Nuprl Lemma : member-rat-complex-subdiv2
∀k,n:ℕ. ∀K:n-dim-complex. ∀c:ℚCube(k). ((c ∈ (K)') ⇐⇒ ∃a:ℚCube(k). ((a ∈ K) ∧ (↑is-half-cube(k;c;a))))
Proof
Definitions occuring in Statement :
rat-complex-subdiv: (K)',
rational-cube-complex: n-dim-complex,
is-half-cube: is-half-cube(k;h;c),
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
rat-cube-dimension: dim(c),
iff: P ⇐⇒ Q,
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
nat: ℕ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
rational-cube-complex: n-dim-complex,
all: ∀x:A. B[x]
Lemmas referenced :
assert_wf,
subtype_rel_list_set,
list-subtype,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
assert_of_bnot,
eqff_to_assert,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
inhabited-rat-cube_wf,
l_member_wf,
le_wf,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
equal-wf-base,
l_all_iff,
istype-nat,
rational-cube-complex_wf,
rational-cube_wf,
member-rat-complex-subdiv
Rules used in proof :
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
cumulativity,
instantiate,
unionElimination,
independent_functionElimination,
setIsType,
because_Cache,
independent_isectElimination,
addEquality,
natural_numberEquality,
minusEquality,
applyEquality,
intEquality,
lambdaEquality_alt,
sqequalRule,
productElimination,
inhabitedIsType,
isectElimination,
universeIsType,
equalitySymmetry,
equalityTransitivity,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
hypothesis,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}K:n-dim-complex. \mforall{}c:\mBbbQ{}Cube(k).
((c \mmember{} (K)') \mLeftarrow{}{}\mRightarrow{} \mexists{}a:\mBbbQ{}Cube(k). ((a \mmember{} K) \mwedge{} (\muparrow{}is-half-cube(k;c;a))))
Date html generated:
2019_10_29-AM-07_59_40
Last ObjectModification:
2019_10_22-AM-10_48_40
Theory : rationals
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