Nuprl Lemma : extend-half-cube-face
∀k:ℕ. ∀a,b,c:ℚCube(k).
(0 < dim(c)
⇒ a ≤ b
⇒ (↑is-half-cube(k;b;c))
⇒ (dim(a) = (dim(b) - 1) ∈ ℤ)
⇒ (((∃!d:ℚCube(k). ((↑is-half-cube(k;a;d)) ∧ d ≤ c))
∧ (∀b':ℚCube(k). ((↑is-half-cube(k;b';c)) ⇒ a ≤ b' ⇒ (b' = b ∈ ℚCube(k)))))
∨ ((∃!b':ℚCube(k). (a ≤ b' ∧ (↑is-half-cube(k;b';c)) ∧ (¬(b' = b ∈ ℚCube(k))))) ∧ has-interior-point(k;a;c))))
Proof
Definitions occuring in Statement :
rat-cube-dimension: dim(c),
has-interior-point: has-interior-point(k;c;a),
is-half-cube: is-half-cube(k;h;c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
nat: ℕ,
assert: ↑b,
less_than: a < b,
exists!: ∃!x:T. P[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
rat-cube-dimension: dim(c),
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
immediate-rc-face: immediate-rc-face(k;f;c),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rational-cube: ℚCube(k),
rational-interval: ℚInterval,
rat-interval-dimension: dim(I),
is-half-interval: is-half-interval(I;J),
cand: A c∧ B,
pi2: snd(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
iff: P ⇐⇒ Q,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
band: p ∧b q,
rat-cube-face: c ≤ d,
pi1: fst(t),
exists!: ∃!x:T. P[x],
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
rat-point-interval: [a],
rat-interval-face: I ≤ J,
has-interior-point: has-interior-point(k;c;a),
rat-point-in-cube: rat-point-in-cube(k;x;c),
label: ...$L... t,
inhabited-rat-interval: Inhabited(I)
Lemmas referenced :
inhabited-rat-cube_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
half-cube-dimension,
istype-assert,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_wf,
immediate-rc-face-implies,
decidable__le,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
rat-cube-dimension_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
subtract_wf,
is-half-cube_wf,
rat-cube-face_wf,
istype-less_than,
rational-cube_wf,
istype-nat,
assert-is-half-cube,
rat-interval-dimension_wf,
rational-interval_wf,
rat-point-interval_wf,
pi1_wf_top,
rationals_wf,
subtype_rel_product,
top_wf,
q_less_wf,
assert-q_less-eq,
iff_weakening_equal,
bool_cases_sqequal,
assert-bnot,
qless_wf,
qless-qavg-iff-1,
qavg_wf,
qavg-qless-iff-1,
assert_wf,
bor_wf,
qeq_wf2,
band_wf,
btrue_wf,
assert-qeq,
bfalse_wf,
equal_wf,
ifthenelse_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_bor,
assert_of_band,
assert-inhabited-rat-cube,
rat-interval-face_wf,
exists!_wf,
int_seg_wf,
not_wf,
has-interior-point_wf,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
assert_functionality_wrt_uiff,
is-half-interval_wf,
member_wf,
qavg-eq-iff-1,
squash_wf,
true_wf,
rat-interval-face-self,
pi2_wf,
istype-universe,
qavg-same,
subtype_rel_self,
nequal_wf,
qavg-eq-iff-4,
decidable__equal_int,
qavg-eq-iff-7,
qavg-eq-iff-3,
int_seg_properties,
pair-eta,
istype-top,
qavg-eq-iff-2,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
decidable__equal_rational-interval,
istype-le,
rat-point-in-cube_wf,
rat-point-in-cube-interior_wf,
qle_reflexivity,
qle_wf,
qle-qavg-iff-1,
qavg-qle-iff-1,
q_le_wf,
assert-q_le-eq,
center-point-in-cube-interior
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
unionElimination,
instantiate,
cumulativity,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
sqequalRule,
imageElimination,
voidElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
independent_pairFormation,
universeIsType,
promote_hyp,
equalityIstype,
applyEquality,
intEquality,
minusEquality,
addEquality,
inhabitedIsType,
sqequalBase,
productIsType,
baseClosed,
functionIsType,
unionIsType,
equalityElimination,
hyp_replacement,
applyLambdaEquality,
unionEquality,
productEquality,
inlFormation_alt,
inrFormation_alt,
functionEquality,
imageMemberEquality,
functionExtensionality,
independent_pairEquality,
universeEquality
Latex:
\mforall{}k:\mBbbN{}. \mforall{}a,b,c:\mBbbQ{}Cube(k).
(0 < dim(c)
{}\mRightarrow{} a \mleq{} b
{}\mRightarrow{} (\muparrow{}is-half-cube(k;b;c))
{}\mRightarrow{} (dim(a) = (dim(b) - 1))
{}\mRightarrow{} (((\mexists{}!d:\mBbbQ{}Cube(k). ((\muparrow{}is-half-cube(k;a;d)) \mwedge{} d \mleq{} c))
\mwedge{} (\mforall{}b':\mBbbQ{}Cube(k). ((\muparrow{}is-half-cube(k;b';c)) {}\mRightarrow{} a \mleq{} b' {}\mRightarrow{} (b' = b))))
\mvee{} ((\mexists{}!b':\mBbbQ{}Cube(k). (a \mleq{} b' \mwedge{} (\muparrow{}is-half-cube(k;b';c)) \mwedge{} (\mneg{}(b' = b))))
\mwedge{} has-interior-point(k;a;c))))
Date html generated:
2020_05_20-AM-09_21_32
Last ObjectModification:
2019_11_02-PM-08_05_29
Theory : rationals
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