Nuprl Lemma : has-interior-point-implies
∀[k:ℕ]. ∀[c,a:ℚCube(k)].
(has-interior-point(k;c;a) ⇒ dim(c) < dim(a) ⇒ (∀d:ℚCube(k). ((↑is-half-cube(k;c;d)) ⇒ (¬d ≤ a))))
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,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
has-interior-point: has-interior-point(k;c;a),
exists: ∃x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
true: True,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T,
guard: {T},
lelt: i ≤ j < k,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q)
Lemmas referenced :
rat-point-in-half-cube,
rat-cube-face_wf,
istype-assert,
is-half-cube_wf,
istype-less_than,
rat-cube-dimension_wf,
has-interior-point_wf,
rational-cube_wf,
istype-nat,
rat-point-in-cube-interior-not-in-face,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
half-cube-dimension,
equal_wf,
squash_wf,
true_wf,
istype-universe,
int_seg_properties,
subtype_rel_self,
iff_weakening_equal,
inhabited-rat-cube-iff-point,
rat-point-in-cube_wf,
inhabited-rat-cube_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
productElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
universeIsType,
independent_functionElimination,
voidElimination,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
dependent_functionElimination,
because_Cache,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
intEquality,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
minusEquality,
addEquality,
applyLambdaEquality,
dependent_set_memberEquality_alt
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c,a:\mBbbQ{}Cube(k)].
(has-interior-point(k;c;a)
{}\mRightarrow{} dim(c) < dim(a)
{}\mRightarrow{} (\mforall{}d:\mBbbQ{}Cube(k). ((\muparrow{}is-half-cube(k;c;d)) {}\mRightarrow{} (\mneg{}d \mleq{} a))))
Date html generated:
2020_05_20-AM-09_19_55
Last ObjectModification:
2019_11_02-PM-07_43_39
Theory : rationals
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