Nuprl Lemma : rat-complex-boundary-subdiv
∀k,n:ℕ. ∀K:n-dim-complex. permutation(ℚCube(k);∂((K)');(∂(K))')
Proof
Definitions occuring in Statement :
rat-complex-subdiv: (K)',
rat-complex-boundary: ∂(K),
rational-cube-complex: n-dim-complex,
rational-cube: ℚCube(k),
permutation: permutation(T;L1;L2),
nat: ℕ,
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},
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
rat-complex-subdiv: (K)',
concat: concat(ll),
reduce: reduce(f;k;as),
list_ind: list_ind,
map: map(f;as),
nil: [],
it: ⋅,
uiff: uiff(P;Q),
cand: A c∧ B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
int_seg: {i..j-},
iff: P ⇐⇒ Q,
rational-cube-complex: n-dim-complex,
rev_implies: P ⇐ Q,
in-complex-boundary: in-complex-boundary(k;f;K),
isOdd: isOdd(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m,
length: ||as||,
bfalse: ff,
cons: [a / b],
squash: ↓T,
l_member: (x ∈ l),
select: L[n],
nat_plus: ℕ+,
rat-cube-dimension: dim(c),
exists!: ∃!x:T. P[x],
sq_stable: SqStable(P),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
has-interior-point: has-interior-point(k;c;a),
rat-point-in-cube: rat-point-in-cube(k;x;c),
rat-point-in-cube-interior: rat-point-in-cube-interior(k;x;a),
rev_uimplies: rev_uimplies(P;Q),
compatible-rat-cubes: Compatible(c;d),
set-equal: set-equal(T;x;y),
l_all: (∀x∈L.P[x]),
lelt: i ≤ j < k
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
rational-cube-complex_wf,
istype-nat,
rat-complex-subdiv_wf,
istype-void,
istype-le,
set_subtype_base,
nat_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
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,
subtype_rel_self,
nat_wf,
le_wf,
permutation-nil,
rational-cube_wf,
rat-complex-boundary-0-dim,
equal-rat-cube-complexes,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
rat-complex-boundary_wf,
assert_elim,
inhabited-rat-cube_wf,
bool_wf,
bool_subtype_base,
istype-assert,
rat-cube-dimension_wf,
lelt_wf,
in-complex-boundary_wf,
is-half-cube_wf,
l_member_wf,
member-rat-complex-boundary-n,
member-rat-complex-subdiv2,
filter_wf5,
is-rat-cube-face_wf,
list-cases,
product_subtype_list,
length_of_cons_lemma,
squash_wf,
true_wf,
list_wf,
istype-universe,
iff_weakening_equal,
add_nat_plus,
length_wf_nat,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
false_wf,
length_wf,
select_wf,
member_filter,
iff_weakening_uiff,
assert_wf,
rat-cube-face_wf,
assert-is-rat-cube-face,
isOdd_wf,
l_all_iff,
equal-wf-base,
equal_wf,
bool_cases,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
half-cube-dimension,
extend-half-cube-face,
inhabited-rat-cube-face,
no_repeats_filter,
sq_stable__no_repeats,
no_repeats-length-equal-by-relation,
l_all_wf2,
l_exists_wf,
same-half-cube-of-compatible,
sq_stable__compatible-rat-cubes,
pairwise-iff,
compatible-rat-cubes_wf,
compatible-rat-cubes-symm,
compatible-rat-cubes-refl,
has-interior-point-implies,
l_exists_iff,
face-of-half-cube,
half-cube-of-face,
assert_functionality_wrt_uiff,
decidable__equal_rc,
rat-point-in-face,
rat-point-in-half-cube,
int_seg_wf,
rat-point-in-intersection,
inhabited-rat-cube-iff-point,
rat-cube-intersection_wf,
rat-point-in-cube_wf,
rat-point-in-cube-interior-not-in-face,
rat-cube-face-dimension-equal,
set-equal-no_repeats-length,
cons_wf,
nil_wf,
no_repeats_cons,
no_repeats_singleton,
member_singleton,
cons_member,
length_of_nil_lemma,
l_exists_filter,
l_all_filter_iff,
sq_stable__all,
sq_stable__assert,
assert_witness
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,
universeIsType,
inhabitedIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
sqequalRule,
voidElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
productElimination,
setIsType,
productIsType,
equalityIstype,
minusEquality,
addEquality,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
promote_hyp,
hypothesis_subsumption,
imageElimination,
universeEquality,
imageMemberEquality,
applyLambdaEquality,
pointwiseFunctionality,
hyp_replacement,
functionIsType,
productEquality,
unionIsType,
inlFormation_alt,
inrFormation_alt,
functionIsTypeImplies
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}K:n-dim-complex. permutation(\mBbbQ{}Cube(k);\mpartial{}((K)');(\mpartial{}(K))')
Date html generated:
2020_05_20-AM-09_24_03
Last ObjectModification:
2019_11_02-PM-07_28_27
Theory : rationals
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