istype-le,
int_seg_properties,
nat_plus_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
itermAdd_wf,
int_term_value_add_lemma,
rless_functionality,
req_weakening,
real-vec-dist-dim0,
real-vec-sep-iff-dot-product,
radd-preserves-rleq,
rsub_wf,
dot-product_wf,
radd_wf,
itermSubtract_wf,
rnexp_wf,
real-vec-norm_wf,
sq_stable__rleq,
real-vec-norm-nonneg,
rleq_weakening_equal,
rleq_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
req_inversion,
real-vec-norm-squared,
rleq_functionality_wrt_implies,
rnexp_functionality_wrt_rleq,
rnexp-one,
dot-product-nonneg,
real-vec-sub_wf,
rmul_preserves_rleq2,
rleq_wf,
rmul_wf,
itermMultiply_wf,
rleq-int,
istype-false,
real-term-nonneg,
real_term_value_mul_lemma,
rsub_functionality_wrt_rleq,
quadratic-formula1,
quadratic1_wf,
req_wf,
quadratic2_wf,
real-vec-norm-eq-iff,
real-vec-add_wf,
real-vec-mul_wf,
dot-product-comm,
rnexp2,
req-implies-req,
req_functionality,
req_transitivity,
dot-product-linearity1,
radd_functionality,
dot-product-linearity2,
rmul_functionality,
real-vec-sub_functionality,
req-vec_weakening,
rsub_functionality,
dot-product_functionality,
req-vec_functionality,
real-vec-add_functionality,
real-vec-mul_functionality,
quadratic1_functionality,
rleq_weakening,
rnexp_functionality,
rmul_preserves_rless,
rless-int,
equal_wf,
dot-product-linearity1-sub,
rdiv_wf,
rminus_wf,
rsqrt_wf,
rdiv_functionality,
radd-preserves-req,
itermMinus_wf,
real_term_value_minus_lemma,
square-nonneg,
rsqrt_functionality,
rsqrt-of-square,
rless_transitivity2,
rleq_weakening_rless,
rinv_wf2,
rinv-of-rmul,
rmul-rinv,
rmul-rinv3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
rename,
extract_by_obid,
setElimination,
dependent_set_memberEquality_alt,
hypothesis,
independent_pairFormation,
functionIsType,
universeIsType,
isectElimination,
applyEquality,
because_Cache,
productIsType,
setIsType,
natural_numberEquality,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
functionExtensionality,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
imageElimination,
imageMemberEquality,
baseClosed,
equalityIstype,
promote_hyp,
inrFormation_alt,
closedConclusion,
inlFormation_alt,
hyp_replacement,
applyLambdaEquality,
functionEquality
Latex:
\mforall{}n:\mBbbN{}. BrouwerFPT(n) supposing NoBallRetraction(n)
Date html generated:
2019_10_30-AM-11_29_51
Last ObjectModification:
2019_07_30-PM-01_05_02
Theory : real!vectors
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