Nuprl Lemma : rat-cube-third-not-in-face
∀[k:ℕ]. ∀[p:ℝ^k]. ∀[c:ℚCube(k)].
∀f:ℚCube(k). (¬in-rat-cube(k;p;f)) supposing ((¬(f = c ∈ ℚCube(k))) and f ≤ c) supposing rat-cube-third(k;p;c) ∧ (↑Inh\000Cabited(c))
Proof
Definitions occuring in Statement :
rat-cube-third: rat-cube-third(k;p;c),
in-rat-cube: in-rat-cube(k;p;c),
real-vec: ℝ^n,
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
equal: s = t ∈ T,
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
rev_uimplies: rev_uimplies(P;Q),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
sq_type: SQType(T),
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
uiff: uiff(P;Q),
less_than': less_than'(a;b),
less_than: a < b,
rneq: x ≠ y,
rat-interval-third: rat-interval-third(p;I),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
pi1: fst(t),
pi2: snd(t),
top: Top,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
rat-point-interval: [a],
rat-interval-face: I ≤ J,
rational-interval: ℚInterval,
real-vec: ℝ^n,
rat-cube-face: c ≤ d,
in-rat-cube: in-rat-cube(k;p;c),
rational-cube: ℚCube(k),
rat-cube-third: rat-cube-third(k;p;c),
and: P ∧ Q,
false: False,
implies: P ⇒ Q,
not: ¬A,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
int-rinv-cancel,
rmul-rinv3,
radd_functionality,
req_transitivity,
req_weakening,
req_functionality,
req-iff-rsub-is-0,
rsub_wf,
req-implies-req,
req-rat2real,
nequal_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-int,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
itermAdd_wf,
rinv_wf2,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
itermSubtract_wf,
rmul_preserves_req,
rless_wf,
rless-int,
int-to-real_wf,
rmul_wf,
radd_wf,
rdiv_wf,
req_fake_le_antisymmetry,
istype-universe,
equal_wf,
iff_weakening_equal,
subtype_rel_self,
real_wf,
true_wf,
squash_wf,
rational-interval_wf,
rat2real_wf,
rleq_wf,
rat-interval-third_wf,
istype-nat,
real-vec_wf,
rational-cube_wf,
inhabited-rat-cube_wf,
istype-assert,
rat-cube-third_wf,
rat-cube-face_wf,
in-rat-cube_wf,
int_seg_wf,
rat-interval-face_wf,
istype-void,
pi1_wf_top,
rationals_wf,
pi2_wf,
in-rat-cube-face
Rules used in proof :
int_eqEquality,
sqequalBase,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
intEquality,
cumulativity,
inrFormation_alt,
closedConclusion,
promote_hyp,
independent_pairFormation,
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
imageElimination,
productIsType,
isectIsTypeImplies,
functionIsType,
functionIsTypeImplies,
because_Cache,
rename,
setElimination,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
universeIsType,
voidElimination,
isect_memberEquality_alt,
independent_pairEquality,
lambdaEquality_alt,
sqequalRule,
applyLambdaEquality,
unionElimination,
inhabitedIsType,
applyEquality,
dependent_functionElimination,
functionExtensionality,
hypothesis,
independent_isectElimination,
hypothesisEquality,
isectElimination,
extract_by_obid,
independent_functionElimination,
productElimination,
sqequalHypSubstitution,
thin,
lambdaFormation_alt,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[p:\mBbbR{}\^{}k]. \mforall{}[c:\mBbbQ{}Cube(k)].
\mforall{}f:\mBbbQ{}Cube(k). (\mneg{}in-rat-cube(k;p;f)) supposing ((\mneg{}(f = c)) and f \mleq{} c)
supposing rat-cube-third(k;p;c) \mwedge{} (\muparrow{}Inhabited(c))
Date html generated:
2019_11_04-PM-04_43_32
Last ObjectModification:
2019_11_04-PM-03_32_34
Theory : real!vectors
Home
Index