Nuprl Lemma : simplex-face-face
∀[n:ℤ]. ∀[v:Δ(n)]. ∀[i,j:ℕn + 2].
simplex-face(simplex-face(v;i);j) = simplex-face(simplex-face(v;j);i + 1) ∈ Δ(n + 2) supposing j ≤ i
Proof
Definitions occuring in Statement :
simplex-face: simplex-face(v;i),
std-simplex: Δ(n),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_seg: {i..j-},
real-vec: ℝ^n,
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
simplex-face: simplex-face(v;i),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
std-simplex: Δ(n),
decidable: Dec(P),
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
istype-le,
int_seg_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
intformnot_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
itermAdd_wf,
itermConstant_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
int-to-real_wf,
simplex-face_wf,
subtype_rel-equal,
std-simplex_wf,
squash_wf,
true_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq_wf,
req_wf,
rsum_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
functionExtensionality,
productElimination,
addEquality,
imageElimination,
natural_numberEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
universeIsType,
applyEquality,
dependent_set_memberEquality_alt,
productIsType,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
functionIsType
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[v:\mDelta{}(n)]. \mforall{}[i,j:\mBbbN{}n + 2].
simplex-face(simplex-face(v;i);j) = simplex-face(simplex-face(v;j);i + 1) supposing j \mleq{} i
Date html generated:
2019_10_30-AM-11_30_53
Last ObjectModification:
2019_08_08-PM-00_59_22
Theory : real!vectors
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