Nuprl Lemma : positive-rat-cube-immediate-face
∀k:ℕ. ∀c:ℚCube(k). (0 < dim(c) ⇒ (∃f:ℚCube(k). immediate-rc-face(k;f;c)))
Proof
Definitions occuring in Statement :
immediate-rc-face: immediate-rc-face(k;f;c),
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
immediate-rc-face: immediate-rc-face(k;f;c),
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rational-cube: ℚCube(k),
lelt: i ≤ j < k,
le: A ≤ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
less_than: a < b,
nat: ℕ,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rat-cube-dimension: dim(c),
less_than': less_than'(a;b)
Lemmas referenced :
positive-rat-cube-dimension,
upper-rc-face_wf,
upper-rc-face-is-face,
equal_wf,
squash_wf,
true_wf,
istype-universe,
upper-rc-face-dimension,
subtract_wf,
rat-cube-dimension_wf,
subtype_rel_self,
iff_weakening_equal,
eq_int_wf,
rat-interval-dimension_wf,
eqtt_to_assert,
assert_of_eq_int,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
immediate-rc-face_wf,
istype-less_than,
rational-cube_wf,
istype-nat,
inhabited-rat-cube_wf,
bool_cases,
assert_of_bnot
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
isectElimination,
independent_pairFormation,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
intEquality,
because_Cache,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
unionElimination,
equalityElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
equalityIstype,
promote_hyp,
cumulativity
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k). (0 < dim(c) {}\mRightarrow{} (\mexists{}f:\mBbbQ{}Cube(k). immediate-rc-face(k;f;c)))
Date html generated:
2020_05_20-AM-09_21_52
Last ObjectModification:
2019_11_13-PM-06_21_52
Theory : rationals
Home
Index