Nuprl Lemma : unit-ball-approx-subtype
∀[k,n,m:ℕ]. unit-ball-approx(m;k) ⊆r unit-ball-approx(n;k) supposing n ≤ m
Proof
Definitions occuring in Statement :
unit-ball-approx: unit-ball-approx(n;k),
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
unit-ball-approx: unit-ball-approx(n;k),
nat: ℕ,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
so_apply: x[s],
lelt: i ≤ j < k,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
less_than: a < b,
uiff: uiff(P;Q)
Lemmas referenced :
subtype_rel_function,
int_seg_wf,
int_seg_subtype,
istype-false,
subtype_rel_self,
istype-le,
sum_wf,
unit-ball-approx_wf,
istype-nat,
sum_split,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_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_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
le_wf,
squash_wf,
true_wf,
iff_weakening_equal,
non_neg_sum,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
int_seg_properties,
square_non_neg,
add-member-int_seg2,
add-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
false_wf,
le_functionality,
add_functionality_wrt_le,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
dependent_set_memberEquality_alt,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesis,
minusEquality,
addEquality,
because_Cache,
independent_isectElimination,
independent_pairFormation,
sqequalRule,
lambdaFormation_alt,
productElimination,
multiplyEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
universeIsType,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
equalityIstype,
closedConclusion,
pointwiseFunctionality,
promote_hyp,
baseApply
Latex:
\mforall{}[k,n,m:\mBbbN{}]. unit-ball-approx(m;k) \msubseteq{}r unit-ball-approx(n;k) supposing n \mleq{} m
Date html generated:
2019_10_30-AM-11_28_08
Last ObjectModification:
2019_06_28-PM-01_56_09
Theory : real!vectors
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