Nuprl Lemma : Degree-implies-BrowerFPT
∀n:ℕ. BrouwerFPT(n + 1) supposing ¬¬DegreeExists(n)
Proof
Definitions occuring in Statement :
DegreeExists: DegreeExists(n),
BrouwerFPT: BrouwerFPT(n),
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
not: ¬A,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
NoBallRetraction: NoBallRetraction(n),
DegreeExists: DegreeExists(n),
real-unit-ball: B(n),
subtype_rel: A ⊆r B,
compose: f o g,
cand: A c∧ B,
req-vec: req-vec(n;x;y),
so_lambda: λ2x.t[x],
real-vec: ℝ^n,
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
i-member: r ∈ I,
rccint: [l, u],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
rge: x ≥ y,
sphere-map-eq: sphere-map-eq(n;f;g),
real-unit-sphere: S(n),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
const-sphere-map: const-sphere-map(p),
real-vec-mul: a*X,
id-sphere-map: id-sphere-map()
Lemmas referenced :
NoBallRetraction-implies-BrouwerFPT,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
real-unit-ball_wf,
req-vec_wf,
req_wf,
real-vec-norm_wf,
int-to-real_wf,
DegreeExists_wf,
istype-nat,
sphere-map-from-ball-map,
compose_wf,
sq_stable__all,
int_seg_wf,
sq_stable__req,
req_witness,
real-vec-mul_wf,
rleq_wf,
real_wf,
i-member_wf,
rccint_wf,
rmul_wf,
rabs_wf,
rmul_preserves_rleq2,
zero-rleq-rabs,
itermSubtract_wf,
itermMultiply_wf,
rleq_functionality,
real-vec-norm-mul,
req_weakening,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rabs-of-nonneg,
real-vec-mul_functionality,
real-unit-sphere-subtype-ball,
real-unit-sphere_wf,
req-vec_weakening,
member_rccint_lemma,
int_subtype_base,
rleq-int,
istype-false,
real-vec-norm-0,
const-sphere-map_wf,
rmul-identity1,
id-sphere-map_wf,
req-vec_functionality,
extensional-discrete-real-fun-is-constant,
intformeq_wf,
int_formula_prop_eq_lemma
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,
dependent_set_memberEquality_alt,
addEquality,
setElimination,
hypothesis,
natural_numberEquality,
isectElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
independent_pairFormation,
universeIsType,
promote_hyp,
productElimination,
because_Cache,
productIsType,
functionIsType,
applyEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
setIsType,
functionExtensionality,
equalityIstype,
sqequalBase
Latex:
\mforall{}n:\mBbbN{}. BrouwerFPT(n + 1) supposing \mneg{}\mneg{}DegreeExists(n)
Date html generated:
2019_10_30-AM-11_30_17
Last ObjectModification:
2019_08_06-PM-01_01_29
Theory : real!vectors
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