Nuprl Lemma : rat-cube-faces_wf
∀[k:ℕ]. ∀[c:ℚCube(k)]. rat-cube-faces(k;c) ∈ {f:ℚCube(k)| f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)} List supposing ↑Inhabit\000Ced(c)
Proof
Definitions occuring in Statement :
rat-cube-faces: rat-cube-faces(k;c),
rat-cube-dimension: dim(c),
inhabited-rat-cube: Inhabited(c),
rat-cube-face: c ≤ d,
rational-cube: ℚCube(k),
list: T List,
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
less_than: a < b,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
le: A ≤ B,
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
int_seg: {i..j-},
implies: P ⇒ Q,
squash: ↓T,
uiff: uiff(P;Q),
cand: A c∧ B,
all: ∀x:A. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
rational-cube: ℚCube(k),
prop: ℙ,
and: P ∧ Q,
rat-cube-faces: rat-cube-faces(k;c),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-nat,
inhabited-rat-cube_wf,
istype-assert,
nil_wf,
upper-rc-face-dimension,
upper-rc-face-is-face,
upper-rc-face_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
eqtt_to_assert,
iff_weakening_equal,
subtype_rel_self,
rat-cube-dimension_wf,
subtract_wf,
lower-rc-face-dimension,
istype-universe,
true_wf,
squash_wf,
equal_wf,
assert_of_eq_int,
lower-rc-face-is-face,
lower-rc-face_wf,
cons_wf,
assert_wf,
list_wf,
rat-interval-dimension_wf,
eq_int_wf,
upto_wf,
int_seg_wf,
mapfilter_wf,
equal-wf-base,
rat-cube-face_wf,
rational-cube_wf,
concat_wf
Rules used in proof :
isectIsTypeImplies,
axiomEquality,
setIsType,
sqequalBase,
addEquality,
minusEquality,
productIsType,
cumulativity,
promote_hyp,
equalityIstype,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
equalityElimination,
unionElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
intEquality,
universeEquality,
instantiate,
equalitySymmetry,
equalityTransitivity,
imageElimination,
independent_isectElimination,
productElimination,
independent_pairFormation,
dependent_functionElimination,
dependent_set_memberEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
rename,
setElimination,
universeIsType,
applyEquality,
lambdaEquality_alt,
natural_numberEquality,
closedConclusion,
because_Cache,
productEquality,
hypothesis,
hypothesisEquality,
setEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. rat-cube-faces(k;c) \mmember{} \{f:\mBbbQ{}Cube(k)| f \mleq{} c \mwedge{} (dim(f) = (dim(c) - 1))\} List su\000Cpposing \muparrow{}Inhabited(c)
Date html generated:
2019_10_29-AM-07_57_16
Last ObjectModification:
2019_10_17-PM-05_26_10
Theory : rationals
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