Nuprl Lemma : rat-half-cube-diameter
∀[k:ℕ]. ∀[c,h:ℚCube(k)]. rat-cube-diameter(k;h) = ((r1/r(2)) * rat-cube-diameter(k;c)) supposing ↑is-half-cube(k;h;c)
Proof
Definitions occuring in Statement :
rat-cube-diameter: rat-cube-diameter(k;c),
rdiv: (x/y),
req: x = y,
rmul: a * b,
int-to-real: r(n),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k)
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
sq_type: SQType(T),
rtermAdd: left "+" right,
rtermConstant: "const",
rtermVar: rtermVar(var),
rtermSubtract: left "-" right,
rtermDivide: num "/" denom,
rat_term_ind: rat_term_ind,
rtermMultiply: left "*" right,
rat_term_to_real: rat_term_to_real(f;t),
nat_plus: ℕ+,
is-half-interval: is-half-interval(I;J),
rev_uimplies: rev_uimplies(P;Q),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
rational-interval: ℚInterval,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
ge: i ≥ j ,
rational-cube: ℚCube(k),
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
rat-cube-diameter: rat-cube-diameter(k;c),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
rat2real-qavg,
rsub_functionality,
rmul-rmax,
real_wf,
true_wf,
squash_wf,
uiff_transitivity,
assert_of_band,
assert_of_bor,
iff_weakening_uiff,
iff_transitivity,
rationals_wf,
equal_wf,
bfalse_wf,
assert-qeq,
btrue_wf,
band_wf,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
qeq_wf2,
bor_wf,
assert_wf,
req-iff-rsub-is-0,
itermMultiply_wf,
rtermMultiply_wf,
rtermConstant_wf,
rtermVar_wf,
rtermAdd_wf,
rtermDivide_wf,
rtermSubtract_wf,
assert-rat-term-eq2,
rmax_functionality,
istype-false,
rleq-int-fractions2,
qavg_wf,
radd_wf,
req_wf,
rsum_linearity2,
req_inversion,
req_weakening,
req_functionality,
rsum_functionality,
int_seg_wf,
istype-less_than,
istype-le,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermAdd_wf,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
rat2real_wf,
rsub_wf,
rmax_wf,
subtract_wf,
rsum_wf,
istype-nat,
rational-cube_wf,
is-half-cube_wf,
istype-assert,
rless_wf,
rless-int,
int-to-real_wf,
rdiv_wf,
rmul_wf,
rat-cube-diameter_wf,
req_witness,
assert-is-half-cube
Rules used in proof :
promote_hyp,
inlFormation_alt,
unionIsType,
productEquality,
unionEquality,
cumulativity,
instantiate,
addEquality,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
lambdaFormation_alt,
productIsType,
voidElimination,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
imageElimination,
dependent_set_memberEquality_alt,
applyEquality,
lambdaEquality_alt,
rename,
setElimination,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
universeIsType,
baseClosed,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
inrFormation_alt,
sqequalRule,
natural_numberEquality,
closedConclusion,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c,h:\mBbbQ{}Cube(k)].
rat-cube-diameter(k;h) = ((r1/r(2)) * rat-cube-diameter(k;c)) supposing \muparrow{}is-half-cube(k;h;c)
Date html generated:
2019_10_31-AM-06_03_30
Last ObjectModification:
2019_10_31-AM-00_00_48
Theory : real!vectors
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