Nuprl Lemma : boundary-singleton-complex
∀[k,n:ℕ]. ∀[c:{c:ℚCube(k)| dim(c) = n ∈ ℤ} ]. (∂(singleton-complex(c)) ~ remove-repeats(rc-deq(k);rat-cube-faces(k;c)))
Proof
Definitions occuring in Statement :
singleton-complex: singleton-complex(c),
rat-complex-boundary: ∂(K),
rat-cube-faces: rat-cube-faces(k;c),
rat-cube-dimension: dim(c),
rc-deq: rc-deq(k),
rational-cube: ℚCube(k),
remove-repeats: remove-repeats(eq;L),
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
int: ℤ,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
uimplies: b supposing a,
singleton-complex: singleton-complex(c),
rat-complex-boundary: ∂(K),
face-complex: face-complex(k;L),
all: ∀x:A. B[x],
top: Top,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
concat: concat(ll),
prop: ℙ,
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
true: True,
guard: {T},
in-complex-boundary: in-complex-boundary(k;f;K),
assert: ↑b,
isOdd: isOdd(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
length: ||as||,
list_ind: list_ind,
cons: [a / b],
nil: [],
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q),
rat-cube-dimension: dim(c),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
rational-cube_wf,
istype-int,
rat-cube-dimension_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
le_wf,
istype-nat,
map_cons_lemma,
istype-void,
map_nil_lemma,
inhabited-rat-cube_wf,
equal-wf-T-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
reduce_cons_lemma,
reduce_nil_lemma,
append_back_nil,
rat-cube-face_wf,
equal-wf-base,
subtract_wf,
rat-cube-faces_wf,
filter_trivial,
in-complex-boundary_wf,
cons_wf,
nil_wf,
remove-repeats_wf,
rc-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
subtype_rel_list,
l_all_iff,
l_member_wf,
squash_wf,
true_wf,
list_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
member-remove-repeats,
member-rat-cube-faces,
filter_cons_lemma,
filter_nil_lemma,
is-rat-cube-face_wf,
length_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
length_of_nil_lemma,
assert-is-rat-cube-face,
bool_cases,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
axiomSqEquality,
setIsType,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
equalityIstype,
applyEquality,
sqequalRule,
intEquality,
lambdaEquality_alt,
minusEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
independent_isectElimination,
sqequalBase,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination,
voidElimination,
because_Cache,
equalityTransitivity,
baseClosed,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_functionElimination,
setEquality,
productEquality,
productIsType,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
applyLambdaEquality,
dependent_pairFormation_alt,
promote_hyp,
cumulativity,
approximateComputation,
int_eqEquality,
independent_pairFormation
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[c:\{c:\mBbbQ{}Cube(k)| dim(c) = n\} ].
(\mpartial{}(singleton-complex(c)) \msim{} remove-repeats(rc-deq(k);rat-cube-faces(k;c)))
Date html generated:
2020_05_20-AM-09_22_31
Last ObjectModification:
2019_11_13-PM-07_08_08
Theory : rationals
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