Nuprl Lemma : lower-rc-face-dimension
∀k:ℕ. ∀c:ℚCube(k). ∀j:ℕk.
((↑Inhabited(c)) ⇒ (dim(lower-rc-face(c;j)) = if (dim(c j) =z 0) then dim(c) else dim(c) - 1 fi ∈ ℤ))
Proof
Definitions occuring in Statement :
lower-rc-face: lower-rc-face(c;j),
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rational-cube: ℚCube(k),
rat-interval-dimension: dim(I),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
inhabited-rat-interval: Inhabited(I),
rat-interval-dimension: dim(I),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nequal: a ≠ b ∈ T ,
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
pi1: fst(t),
rational-interval: ℚInterval,
not: ¬A,
less_than': less_than'(a;b),
lower-rc-face: lower-rc-face(c;j),
true: True,
so_apply: x[s],
squash: ↓T,
less_than: a < b,
so_lambda: λ2x.t[x],
nat: ℕ,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
rat-cube-dimension: dim(c),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
rational-cube: ℚCube(k),
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__equal_int,
general_arith_equation1,
int_term_value_constant_lemma,
itermConstant_wf,
qless_wf,
nequal_wf,
q_le_wf,
assert-q_less-eq,
q_less_wf,
istype-nat,
rat-interval-dimension-single,
iff_weakening_equal,
inhabited-lower-rc-face,
istype-universe,
true_wf,
squash_wf,
equal_wf,
subtract_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
int_seg_properties,
rat-point-interval_wf,
istype-false,
int_seg_subtype_nat,
Error :isolate_summand2,
ifthenelse_wf,
rational-interval_wf,
subtype_rel_self,
lower-rc-face_wf,
sum_wf,
int_seg_wf,
inhabited-rat-cube_wf,
istype-assert,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
rat-interval-dimension_wf,
eq_int_wf,
assert-inhabited-rat-cube
Rules used in proof :
functionIsType,
addEquality,
baseClosed,
imageMemberEquality,
universeEquality,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
independent_pairFormation,
minusEquality,
imageElimination,
functionEquality,
lambdaEquality_alt,
intEquality,
natural_numberEquality,
universeIsType,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
equalityIstype,
dependent_pairFormation_alt,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
inhabitedIsType,
rename,
setElimination,
sqequalRule,
because_Cache,
applyEquality,
independent_isectElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}. \mforall{}c:\mBbbQ{}Cube(k). \mforall{}j:\mBbbN{}k.
((\muparrow{}Inhabited(c))
{}\mRightarrow{} (dim(lower-rc-face(c;j)) = if (dim(c j) =\msubz{} 0) then dim(c) else dim(c) - 1 fi ))
Date html generated:
2019_10_29-AM-07_56_23
Last ObjectModification:
2019_10_17-PM-05_13_17
Theory : rationals
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