Nuprl Lemma : rat-complex-boundary-boundary
∀[k,n:ℕ]. ∀[K:n-dim-complex]. (∂(∂(K)) ~ [])
Proof
Definitions occuring in Statement :
rat-complex-boundary: ∂(K),
rational-cube-complex: n-dim-complex,
nil: [],
nat: ℕ,
uall: ∀[x:A]. B[x],
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
rational-cube-complex: n-dim-complex,
and: P ∧ Q,
rat-complex-boundary: ∂(K),
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
face-complex: face-complex(k;L),
remove-repeats: remove-repeats(eq;L),
concat: concat(ll),
map: map(f;as),
nil: [],
it: ⋅,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
so_apply: x[s],
prop: ℙ,
sq_stable: SqStable(P),
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
squash: ↓T,
iff: P ⇐⇒ Q,
in-complex-boundary: in-complex-boundary(k;f;K),
nat_plus: ℕ+,
true: True,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
cons: [a / b],
assert: ↑b,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
btrue: tt,
l_member: (x ∈ l),
le: A ≤ B,
less_than': less_than'(a;b),
select: L[n],
cand: A c∧ B,
immediate-rc-face: immediate-rc-face(k;f;c),
exists!: ∃!x:T. P[x],
bool: 𝔹,
unit: Unit,
rat-cube-dimension: dim(c),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
rational-cube-complex_wf,
istype-nat,
boundary-of-0-dim-is-nil,
rat-complex-boundary_wf,
sq_stable__l_all,
rational-cube_wf,
equal-wf-base,
rat-cube-dimension_wf,
set_subtype_base,
lelt_wf,
l_member_wf,
sq_stable__equal,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
no-member-sq-nil,
subtract_wf,
istype-le,
member-rat-complex-boundary-n,
filter_wf5,
face-complex_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
is-rat-cube-face_wf,
lsum-split,
in-complex-boundary_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
length_wf,
subtype_rel_self,
iff_weakening_equal,
count-related-pairs,
swap-filter-filter,
lsum-of-even,
l_all_iff,
assert_wf,
isEven_wf,
filter-filter,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
bfalse_wf,
list-cases,
product_subtype_list,
length_of_nil_lemma,
list_wf,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
select_wf,
member_filter_2,
assert_of_band,
istype-assert,
assert-is-rat-cube-face,
le_wf,
member-face-complex,
member-rat-cube-faces,
face-of-face-pairity,
rat-cube-face_wf,
no_repeats_wf,
no_repeats_filter,
remove-repeats-no_repeats,
rc-deq_wf,
concat_wf,
map_wf,
inhabited-rat-cube_wf,
rat-cube-faces_wf,
subtype_rel_list,
nil_wf,
eqff_to_assert,
assert_of_bnot,
iff_weakening_uiff,
member_filter,
cons_wf,
decidable__equal_rc,
cons_member,
no_repeats_cons,
list-is-singleton-iff,
odd-iff-not-even,
odd-plus-even,
lsum_wf,
bnot_wf,
odd-lsum-of-odd,
isOdd_wf,
member-rat-complex-boundary,
even-iff-not-odd
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
axiomSqEquality,
universeIsType,
sqequalRule,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
productElimination,
applyEquality,
lambdaEquality_alt,
lambdaFormation_alt,
minusEquality,
addEquality,
baseClosed,
setIsType,
axiomEquality,
functionIsTypeImplies,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
imageMemberEquality,
imageElimination,
equalityIstype,
dependent_set_memberEquality_alt,
universeEquality,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
productIsType,
sqequalBase,
functionIsType,
equalityElimination,
setEquality,
productEquality,
inlFormation_alt,
hyp_replacement,
unionIsType,
inrFormation_alt
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[K:n-dim-complex]. (\mpartial{}(\mpartial{}(K)) \msim{} [])
Date html generated:
2020_05_20-AM-09_23_17
Last ObjectModification:
2019_11_13-PM-03_11_03
Theory : rationals
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