Nuprl Lemma : simplex-face_wf
∀[n:ℤ]. ∀[v:Δ(n)]. ∀[i:ℕn + 2]. (simplex-face(v;i) ∈ Δ(n + 1))
Proof
Definitions occuring in Statement :
simplex-face: simplex-face(v;i),
std-simplex: Δ(n),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
std-simplex: Δ(n),
simplex-face: simplex-face(v;i),
real-vec: ℝ^n,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
subtract: n - m,
true: True,
rev_uimplies: rev_uimplies(P;Q),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
req_int_terms: t1 ≡ t2,
sq_stable: SqStable(P),
nequal: a ≠ b ∈ T
Lemmas referenced :
decidable__le,
std-simplex-properties,
istype-le,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
subtract_wf,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int-to-real_wf,
int_seg_wf,
rleq_weakening_equal,
rleq_wf,
req_wf,
rsum_wf,
std-simplex-void,
std-simplex_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
radd_wf,
add-subtract-cancel,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
req_functionality,
rsum-split-last,
req_weakening,
rsum_functionality,
req-implies-req,
rsub_wf,
req-iff-rsub-is-0,
radd_functionality,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
rsum-split2,
not-equal-2,
zero-add,
add_functionality_wrt_le,
sq_stable__le,
less-iff-le,
ifthenelse_wf,
eq_int_wf,
real_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
subtract-add-cancel,
equal_wf,
eq_int_eq_true,
btrue_wf,
subtype_rel_self,
iff_weakening_equal,
rsum-shift,
int_subtype_base,
rsum-empty,
rsum_functionality2,
req_inversion
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
productElimination,
hypothesis,
hypothesisEquality,
unionElimination,
isectElimination,
dependent_set_memberEquality_alt,
because_Cache,
lambdaEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
sqequalRule,
independent_isectElimination,
applyEquality,
independent_pairFormation,
imageElimination,
addEquality,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
functionIsType,
axiomEquality,
isectIsTypeImplies,
closedConclusion,
applyLambdaEquality,
minusEquality,
multiplyEquality,
imageMemberEquality,
baseClosed,
intEquality
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[v:\mDelta{}(n)]. \mforall{}[i:\mBbbN{}n + 2]. (simplex-face(v;i) \mmember{} \mDelta{}(n + 1))
Date html generated:
2019_10_30-AM-11_30_49
Last ObjectModification:
2019_07_31-PM-03_40_34
Theory : real!vectors
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