Nuprl Lemma : length-rat-complex-boundary-even
∀k,n:ℕ. ∀K:n-dim-complex. (↑isEven(||∂(K)||))
Proof
Definitions occuring in Statement :
rat-complex-boundary: ∂(K),
rational-cube-complex: n-dim-complex,
isEven: isEven(n),
length: ||as||,
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
less_than': less_than'(a;b),
assert: ↑b,
ifthenelse: if b then t else f fi ,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
btrue: tt,
true: True,
rational-cube-complex: n-dim-complex,
less_than: a < b,
squash: ↓T,
cons: [a / b],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
rat-complex-boundary: ∂(K),
filter: filter(P;l),
reduce: reduce(f;k;as),
face-complex: face-complex(k;L),
remove-repeats: remove-repeats(eq;L),
concat: concat(ll),
map: map(f;as),
l_member: (x ∈ l),
select: L[n],
nat_plus: ℕ+,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
rat-cube-dimension: dim(c),
bfalse: ff,
rev_uimplies: rev_uimplies(P;Q),
sq_stable: SqStable(P),
isl: isl(x),
set-equal: set-equal(T;x;y),
l_disjoint: l_disjoint(T;l1;l2),
bnot: ¬bb
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
rat-complex-boundary-0-dim,
nat_wf,
set_subtype_base,
le_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_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_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
istype-le,
subtype_rel_self,
rational-cube-complex_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
istype-less_than,
assert_witness,
le_weakening2,
length_wf,
rational-cube_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
isEven_wf,
rat-complex-boundary_wf,
subtract-1-ge-0,
non_neg_length,
add_nat_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
istype-nat,
rat-complex-boundary-remove1,
rat-cube-sub-complex_wf,
bnot_wf,
rceq_wf,
l_member_wf,
filter-length-less,
iff_weakening_uiff,
assert_wf,
not_wf,
assert_of_bnot,
istype-assert,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add_nat_plus,
nat_plus_properties,
cons_wf,
select_wf,
subtract-add-cancel,
rat-cube-face_wf,
rat-cube-dimension_wf,
lelt_wf,
member_wf,
assert-rceq,
inhabited-rat-cube_wf,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
member-rat-cube-faces,
add-subtract-cancel,
l_all_iff,
equal-wf-base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal,
rat-cube-faces_wf,
subtype_rel_list,
assert_functionality_wrt_uiff,
length-rat-cube-faces,
isEven-2n,
assert_of_tt,
no_repeats-rat-cube-faces,
sq_stable__assert,
no_repeats_wf,
list_wf,
decidable__l_member,
decidable__equal_rc,
filter-split-length,
btrue_wf,
bfalse_wf,
set-equal-no_repeats-length,
filter_wf5,
no_repeats_filter,
istype-true,
member_filter,
append_wf,
no_repeats-append,
isl_wf,
assert_elim,
btrue_neq_bfalse,
member_append,
length-append,
itermMultiply_wf,
itermMinus_wf,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
even-plus-even
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
sqequalRule,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
dependent_set_memberEquality_alt,
intWeakElimination,
functionIsTypeImplies,
imageElimination,
productElimination,
promote_hyp,
hypothesis_subsumption,
addEquality,
minusEquality,
equalityIstype,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
setIsType,
productIsType,
functionIsType,
unionIsType,
sqequalBase,
universeEquality,
imageMemberEquality,
inlFormation_alt,
inrFormation_alt,
setEquality,
productEquality,
multiplyEquality,
functionExtensionality
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}K:n-dim-complex. (\muparrow{}isEven(||\mpartial{}(K)||))
Date html generated:
2020_05_20-AM-09_22_42
Last ObjectModification:
2019_11_27-AM-11_45_00
Theory : rationals
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